update in lambdadelta

+ sources for λδ-2A added (with a correction)
+ sources for λδ-1A moved
+ λδ-related binaries moved here from web site directory

diff --git a/.gitignore b/.gitignore
index 224bc1923ba75a26ac8ba568345427a59e1e02a4..c0ab80566338334fcc3df82d7d9182d1e207e842 100644 (file)
--- a/.gitignore
+++ b/.gitignore
@@ -76,9 +76,8 @@ helm/www/lambdadelta/xslt/chc_45.xsl
 helm/www/lambdadelta/xslt/xhtbl.xsl
 
 matita/matita/contribs/lambdadelta/.depend
 helm/www/lambdadelta/xslt/xhtbl.xsl
 
 matita/matita/contribs/lambdadelta/.depend

-matita/matita/contribs/lambdadelta/nodes
-matita/matita/contribs/lambdadelta/token
-matita/matita/contribs/lambdadelta/2A
+matita/matita/contribs/lambdadelta/bin/nodes
+matita/matita/contribs/lambdadelta/bin/token

 matita/matita/contribs/lambdadelta/*/probe.txt
 matita/matita/contribs/lambdadelta/*/deps.txt
 matita/matita/contribs/lambdadelta/*/web/*_sum.tbl
 matita/matita/contribs/lambdadelta/*/probe.txt
 matita/matita/contribs/lambdadelta/*/deps.txt
 matita/matita/contribs/lambdadelta/*/web/*_sum.tbl

diff --git a/helm/www/lambdadelta/Makefile b/helm/www/lambdadelta/Makefile
index c58e3f597e013159f5d17b10f71676bf011a9b4d..1acebccc52d8f7354264b9d9e7295f06e67c64fa 100644 (file)
--- a/helm/www/lambdadelta/Makefile
+++ b/helm/www/lambdadelta/Makefile
@@ -15,9 +15,9 @@ SITEDIR    = html
 HTMLDIR    = html/lddl
 SRCDIR     = web/home
 LDDLDIR    = web/lddl
 HTMLDIR    = html/lddl
 SRCDIR     = web/home
 LDDLDIR    = web/lddl

-XHTBLDIR   = bin/xhtbl
-INDEXDIR   = bin/index

 ETCDIR     = etc
 ETCDIR     = etc

+XHTBLDIR   = $(ETCDIR)/lambdadelta/bin/xhtbl
+INDEXDIR   = $(ETCDIR)/lambdadelta/bin/index

 DOWNDIR    = download
 XSLTDIR    = xslt
 XMLDIR     = xml
 DOWNDIR    = download
 XSLTDIR    = xslt
 XMLDIR     = xml

diff --git a/helm/www/lambdadelta/bin/Makefile.common b/helm/www/lambdadelta/bin/Makefile.common
deleted file mode 100644 (file)
index 7f95f4a..0000000
--- a/helm/www/lambdadelta/bin/Makefile.common
+++ /dev/null
@@ -1,36 +0,0 @@
-H=@
-
-OCAMLOPTIONS = -linkpkg -package \"$(REQUIRES)\" $(CAMLOPTIONS)
-OCAMLFIND = OCAMLPATH=$(OCAMLPATH) ocamlfind
-OCAMLC = $(OCAMLFIND) ocamlc -g $(OCAMLOPTIONS)
-OCAMLOPT = $(OCAMLFIND) opt $(OCAMLOPTIONS)
-
-CAMLP_FEATURES = $(F:%=-D%)
-
-AMLS = $(wildcard *.aml)
-
-define BUILD_TEMPLATE
-$(1).all:
-       @echo "  OCAMLBUILD $(1)"
-       $(H)ocamlbuild $$(BUILDOPTIONS) -ocamlc "$$(OCAMLC)" -ocamlopt "$$(OCAMLOPT)" -yaccflags "-v" -pp "$$(CAMLP)" $(1)
-
-.PHONY: $(1).all
-endef
-
-all:: $(AMLS:%.aml=%.ml) $(EXECS:%=%.native.all)
-
-byte:: $(AMLS:%.aml=%.ml) $(EXECS:%=%.byte.all)
-
-$(foreach EXEC, $(EXECS), $(eval $(call BUILD_TEMPLATE,$(EXEC:%=%.native))))
-
-$(foreach EXEC, $(EXECS), $(eval $(call BUILD_TEMPLATE,$(EXEC:%=%.byte))))
-
-clean::
-       @echo "  OCAMLBUILD -clean"
-       $(H)ocamlbuild -clean
-       $(H)$(RM) $(AMLS:%.aml=%.ml) *~
-
-.PHONY: all clean
-
-%.ml: %.aml
-       $(H)$(ALPHA) < $< > $@

diff --git a/helm/www/lambdadelta/bin/a.ml b/helm/www/lambdadelta/bin/a.ml
deleted file mode 100644 (file)
index 4f310c8..0000000
--- a/helm/www/lambdadelta/bin/a.ml
+++ /dev/null
@@ -1,17 +0,0 @@
-let f = "0123456789abcdef"
-
-let r, g, b = 1.0, 0.5, 0.0
-
-let h = 1. /. 2.
-
-let mk_h x = x +. (1. -. x) *. h
-
-let rr, gg, bb = mk_h r, mk_h g, mk_h b 
-
-let mk_f x = 
-   let x = int_of_float x in
-   print_char f.[x / 16]; print_char f.[x mod 16]  
-
-let _ = 
-   mk_f (rr *. 255.); mk_f (gg *. 255.); mk_f (bb *. 255.);
-   print_newline ()

diff --git a/helm/www/lambdadelta/bin/index/Makefile b/helm/www/lambdadelta/bin/index/Makefile
deleted file mode 100644 (file)
index faf8860..0000000
--- a/helm/www/lambdadelta/bin/index/Makefile
+++ /dev/null
@@ -1,10 +0,0 @@
-EXECS = index
-
-REQUIRES = unix
-
-include ../Makefile.common
-
-test:
-#      @$(MAKE) --no-print-directory -C ../../ www
-
-.PHONY: test

diff --git a/helm/www/lambdadelta/bin/index/index.ml b/helm/www/lambdadelta/bin/index/index.ml
deleted file mode 100644 (file)
index 9496cc7..0000000
--- a/helm/www/lambdadelta/bin/index/index.ml
+++ /dev/null
@@ -1,123 +0,0 @@
-module KF = Filename
-module KP = Printf
-module KU = Unix
-
-type status = {
-(* base directory *)
-  bd: string;
-(* input prefix *)
-  ip: string;
-(* output prefix *)
-  op: string;
-(* current path *)
-  cp: string list
-}
-
-let initial_status = {
-  bd = ""; ip = ""; op = "";
-  cp = [];
-}
-
-let imp_st = ref initial_status
-
-let i_ext = ".ld.ldw.xml"
-let o_ext = ".ld.html"
-
-let concats l =
-  List.fold_left KF.concat "" l
-
-let concat st dname = {st with
-  ip = KF.concat st.ip dname; op = KF.concat st.op dname;
-}
-
-let normalize dname =
-  if dname = KF.current_dir_name then "" else dname
-
-let mk_rlink s_to s_body =
-  KP.sprintf "<rlink to=\"%s\">%s</rlink>" s_to s_body
-
-let out_entry st dname och dirs name =
-  let iname = concats [st.bd; st.ip; dname; name] in
-  let stats = KU.lstat iname in
-  match stats.KU.st_kind with
-  | KU.S_REG when KF.check_suffix name i_ext ->
-    let base = KF.chop_suffix name i_ext in 
-    let oname = concats [st.bd; st.op; dname; base^o_ext] in
-    KP.fprintf och "    <file class=\"global emph\" type=\"&#x1F5CF;\" to=\"%s\" name=\"%s.ld\"/>\n" oname base;
-    dirs
-  | KU.S_DIR ->
-    let oname = concats [st.bd; st.op; dname; name] in
-    KP.fprintf och "    <file class=\"alpha emph\" type=\"&#x1F5C1;\" to=\"%s\" name=\"%s/\"/>\n" oname name;
-    name :: dirs
-  | _        ->
-    dirs
-
-let mk_path st och =
-  let path = String.concat "/" (List.rev st.cp) in
-  KP.fprintf och "    Contents of %s/\n" path
-
-let list_dir st dname och =
-  let iname = concats [st.bd; st.ip; dname] in
-  let dir = Sys.readdir iname in
-  Array.sort String.compare dir;
-  KP.fprintf och "   <index>\n";
-  let dirs = Array.fold_left (out_entry st dname och) [] dir in
-  KP.fprintf och "   </index>\n";
-  dirs
-
-let out_index st dname och =
-  KP.fprintf och "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n\n";
-  KP.fprintf och "<page xmlns=\"http://lambdadelta.info/\"\n";
-  KP.fprintf och "      description=\"\\lambda\\delta home page\"\n";
-  KP.fprintf och "      title=\"\\lambda\\delta home page\"\n";
-  KP.fprintf och "      head=\"λδ digital library (LDDL)\"\n";
-  KP.fprintf och ">\n";
-  KP.fprintf och "  <sitemap name=\"sitemap\"/>\n";
-  KP.fprintf och "  <section5 name=\"index\">Index</section5>\n";
-  KP.fprintf och "  <subsection name=\"path\">\n";
-  mk_path st och;
-  KP.fprintf och "  </subsection>\n";
-  KP.fprintf och "  <body>\n";
-  let dirs = list_dir st dname och in
-  KP.fprintf och "  </body>\n";
-  KP.fprintf och "  <footer><img label=\"helena\"/></footer>\n";
-  KP.fprintf och "</page>\n";
-  dirs
-
-let rec out_dir st dname =
-  let s_to, s_body =
-    if dname = ""
-    then concats [st.bd; st.op], "ld:" 
-    else concats [st.bd; st.op; dname], dname
-  in
-  let st = {st with cp = mk_rlink s_to s_body :: st.cp} in
-  let oname = concats [st.bd; st.ip; dname; "index.ldw.xml"] in
-  let och = open_out oname in
-  let dirs = out_index st dname och in
-  close_out och;
-  let map st = out_dir (concat st dname) in
-  List.iter (map st) dirs
-
-let help_b = "<dir>  Set this base directory"
-let help_i = "<dir>  Set this input prefix"
-let help_o = "<dir>  Set this output prefix"
-let help = "Usage: index [ -bio <dir> | <dir> ]*"
-
-let set_b bd =
-  imp_st := {!imp_st with bd = normalize bd}
-
-let set_i ip =
-  imp_st := {!imp_st with ip = normalize ip}
-
-let set_o op =
-  imp_st := {!imp_st with op = normalize op}
-
-let process dname =
-  out_dir !imp_st (normalize dname)
-
-let main =
-  Arg.parse [
-    "-b", Arg.String set_b, help_b;
-    "-i", Arg.String set_i, help_i;
-    "-o", Arg.String set_o, help_o;
-  ] process help

diff --git a/helm/www/lambdadelta/bin/inline/Makefile b/helm/www/lambdadelta/bin/inline/Makefile
deleted file mode 100644 (file)
index 60ad8b7..0000000
--- a/helm/www/lambdadelta/bin/inline/Makefile
+++ /dev/null
@@ -1,12 +0,0 @@
-EXECS = inline
-
-REQUIRES =
-
-include ../Makefile.common
-
-test:
-       @./inline.native -p ../lambdadelta/*/deps.txt > deps.txt
-       @../matitadep/matitadep.native -c ../lambdadelta/.depend deps.txt > redundant.txt
-       @./inline.native -i -b ../lambdadelta redundant.txt ../lambdadelta/*/deps.txt
-
-.PHONY: test

diff --git a/helm/www/lambdadelta/bin/inline/inline.ml b/helm/www/lambdadelta/bin/inline/inline.ml
deleted file mode 100644 (file)
index 905f7ec..0000000
--- a/helm/www/lambdadelta/bin/inline/inline.ml
+++ /dev/null
@@ -1,162 +0,0 @@
-module Deps  = Set.Make(String)
-module Table = Map.Make(String)
-
-let opt_map f = function
-  | None   -> None
-  | Some a -> Some (f a)
-
-let rec filename_split l s =
-  let dir, base = Filename.dirname s, Filename.basename s in
-  if dir = Filename.current_dir_name then base::l else filename_split (base::l) dir
-
-let filename_concat l =
-  String.concat Filename.dir_sep l
-
-let relative s =
-  match filename_split [] s with
-  | "cic:" :: "matita" :: "lambdadelta" :: tl -> List.rev tl
-  | _                                         -> []
-
-let to_string l =
-  filename_concat (List.rev l)
-
-let table = ref (Table.empty: Deps.t Table.t) 
-
-let add src dep =
-  let deps = match Table.find_opt src !table with
-  | None      -> Deps.singleton dep
-  | Some deps -> Deps.add dep deps
-  in
-  table := Table.add src deps !table
-
-let split_or s =
-  let map m = Printf.sprintf "or_%u" m in 
-  try Scanf.sscanf s "or%u" map
-  with Scanf.Scan_failure _ | End_of_file -> ""
-
-let split_and s =
-  let map m = Printf.sprintf "and_%u" m in 
-  try Scanf.sscanf s "and%u" map
-  with Scanf.Scan_failure _ | End_of_file -> ""
-
-let split_ex s =
-  let map m n = Printf.sprintf "ex_%u_%u" m n in 
-  try Scanf.sscanf s "ex%u=%u" map
-  with Scanf.Scan_failure _ | End_of_file -> ""
-
-let split_ex1 s =
-  let map m = Printf.sprintf "ex_%u_1" m in 
-  try Scanf.sscanf s "ex%u" map
-  with Scanf.Scan_failure _ | End_of_file -> ""
-
-let map_deps s1 s2 =
-  match relative s2 with
-  | [b2;"xoa";"xoa";"ground_2"] ->
-    let r1 = List.tl (relative s1) in
-    let r1 = to_string r1 in
-    let b2 = Filename.remove_extension b2 in
-(* '_' is accepted (and ignored) within integer literals *)
-    let b2 = String.concat "=" (String.split_on_char '_' b2) in
-    let r2 =
-      let cx = split_ex b2 in
-      let cy = split_ex1 b2 in
-      let ca = split_and b2 in
-      let co = split_or b2 in
-      if cx <> "" then cx else
-      if cy <> "" then cy else
-      if ca <> "" then ca else
-      if co <> "" then co else
-      failwith (Printf.sprintf "unrecognized xoa: %S\n" b2)
-    in
-    if r1 <> "ground_2/xoa/xoa" then add r1 r2
-  | _                           -> ()
-
-let reds = ref []
-
-let map_reds s1 s2 =
-  reds := (s1,s2) :: !reds
-
-let rec read map_deps map_reds ich =
-  let line = input_line ich in
-  begin try Scanf.sscanf line "%S: %S" map_deps
-  with Scanf.Scan_failure _ | End_of_file ->
-    begin try Scanf.sscanf line "%S: redundant %S" map_reds
-    with Scanf.Scan_failure _ | End_of_file ->
-      Printf.eprintf "unknown line: %s.\n" line
-    end
-  end; 
-  read map_deps map_reds ich
-
-let xoadir = ref "ground_2/xoa"
-
-let print_deps () =
-  let map_d src dep =
-    let src = src^".ma" in
-    let dep = Filename.concat !xoadir (dep^".ma") in
-    if List.mem (src,dep) !reds then ()
-    else Printf.printf "%S: %S\n" src dep
-  in
-  let map_t src deps =
-    Deps.iter (map_d src) deps
-  in
-  Table.iter map_t !table
-
-let rec copy xn ich och =
-  if xn = Some 0 then ()
-  else begin
-    Printf.fprintf och "%s\n" (input_line ich); 
-    copy (opt_map pred xn) ich och
-  end
-
-let base_dir = ref ""
-
-let preamble = ref 14
-
-let insert_deps () =
-  let map_d src dep rdeps =
-    let dep = Filename.concat !xoadir (dep^".ma") in
-    if List.mem (src,dep) !reds then rdeps else dep::rdeps
-  in
-  let map_r och rdep =
-    Printf.fprintf och "include %S.\n" rdep;
-  in
-  let map_t src deps =
-    let src = src^".ma" in
-    let rdeps = Deps.fold (map_d src) deps [] in
-    if rdeps <> [] then begin    
-      let ma = Filename.concat !base_dir src in
-      let old = ma^".old" in
-      Sys.rename ma old;
-      let och = open_out ma in
-      let ich = open_in old in
-      copy (Some !preamble) ich och;
-      List.iter (map_r och) (List.rev rdeps);
-      try copy None ich och
-      with End_of_file -> close_in ich; close_out och
-    end 
-  in
-  Table.iter map_t !table
-
-let process fname =
-  let ich = open_in fname in
-  try read map_deps map_reds ich with 
-  | End_of_file -> close_in ich
-
-let help_b = "<dir>  Set this base directory (default: current directory)"
-let help_i = "  Insert the dependences (default: no)"
-let help_l = "<int>  .ma preamble has theese lines (default: 14)"
-let help_p = "  Print the dependences to be inserted (default: no)"
-let help = "inline [ -ip | -b <dir> | -l <int> | <file> ]*"
-
-let print = ref false
-let insert = ref false
-
-let _ =
-  Arg.parse [
-    "-b", Arg.String ((:=) base_dir), help_b;
-    "-l", Arg.Int ((:=) preamble), help_l;
-    "-i", Arg.Set insert, help_i; 
-    "-p", Arg.Set print, help_p; 
-  ] process help;
-  if !print then print_deps ();
-  if !insert then insert_deps ()

diff --git a/helm/www/lambdadelta/bin/xhtbl/Makefile b/helm/www/lambdadelta/bin/xhtbl/Makefile
deleted file mode 100644 (file)
index c56f2d8..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/Makefile
+++ /dev/null
@@ -1,10 +0,0 @@
-EXECS = xhtbl
-
-REQUIRES = str
-
-include ../Makefile.common
-
-test:
-       @$(MAKE) --no-print-directory -C ../../ www
-
-.PHONY: test

diff --git a/helm/www/lambdadelta/bin/xhtbl/attr.ml b/helm/www/lambdadelta/bin/xhtbl/attr.ml
deleted file mode 100644 (file)
index 36b3d00..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/attr.ml
+++ /dev/null
@@ -1,20 +0,0 @@
-module L = List
-
-module T = Table
-
-(* true for a row specification *)
-type 'a atom = 'a * bool * int option * int option
-
-type 'a atoms = 'a atom list
-
-let get_attr concat null a y x =
-   let map y x (c, b, x1, x2) = match b, x1, x2 with
-      | _    , None, None       -> c
-      | false, None, Some c2    -> if x <= c2 then c else null
-      | false, Some c1, None    -> if x >= c1 then c else null
-      | false, Some c1, Some c2 -> if x >= c1 && x <= c2 then c else null
-      | true , None, Some r2    -> if y <= r2 then c else null
-      | true , Some r1, None    -> if y >= r1 then c else null
-      | true , Some r1, Some r2 -> if y >= r1 && y <= r2 then c else null
-   in
-   concat (L.map (map y x) a)

diff --git a/helm/www/lambdadelta/bin/xhtbl/fold.ml b/helm/www/lambdadelta/bin/xhtbl/fold.ml
deleted file mode 100644 (file)
index 752b06d..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/fold.ml
+++ /dev/null
@@ -1,25 +0,0 @@
-module T = Table
-
-type 'a fold_cb = {
-   open_table : 'a -> T.table -> 'a;
-   close_table: 'a -> T.table -> 'a;
-   map_key    : 'a -> T.key -> 'a;
-   open_line  : bool -> 'a -> 'a;
-   close_line : bool -> 'a -> 'a;
-   open_entry : bool -> 'a -> 'a;   
-   close_entry: bool -> 'a -> 'a -> 'a;
-}
-
-let map h g f a b = h a (g (f a) b)
-
-let rec fold_table cb a t =
-   let a = cb.open_table a t in
-   let a = fold_entry cb a t.T.te in
-   cb.close_table a t
-
-and fold_entry cb a = function
-   | T.Key k        -> cb.map_key a k
-   | T.Line (r, ts) ->
-      let a = cb.open_line r a in
-      let a = List.fold_left (map (cb.close_entry r) (fold_table cb) (cb.open_entry r)) a ts in
-      cb.close_line r a

diff --git a/helm/www/lambdadelta/bin/xhtbl/matrix.ml b/helm/www/lambdadelta/bin/xhtbl/matrix.ml
deleted file mode 100644 (file)
index 1c65c50..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/matrix.ml
+++ /dev/null
@@ -1,63 +0,0 @@
-module A = Array
-module N = Filename
-
-module T = Table
-
-type cell = {
-   ck: T.text list; (* contents *)
-   cc: T.css;       (* css classes *)
-   cu: T.uri;       (* uri *)
-   cx: T.ext;       (* extension *)
-   cn: T.anchor;    (* named anchor *)   
-   cb: T.border;    (* border *)
-}
-
-type matrix = {
-   r: int;              (* rows *)
-   c: int;              (* columns *)
-   m: cell array array; (* matrix *)
-}
-
-let strand a b = if a = "" then b else a
-
-let empty = {
-   ck = []; cc = []; cu = ""; cx = ""; cn = ""; cb = T.border false;
-}
-
-let make ts = {
-   r = ts.T.rf; c = ts.T.cf;
-   m = A.make_matrix ts.T.rf ts.T.cf empty;
-}
-
-let set_key m y x kl = 
-   m.m.(y).(x) <- {m.m.(y).(x) with ck = kl}
-
-let set_attrs m y x c u e n = 
-   m.m.(y).(x) <- {m.m.(y).(x) with
-      cc = c @ m.m.(y).(x).cc;
-      cu = u ^ m.m.(y).(x).cu;
-      cx = m.m.(y).(x).cx ^ e;
-      cn = strand m.m.(y).(x).cn n;
-   }
-
-let set_west m y x b =
-   let c = m.m.(y).(x) in
-   let cb = {c.cb with T.w = c.cb.T.w || b.T.w} in 
-   m.m.(y).(x) <- {c with cb = cb}
-
-let set_north m y x b =
-   let c = m.m.(y).(x) in
-   let cb = {c.cb with T.n = c.cb.T.n || b.T.n} in 
-   m.m.(y).(x) <- {c with cb = cb}
-
-let set_east m y x b =
-   if x < pred m.c then set_west m y (succ x) {b with T.w = b.T.e} else
-   let c = m.m.(y).(x) in
-   let cb = {c.cb with T.e = c.cb.T.e || b.T.e} in 
-   m.m.(y).(x) <- {c with cb = cb}
-
-let set_south m y x b =
-   if y < pred m.r then set_north m (succ y) x {b with T.n = b.T.s} else
-   let c = m.m.(y).(x) in
-   let cb = {c.cb with T.s = c.cb.T.s || b.T.s} in 
-   m.m.(y).(x) <- {c with cb = cb}

diff --git a/helm/www/lambdadelta/bin/xhtbl/options.ml b/helm/www/lambdadelta/bin/xhtbl/options.ml
deleted file mode 100644 (file)
index 21ebec1..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/options.ml
+++ /dev/null
@@ -1,39 +0,0 @@
-let output_dir_default = ""
-
-let baseuri_default = ""
-
-let debug_lexer_default = false
-
-let debug_pass_default = false
-
-let pass_default = false
-
-let output_dir = ref output_dir_default
-
-let baseuri = ref baseuri_default
-
-let debug_lexer = ref debug_lexer_default
-
-let d0 = ref debug_pass_default
-
-let d1 = ref debug_pass_default
-
-let d2 = ref debug_pass_default
-
-let e1 = ref debug_pass_default
-
-let e2 = ref debug_pass_default
-
-let p0 = ref pass_default
-
-let p1 = ref pass_default
-
-let p2 = ref pass_default
-
-let clear () =
-   output_dir := output_dir_default;
-   baseuri := baseuri_default;
-   debug_lexer := debug_lexer_default;
-   d0 := debug_pass_default; d1 := debug_pass_default; d2 := debug_pass_default; 
-   e1 := debug_pass_default; e2 := debug_pass_default;
-   p0 := pass_default; p1 := pass_default; p2 := pass_default

diff --git a/helm/www/lambdadelta/bin/xhtbl/pass1.ml b/helm/www/lambdadelta/bin/xhtbl/pass1.ml
deleted file mode 100644 (file)
index bedd961..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/pass1.ml
+++ /dev/null
@@ -1,88 +0,0 @@
-module L = List
-
-module T = Table
-module F = Fold
-
-type status = {
-   ts: T.size; (* current dimensions *)
-   tc: T.css;  (* current class *)
-   tu: T.uri;  (* current uri *)
-   tx: T.ext;  (* current extension *)
-}
-
-let empty = {
-   ts = T.no_size; tc = []; tu = ""; tx = ""
-}
-
-let init b ts =
-   if b then
-      {ts with T.ri = max_int; T.ci = 0}
-   else
-      {ts with T.ri = 0; T.ci = max_int}
-
-let combine b ts1 ts2 =
-   if b then    
-      {ts1 with 
-         T.rf = max ts1.T.rf ts2.T.rf; T.ri = min ts1.T.ri ts2.T.ri; 
-        T.cf = ts1.T.cf + ts2.T.cf; T.ci = ts1.T.ci + ts2.T.ci;
-      }
-   else
-      {ts1 with
-         T.cf = max ts1.T.cf ts2.T.cf; T.ci = min ts1.T.ci ts2.T.ci;
-        T.rf = ts1.T.rf + ts2.T.rf; T.ri = ts1.T.ri + ts2.T.ri; 
-      }
-
-let deinit ts = {ts with
-   T.ri = if ts.T.ri = max_int then 0 else ts.T.ri;
-   T.ci = if ts.T.ci = max_int then 0 else ts.T.ci;
-}
-
-(****************************************************************************)
-
-let open_table st t =
-   t.T.tc <- t.T.tc @ st.tc; t.T.tu <- st.tu ^ t.T.tu; t.T.tx <- st.tx ^ t.T.tx; 
-   {st with tc = t.T.tc; tu = t.T.tu; tx = t.T.tx}
-
-let close_table st t =
-   t.T.ts <- st.ts; st
-
-let map_key st k = 
-   let ts = match k, st.ts.T.p with
-      | T.Text _     , _          ->
-         {st.ts with T.rf = 1; T.cf = 1; T.ri = 0; T.ci = 0}
-      | T.Glue None  , _          ->
-         {st.ts with T.rf = 0; T.cf = 0; T.ri = 1; T.ci = 1}
-      | T.Glue Some g, Some false ->
-         {st.ts with T.rf = g; T.cf = 0; T.ri = 0; T.ci = 1}
-      | T.Glue Some g, Some true  ->
-         {st.ts with T.rf = 0; T.cf = g; T.ri = 1; T.ci = 0}
-      | T.Glue Some g, None       ->
-         {st.ts with T.rf = g; T.cf = g; T.ri = 0; T.ci = 0}
-   in
-   {st with ts = ts}
-
-let open_line b st =
-   let ts = init b st.ts in
-   let ts = {ts with T.rf = 0; T.cf = 0} in
-   {st with ts = ts}
-
-let open_entry b st =
-   let ts = {st.ts with T.p = Some b} in
-   {st with ts = ts}
-
-let close_entry b st sst =
-   {st with ts = combine b st.ts sst.ts}
-
-let close_line b st =
-   {st with ts = deinit st.ts}
-
-let cb = {
-   F.open_table = open_table; F.close_table = close_table;   
-   F.open_line = open_line; F.close_line = close_line;
-   F.open_entry = open_entry; F.close_entry = close_entry;
-   F.map_key = map_key;
-}
-
-let process t =
-   let st = F.fold_table cb empty t in
-   st.ts

diff --git a/helm/www/lambdadelta/bin/xhtbl/pass2.ml b/helm/www/lambdadelta/bin/xhtbl/pass2.ml
deleted file mode 100644 (file)
index 549d765..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/pass2.ml
+++ /dev/null
@@ -1,139 +0,0 @@
-module O = Options
-module T = Table
-module M = Matrix
-module F = Fold
-
-type status = {
-   ts: T.size;   (* current dimensions *)
-   tm: M.matrix; (* current matrix *)
-}
-
-let initial t m = {
-   ts = {t.T.ts with T.ri = 0; T.ci = 0};
-   tm = m;
-}
-
-let resize b sts tts = 
-   if b then begin (* parent is a row *) 
-      if tts.T.rf < sts.T.rf && tts.T.ri = 0 then
-         failwith "underful column";
-      {tts with T.rf = sts.T.rf; T.cf = tts.T.cf + sts.T.ci * tts.T.ci}
-   end else begin (* parent is a column *)
-      if tts.T.cf < sts.T.cf && tts.T.ci = 0 then
-         failwith "underful row";
-      {tts with T.cf = sts.T.cf; T.rf = tts.T.rf + sts.T.ri * tts.T.ri}
-   end
-
-let fill b sts tts =
-   if b then (* parent is a row *) 
-      {sts with T.ri = 
-         let rf, ri = sts.T.rf - tts.T.rf, tts.T.ri in
-        if ri = 0 then 0 else
-        if rf mod ri = 0 then rf / ri else
-        failwith "fracted column"
-      }
-   else (* parent is a column *)
-      {sts with T.ci = 
-         let cf, ci = sts.T.cf - tts.T.cf, tts.T.ci in
-        if ci = 0 then 0 else
-        if cf mod ci = 0 then cf / ci else
-        failwith "fracted row"
-      }
-
-let place b sts tts =
-   if b then (* parent is a row *)
-      {sts with T.x = sts.T.x + tts.T.cf}
-   else (* parent is a column *)
-      {sts with T.y = sts.T.y + tts.T.rf}
-
-let set_key st t = match t.T.te with
-   | T.Key (T.Text sl) -> M.set_key st.tm t.T.ts.T.y t.T.ts.T.x sl  
-   | _                 -> ()
-
-let set_attrs st t =
-   let rec aux y x =
-      if y >= t.T.ts.T.rf then () else
-      if x >= t.T.ts.T.cf then aux (succ y) 0 else begin
-        M.set_attrs st.tm (t.T.ts.T.y + y) (t.T.ts.T.x + x) t.T.tc t.T.tu t.T.tx t.T.tn;
-         aux y (succ x)
-      end
-   in
-   match t.T.te with 
-      | T.Key _ -> aux 0 0 
-      | _       -> ()
-
-let set_borders st t =
-   let rec aux_we y =
-      if y >= t.T.ts.T.rf then () else begin
-        M.set_west st.tm (t.T.ts.T.y + y) t.T.ts.T.x t.T.tb;
-        if t.T.ts.T.cf > 0 then 
-           M.set_east st.tm (t.T.ts.T.y + y) (t.T.ts.T.x + pred t.T.ts.T.cf) t.T.tb;
-         aux_we (succ y)
-      end      
-   in
-   let rec aux_ns x =
-      if x >= t.T.ts.T.cf then () else begin
-        M.set_north st.tm t.T.ts.T.y (t.T.ts.T.x + x) t.T.tb;
-        if t.T.ts.T.rf > 0 then 
-           M.set_south st.tm (t.T.ts.T.y + pred t.T.ts.T.rf) (t.T.ts.T.x + x) t.T.tb;
-         aux_ns (succ x)
-      end      
-   in
-   match t.T.te with 
-      | T.Line (true, _) -> aux_we 0; aux_ns 0
-      | _                -> ()
-
-let print st t = 
-   if !O.e2 then
-      Printf.printf "#%u: (%u+%u, %u+%u) - (%u+%u, %u+%u)\n"
-         t.T.ti 
-         t.T.ts.T.rf t.T.ts.T.ri 
-         t.T.ts.T.cf t.T.ts.T.ci
-         st.ts.T.rf st.ts.T.ri
-         st.ts.T.cf st.ts.T.ci
-
-(****************************************************************************)
-
-let open_table st t =
-   print st t;
-   let ts = match t.T.ts.T.p with
-      | None   ->
-         let ts = fill false st.ts t.T.ts in
-         let ts = fill true ts t.T.ts in
-        t.T.ts <- resize false st.ts t.T.ts;
-         t.T.ts <- resize true st.ts t.T.ts;
-        ts
-      | Some b ->
-         let ts = fill b st.ts t.T.ts in
-        t.T.ts <- resize b st.ts t.T.ts;
-        ts
-   in
-   t.T.ts <- {t.T.ts with T.ri = 0; T.ci = 0; T.x = st.ts.T.x; T.y = st.ts.T.y};
-   let ts = {ts with T.rf = t.T.ts.T.rf; T.cf = t.T.ts.T.cf} in
-   let st = {st with ts = ts} in 
-   print st t; st
-
-let close_table st t =
-   set_key st t; set_attrs st t; set_borders st t; st
-
-let map_key st k = st
-
-let open_line b st = st
-
-let open_entry b st = st
-
-let close_entry b st sst =
-   let ts = place b st.ts sst.ts in
-   {st with ts = ts}
-
-let close_line b st = st
-
-let cb = {
-   F.open_table = open_table; F.close_table = close_table;   
-   F.open_line = open_line; F.close_line = close_line;
-   F.open_entry = open_entry; F.close_entry = close_entry;
-   F.map_key = map_key;
-}
-
-let process t m =
-   let _ = F.fold_table cb (initial t m) t in ()

diff --git a/helm/www/lambdadelta/bin/xhtbl/pass3.ml b/helm/www/lambdadelta/bin/xhtbl/pass3.ml
deleted file mode 100644 (file)
index d2455a3..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/pass3.ml
+++ /dev/null
@@ -1,32 +0,0 @@
-module L = List
-module S = String
-module V = Array
-
-module T = Table
-module M = Matrix
-module A = Attr
-
-type status = {
-   m: M.matrix;
-   c: T.css A.atoms;
-   u: T.uri A.atoms;
-   x: T.ext A.atoms;
-}
-
-let initial c u x m = {
-   m = m; c = c; u = u; x = x
-}
-
-let process_cell st y x c =
-   M.set_attrs st.m y x
-      (A.get_attr L.concat [] st.c y x)
-      (A.get_attr (S.concat "") "" st.u y x)
-      (A.get_attr (S.concat "") "" st.x y x)
-      ""
-
-let process_row st y row =
-   V.iteri (process_cell st y) row
-
-let process css uri ext matrix =
-   let st = initial css uri ext matrix in
-   V.iteri (process_row st) matrix.M.m

diff --git a/helm/www/lambdadelta/bin/xhtbl/table.ml b/helm/www/lambdadelta/bin/xhtbl/table.ml
deleted file mode 100644 (file)
index d3ee13b..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/table.ml
+++ /dev/null
@@ -1,70 +0,0 @@
-type css = string list
-
-type uri = string
-
-type ext = string
-
-type anchor = string
-
-type absolute = bool
-
-type size = {
-   y : int;         (* first row *)
-   x : int;         (* first column *)
-   rf: int;         (* finite rows *)
-   cf: int;         (* finite columns *)
-   ri: int;         (* infinite rows *)
-   ci: int;         (* infinite columns *)
-   p : bool option; (* parent kind *)
-}
-
-type border = {
-   n: bool; (* north *)
-   s: bool; (* south *)
-   e: bool; (* east *)
-   w: bool; (* west *)
-}
-
-type text = Plain of string
-          | Link of absolute * string * string
-
-type key = Text of text list
-         | Glue of int option
-
-type table = {
-           tn: anchor; (* named anchor *)
-   mutable tc: css;    (* css classes *)
-   mutable tu: uri;    (* uri *)
-   mutable tx: ext;    (* uri extension *)
-   mutable ts: size;   (* dimension *)
-           tb: border; (* border *)
-           te: entry;  (* contents *)
-          ti: int;    (* table identifier *)
-}
-
-and entry = Key  of key
-          | Line of bool * table list (* true for a row *)
-
-let id =
-   let current = ref 0 in
-   fun () -> incr current; !current
-
-let no_size = {
-   y = 0; x = 0; rf = 0; cf = 0; ri = 0; ci = 0; p = None;
-}
-
-let border b = {
-   n = b; s = b; e = b; w = b;
-}
-
-let mk_key k tc tu tx tn = {
-   ts = no_size; tb = border false; te = Key k;
-   tc = tc; tu = tu; tx = tx; tn = tn;
-   ti = id ();
-}
-
-let mk_line b tl tc tu tx tn = {
-   ts = no_size; tb = border b; te = Line (b, tl);
-   tc = tc; tu = tu; tx = tx; tn = tn;
-   ti = id ();
-}

diff --git a/helm/www/lambdadelta/bin/xhtbl/textLexer.mll b/helm/www/lambdadelta/bin/xhtbl/textLexer.mll
deleted file mode 100644 (file)
index 4b06e4c..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/textLexer.mll
+++ /dev/null
@@ -1,47 +0,0 @@
-{
-   module S  = String
-   
-   module O  = Options
-   module TP = TextParser
-   
-   let out s = if !O.debug_lexer then prerr_endline s
-}
-
-let SPC = ['\r' '\n' '\t' ' ']+
-let QT  = "\""
-let NUM = ['0'-'9']+
-
-rule token = parse
-   | SPC      { token lexbuf                    }
-   | QT       { let s = str lexbuf in 
-                out s; TP.TEXT s                }
-   | NUM as s { out s; TP.NUM (int_of_string s) }
-   | "(*"     { block lexbuf; token lexbuf      }
-   | "{"      { out "{"; TP.OC                  }
-   | "}"      { out "}"; TP.CC                  }
-   | "["      { out "["; TP.OB                  }
-   | "]"      { out "]"; TP.CB                  }   
-   | "*"      { out "*"; TP.SR                  }
-   | "^"      { out "^"; TP.CF                  }
-   | "+"      { out "+"; TP.PS                  }
-   | "("      { out "("; TP.OP                  }
-   | ")"      { out ")"; TP.CP                  }   
-   | "@"      { out ")"; TP.AT                  }    
-   | "space"  { out "space"; TP.SPACE           }
-   | "name"   { out "name"; TP.NAME             }   
-   | "table"  { out "table"; TP.TABLE           }
-   | "class"  { out "class"; TP.CSS             }
-   | "uri"    { out "uri"; TP.URI               }
-   | "ext"    { out "ext"; TP.EXT               }
-   | eof      { TP.EOF                          }
-and str = parse
-   | QT       { ""                              }
-   | "\\\\"   { "\\" ^ str lexbuf               }
-   | "\\\""   { "\"" ^ str lexbuf               }
-   | _ as c   { S.make 1 c ^ str lexbuf         }
-and block = parse
-   | "*)"     { ()                              }
-   | "(*"     { block lexbuf; block lexbuf      }
-   | QT       { let _ = str lexbuf in
-                block lexbuf                    }
-   | _        { block lexbuf                    }

diff --git a/helm/www/lambdadelta/bin/xhtbl/textParser.mly b/helm/www/lambdadelta/bin/xhtbl/textParser.mly
deleted file mode 100644 (file)
index 9072c2b..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/textParser.mly
+++ /dev/null
@@ -1,147 +0,0 @@
-%{
-
-module S = Str
-module L = List
-module T = Table
-
-let split s =
-   S.split (S.regexp "[ \r\n\t]+") s
-
-let mk_css_atom s rs =
-   let cs = split s in
-   let map (b, (x1, x2)) = cs, b, x1, x2 in 
-   L.map map rs
-
-let mk_string_atom s rs =
-   let map (b, (x1, x2)) = s, b, x1, x2 in 
-   L.map map rs
-
-%}
-
-%token <int> NUM
-%token <string> TEXT 
-%token SPACE NAME TABLE CSS URI EXT SR OC CC OB CB PS CF OP CP AT EOF
-
-%start script
-%type <(string * string) list * (string * Table.table * Table.css Attr.atoms * Table.uri Attr.atoms * Table.ext Attr.atoms) list> script
-
-%%
-
-space:
-   | SPACE TEXT TEXT { $2, $3 }
-;
-
-spaces:
-   |              { []       }
-   | space spaces { $1 :: $2 }
-;
-
-text:
-   | TEXT                  { T.Plain $1             }
-   | AT OP TEXT TEXT CP    { T.Link (true, $3, $4)  }
-   | AT AT OP TEXT TEXT CP { T.Link (false, $4, $5) }
-   | AT TEXT               { T.Link (true, $2, $2)  }
-   | AT AT TEXT            { T.Link (false, $3, $3) }   
-;
-
-texts:
-  | text          { [$1]                    }
-  | text PS texts { $1 :: T.Plain " " :: $3 }
-  | text CF texts { $1 :: $3                }
-;
-
-key:
-   | texts { T.Text $1        }
-   | SR    { T.Glue None      }
-   | NUM   { T.Glue (Some $1) }
-;
-
-css:
-   |          { []       }
-   | CSS TEXT { split $2 }
-;
-
-uri:
-   |          { "" }
-   | URI TEXT { $2 }
-;
-
-ext:
-   |          { "" }
-   | EXT TEXT { $2 }
-;
-
-table:
-   | css uri ext name key     { T.mk_key        $5 $1 $2 $3 $4 }
-   | css uri ext OC tables CC { T.mk_line false $5 $1 $2 $3 "" }
-   | css uri ext OB tables CB { T.mk_line true  $5 $1 $2 $3 "" }
-;
-
-tables:
-   |              { []       }
-   | table tables { $1 :: $2 }
-;
-
-name:
-   |           { "" }
-   | NAME TEXT { $2 }
-;
-
-interval:
-   | NUM     { Some $1, Some $1 }
-   | SR      { None, None       } 
-   | NUM NUM { Some $1, Some $2 }
-   | NUM SR  { Some $1, None    }
-   | SR NUM  { None, Some $2    }
-   | SR SR   { None, None       }
-;
-
-range:
-   | OB interval CB { true, $2  }
-   | OC interval CC { false, $2 }
-;
-
-ranges:
-   |              { []       }
-   | range ranges { $1 :: $2 }
-;
-
-catom:
-   | CSS TEXT ranges { mk_css_atom $2 $3 }
-;
-
-catoms:
-   |              { []      }
-   | catom catoms { $1 @ $2 }
-;
-
-uatom:
-   | URI TEXT ranges { mk_string_atom $2 $3 }
-;
-
-uatoms:
-   |              { []      }
-   | uatom uatoms { $1 @ $2 }
-;
-
-xatom:
-   | EXT TEXT ranges { mk_string_atom $2 $3 }
-;
-
-xatoms:
-   |              { []      }
-   | xatom xatoms { $1 @ $2 }
-;
-
-directive:
-   | name TABLE table catoms uatoms xatoms { $1, $3, $4, $5, $6 }
-;
-
-directives:
-   |                      { []       }
-   | directive directives { $1 :: $2 }
-;
-
-script:
-   | spaces directives EOF { $1, $2 }
-;

diff --git a/helm/www/lambdadelta/bin/xhtbl/textUnparser.ml b/helm/www/lambdadelta/bin/xhtbl/textUnparser.ml
deleted file mode 100644 (file)
index cf7724c..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/textUnparser.ml
+++ /dev/null
@@ -1,101 +0,0 @@
-module L = List
-module P = Printf
-module S = String
-
-module T = Table
-module F = Fold
-
-type status = {
-   i: int;              (* indentation *)
-   out: string -> unit; (* output function *)
-}
-
-let home = {
-   i = 0; out = print_string
-}
-
-let indent st =
-   S.make st.i ' '
-
-let add st = {st with i = st.i + 3}
-
-let sub st = {st with i = st.i - 3}
-
-let parent = function
-   | None       -> "key"
-   | Some false -> "col"
-   | Some true  -> "row"
-
-let size ts =
-   P.sprintf "(%u, %u); (%u+%u, %u+%u); %s"
-      ts.T.y ts.T.x ts.T.rf ts.T.ri ts.T.cf ts.T.ci (parent ts.T.p)
-
-let border tb =
-   let str = S.make 4 ' ' in
-   if tb.T.w then str.[0] <- 'W';
-   if tb.T.n then str.[1] <- 'N';
-   if tb.T.e then str.[2] <- 'E';
-   if tb.T.s then str.[3] <- 'S';
-   str
-
-let css tc =
-   P.sprintf "\"%s\"" (S.concat " " tc)
-
-let uri tu tx =
-   P.sprintf "@\"%s\" \"%s\"" tu tx
-
-let name tn =
-   P.sprintf "$\"%s\"" tn
-
-
-let text = function
-   | T.Plain s              -> P.sprintf "\"%s\"" s
-   | T.Link (true, uri, s)  -> P.sprintf "@(\"%s\" \"%s\")" uri s
-   | T.Link (false, uri, s) -> P.sprintf "@@(\"%s\" \"%s\")" uri s
-
-let key = function
-   | T.Text sl       -> S.concat " ^ " (L.map text sl)
-   | T.Glue None     -> "*"
-   | T.Glue (Some i) -> P.sprintf "%u" i
-
-let entry = function
-   | false -> "column"
-   | true  -> "row"
-
-(****************************************************************************)
-
-let open_table st t =
-   let str = 
-      P.sprintf "%s[{#%u: %s; %s; %s; %s; %s}\n"    
-         (indent st) t.T.ti (size t.T.ts) (border t.T.tb) (css t.T.tc) (uri t.T.tu t.T.tx) (name t.T.tn)
-   in
-   st.out str; add st
-
-let close_table st t =
-   let st = sub st in
-   let str = P.sprintf "%s]\n" (indent st) in
-   st.out str; st
-
-let map_key st k =
-   let str = P.sprintf "%s%s\n" (indent st) (key k) in
-   st.out str; st
-   
-let open_line b st =
-   let str = P.sprintf "%s%s\n" (indent st) (entry b) in
-   st.out str; st
-
-let close_line b st = st
-
-let open_entry b st = st
-
-let close_entry b st sst = st
-
-let cb = {
-   F.open_table = open_table; F.close_table = close_table;   
-   F.open_line = open_line; F.close_line = close_line;
-   F.open_entry = open_entry; F.close_entry = close_entry;
-   F.map_key = map_key;
-}
-
-let debug t =
-   let _ = F.fold_table cb home t in ()

diff --git a/helm/www/lambdadelta/bin/xhtbl/xhtbl.ml b/helm/www/lambdadelta/bin/xhtbl/xhtbl.ml
deleted file mode 100644 (file)
index 6c5f8b0..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/xhtbl.ml
+++ /dev/null
@@ -1,77 +0,0 @@
-module A = Arg
-module F = Filename
-module L = List
-
-module O  = Options 
-module TP = TextParser
-module TL = TextLexer
-module TU = TextUnparser
-module P1 = Pass1
-module P2 = Pass2
-module P3 = Pass3
-module M  = Matrix
-module XU = XmlUnparser
-
-let help    = "Usage: xhtbl [ -LX | -O <dir> | -d0 | -d1 | -d2 | -e1 | -e2 | -p0 | -p1 | -p2 | <file> ]*"
-let help_L  = " Output lexer tokens"
-let help_O  = "<dir>  Set this output directory"
-let help_X  = " Clear all options"
-let help_b  = "<uri>  Set this base uri for relative links"
-let help_d0 = " Output table contents after phase zero (parsing)"
-let help_d1 = " Output table contents after phase one (sizing)"
-let help_d2 = " Output table contents after phase two (filling)"
-let help_e1 = " Disabled"
-let help_e2 = " Output debug information during phase two (filling)"
-let help_p0 = " Process until phase zero (parsing)"
-let help_p1 = " Process until phase one (sizing)"
-let help_p2 = " Process until phase two (filling)"
-
-let hook = "xhtbl"
-
-let includes, tables = ref [], ref []
-
-let process_directive och bname (name, table, css, uri, ext) =
-   tables := name :: !tables;
-   if !O.d0 then TU.debug table;
-   if not !O.p0 then begin
-      let size = P1.process table in
-      if !O.d1 then TU.debug table;
-      if not !O.p1 then begin
-         let matrix = M.make size in
-         let _ = P2.process table matrix in
-         if !O.d2 then TU.debug table;
-         if not !O.p2 then P3.process css uri ext matrix;
-        let name = if name = "" then bname else name in
-         XU.output och name matrix
-      end
-   end
-
-let process_file fname =
-   let bname = F.chop_extension (F.basename fname) in
-   let ich = open_in fname in
-   let lexbuf = Lexing.from_channel ich in
-   let ns, ds = TP.script TL.token lexbuf in
-   close_in ich; includes := bname :: !includes;
-   let ns = ("", "http://www.w3.org/1999/xhtml") :: ns in
-   let och = XU.open_out bname ns in 
-   L.iter (process_directive och bname) ds;
-   XU.close_out och
-
-let main () =
-   A.parse [
-      "-L", A.Set O.debug_lexer, help_L;
-      "-O", A.String ((:=) O.output_dir), help_O; 
-      "-X", A.Unit O.clear, help_X;
-      "-b", A.String ((:=) O.baseuri), help_b;
-      "-d0", A.Set O.d0, help_d0;  
-      "-d1", A.Set O.d1, help_d1;  
-      "-d2", A.Set O.d2, help_d2;  
-      "-e1", A.Set O.e1, help_e1;  
-      "-e2", A.Set O.e2, help_e2;  
-      "-p0", A.Set O.p0, help_p0;  
-      "-p1", A.Set O.p1, help_p1;  
-      "-p2", A.Set O.p2, help_p2;  
-   ] process_file help;
-   XU.write_hook hook !includes !tables 
-
-let _ = main ()

diff --git a/helm/www/lambdadelta/bin/xhtbl/xmlUnparser.ml b/helm/www/lambdadelta/bin/xhtbl/xmlUnparser.ml
deleted file mode 100644 (file)
index 2f29e4b..0000000
--- a/helm/www/lambdadelta/bin/xhtbl/xmlUnparser.ml
+++ /dev/null
@@ -1,104 +0,0 @@
-module A = Array
-module F = Filename
-module L = List
-module P = Printf
-module S = String
-
-module O = Options
-module T = Table
-module M = Matrix
-
-let xhtbl = "xhtbl"
-
-let i = 0
-
-let myself = F.basename (Sys.argv.(0))
-
-let msg = P.sprintf "This file was generated by %s, do not edit" myself
-
-let compose uri ext =
-   if uri.[pred (S.length uri)] = '/' then uri else
-   try
-      let i = S.index uri '#' in
-      let uri, fragment = S.sub uri 0 i, S.sub uri i (S.length uri - i) in
-      uri ^ ext ^ fragment
-   with Not_found -> uri ^ ext
-
-let border cell =
-   let str = S.make 4 'n' in
-   if cell.M.cb.T.n then str.[0] <- 's';   
-   if cell.M.cb.T.e then str.[1] <- 's';
-   if cell.M.cb.T.s then str.[2] <- 's';
-   if cell.M.cb.T.w then str.[3] <- 's';
-   str :: cell.M.cc
-
-let text baseuri ext = function
-   | T.Plain s              -> s
-   | T.Link (true, uri, s)  -> P.sprintf "<a href=\"%s\">%s</a>" uri s
-   | T.Link (false, uri, s) -> 
-      let uri = !O.baseuri ^ baseuri ^ compose uri ext in
-      P.sprintf "<a href=\"%s\">%s</a>" uri s
-
-let name cell =
-   if cell.M.cn = "" then "" else P.sprintf " id=\"%s\"" cell.M.cn
-
-let key cell =
-   if cell.M.ck = [] then "<br/>" else S.concat "" (L.map (text cell.M.cu cell.M.cx) cell.M.ck)
-
-let ind i = S.make (2 * i) ' '
-
-let out_cell och cell =
-   let cc = xhtbl :: border cell in
-   P.fprintf och "%s<td class=\"%s\"%s>%s</td>\n"
-      (ind (i+3)) (S.concat " " cc) (name cell) (key cell)
-
-let out_row och row =
-   P.fprintf och "%s<tr class=\"%s\">\n" (ind (i+2)) xhtbl;
-   A.iter (out_cell och) row;
-   P.fprintf och "%s</tr>\n" (ind (i+2))
-
-let out_space och (name, uri) =
-   let name = if name = "" then name else ":" ^ name in
-   P.fprintf och "                xmlns%s=\"%s\"\n" name uri
-
-(****************************************************************************)
-
-let open_out name spaces =
-   let fname = F.concat !O.output_dir (P.sprintf "%s.xsl" name) in
-   let spaces = ("xsl", "http://www.w3.org/1999/XSL/Transform") :: spaces in
-   let och = open_out fname in
-   P.fprintf och "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n\n";
-   P.fprintf och "<!-- %s -->\n\n" msg;   
-   P.fprintf och "<xsl:stylesheet version=\"1.0\"\n";
-   L.iter (out_space och) spaces;
-   P.fprintf och ">\n\n";
-   och
-
-let output och name matrix =
-   P.fprintf och "<xsl:template name=\"%s\">\n" name;
-   P.fprintf och "%s<table class=\"%s\" cellpadding=\"4\" cellspacing=\"0\">\n" (ind (i+1)) xhtbl;
-   A.iter (out_row och) matrix.M.m;
-   P.fprintf och "%s</table>\n" (ind (i+1));
-   P.fprintf och "</xsl:template>\n\n"
-
-let close_out och =
-   P.fprintf och "</xsl:stylesheet>\n";
-   close_out och
-
-let map_incs och name =
-   P.fprintf och "<xsl:include href=\"%s.xsl\"/>\n" name
-
-let map_tbls och name =
-   P.fprintf och "%s<xsl:when test=\"@name='%s'\">\n" (ind (i+2)) name;
-   P.fprintf och "%s<xsl:call-template name=\"%s\"/>\n" (ind (i+3)) name;
-   P.fprintf och "%s</xsl:when>\n" (ind (i+2))
-
-let write_hook name incs tbls =
-   let och = open_out name [] in
-   L.iter (map_incs och) incs;
-   P.fprintf och "\n<xsl:template name=\"%s\">\n" name;
-   P.fprintf och "%s<xsl:choose>\n" (ind (i+1));
-   L.iter (map_tbls och) tbls;   
-   P.fprintf och "%s</xsl:choose>\n" (ind (i+1));
-   P.fprintf och "</xsl:template>\n\n";
-   close_out och

diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma
deleted file mode 100644 (file)
index dc435ca..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma
+++ /dev/null
@@ -1,22 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/preamble.ma".
-
-inductive A: Type[0] \def
-| ASort: nat \to (nat \to A)
-| AHead: A \to (A \to A).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma
deleted file mode 100644 (file)
index 9e2eb7d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/A/defs.ma".
-
-implied rec lemma A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall 
-(n0: nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: 
-A).((P a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with 
-[(ASort n n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (f0 a0 
-((A_rect P f f0) a0) a1 ((A_rect P f f0) a1))].
-
-implied lemma A_ind:
- \forall (P: ((A \to Prop))).(((\forall (n: nat).(\forall (n0: nat).(P (ASort 
-n n0))))) \to (((\forall (a: A).((P a) \to (\forall (a0: A).((P a0) \to (P 
-(AHead a a0))))))) \to (\forall (a: A).(P a))))
-\def
- \lambda (P: ((A \to Prop))).(A_rect P).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma
deleted file mode 100644 (file)
index 9d41d8c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/C/defs.ma
+++ /dev/null
@@ -1,39 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-inductive C: Type[0] \def
-| CSort: nat \to C
-| CHead: C \to (K \to (T \to C)).
-
-rec definition cweight (c: C) on c: nat \def match c with [(CSort _) 
-\Rightarrow O | (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))].
-
-definition clt:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
-
-definition cle:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
-
-rec definition CTail (k: K) (t: T) (c: C) on c: C \def match c with [(CSort 
-n) \Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead 
-(CTail k t d) h u)].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma
deleted file mode 100644 (file)
index 43d43fc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/C/fwd.ma
+++ /dev/null
@@ -1,58 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-implied rec lemma C_rect (P: (C \to Type[0])) (f: (\forall (n: nat).(P (CSort 
-n)))) (f0: (\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P 
-(CHead c k t))))))) (c: C) on c: P c \def match c with [(CSort n) \Rightarrow 
-(f n) | (CHead c0 k t) \Rightarrow (f0 c0 ((C_rect P f f0) c0) k t)].
-
-implied lemma C_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
-(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k 
-t))))))) \to (\forall (c: C).(P c))))
-\def
- \lambda (P: ((C \to Prop))).(C_rect P).
-
-fact clt_wf__q_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to 
-Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 
-c))))) P n))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: 
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) 
-n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight 
-c)))))).
-
-lemma clt_wf_ind:
- \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) 
-\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: 
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to 
-Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) 
-(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind 
-(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: 
-C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
-\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat 
-(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
-(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P 
-c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt 
-(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight 
-d))))))))))))) c)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/C/props.ma b/matita/matita/contribs/lambdadelta/basic_1/C/props.ma
deleted file mode 100644 (file)
index dacd548..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/C/props.ma
+++ /dev/null
@@ -1,115 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/fwd.ma".
-
-include "basic_1/T/props.ma".
-
-lemma cle_r:
- \forall (c: C).(cle c c)
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(le (cweight c0) (cweight c0))) 
-(\lambda (_: nat).(le_O_n O)) (\lambda (c0: C).(\lambda (_: (le (cweight c0) 
-(cweight c0))).(\lambda (_: K).(\lambda (t: T).(le_n (plus (cweight c0) 
-(tweight t))))))) c).
-
-lemma cle_head:
- \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (u1: T).(\forall 
-(u2: T).((tle u1 u2) \to (\forall (k: K).(cle (CHead c1 k u1) (CHead c2 k 
-u2))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
-c2))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (le (tweight u1) 
-(tweight u2))).(\lambda (_: K).(le_plus_plus (cweight c1) (cweight c2) 
-(tweight u1) (tweight u2) H H0))))))).
-
-lemma cle_trans_head:
- \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (k: K).(\forall 
-(u: T).(cle c1 (CHead c2 k u))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
-c2))).(\lambda (_: K).(\lambda (u: T).(le_plus_trans (cweight c1) (cweight 
-c2) (tweight u) H))))).
-
-lemma clt_cong:
- \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: 
-T).(clt (CHead c k t) (CHead d k t))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight 
-d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) 
-(tweight t) H))))).
-
-lemma clt_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight 
-c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_reg_l O 
-(tweight u) (cweight c) (tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
-
-lemma chead_ctail:
- \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: 
-K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) 
-(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: 
-K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
-(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead 
-(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall 
-(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
-(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def 
-H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
-(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: 
-C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) 
-(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead 
-c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: 
-C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead 
-c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: 
-C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 
-(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 
-k t) H1))))) H0))))))))) c).
-
-lemma clt_thead:
- \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt 
-c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: 
-C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: 
-T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
-
-lemma c_tail_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
-(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t 
-c))))))) \to (\forall (c: C).(P c))))
-\def
- \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort 
-n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: 
-T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P 
-c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) 
-\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: 
-C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: 
-((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to 
-(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind 
-K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) 
-(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 
-x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def 
-(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P 
-d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) 
-(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma
deleted file mode 100644 (file)
index ae81c78..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/G/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/preamble.ma".
-
-record G : Type[0] \def {
- next: (nat \to nat);
- next_lt: (\forall (n: nat).(lt n (next n)))
-}.
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
deleted file mode 100644 (file)
index ae783b8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
+++ /dev/null
@@ -1,411 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/fwd.ma".
-
-fact terms_props__bind_dec:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
-(P: Prop).P))))
-\def
- \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq 
-B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: 
-B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl 
-(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B 
-Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: 
-Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
-\Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind P 
-H0))))) (or_intror (eq B Abbr Void) ((eq B Abbr Void) \to (\forall (P: 
-Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
-\Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind P 
-H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B 
-Abst b) \to (\forall (P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst 
-Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: 
-Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match ee with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False])) I Abbr 
-H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B Abst Abst) \to 
-(\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B Abst Void) ((eq 
-B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst 
-Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match 
-ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow 
-False])) I Void H) in (False_ind P H0))))) b2)) (\lambda (b2: B).(B_ind 
-(\lambda (b: B).(or (eq B Void b) ((eq B Void b) \to (\forall (P: Prop).P)))) 
-(or_intror (eq B Void Abbr) ((eq B Void Abbr) \to (\forall (P: Prop).P)) 
-(\lambda (H: (eq B Void Abbr)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void 
-(\lambda (ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow 
-False | Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror 
-(eq B Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: 
-(eq B Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda 
-(ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | 
-Void \Rightarrow True])) I Abst H) in (False_ind P H0))))) (or_introl (eq B 
-Void Void) ((eq B Void Void) \to (\forall (P: Prop).P)) (refl_equal B Void)) 
-b2)) b1).
-
-lemma bind_dec_not:
- \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
-\def
- \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
-in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: 
-Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 
-b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: 
-(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
-b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
-
-fact terms_props__flat_dec:
- \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
-(P: Prop).P))))
-\def
- \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq 
-F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: 
-F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl 
-(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F 
-Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: 
-Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind F Appl (\lambda (ee: F).(match ee with [Appl \Rightarrow True | Cast 
-\Rightarrow False])) I Cast H) in (False_ind P H0))))) f2)) (\lambda (f2: 
-F).(F_ind (\lambda (f: F).(or (eq F Cast f) ((eq F Cast f) \to (\forall (P: 
-Prop).P)))) (or_intror (eq F Cast Appl) ((eq F Cast Appl) \to (\forall (P: 
-Prop).P)) (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind F Cast (\lambda (ee: F).(match ee with [Appl \Rightarrow False | Cast 
-\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast 
-Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) 
-f1).
-
-fact terms_props__kind_dec:
- \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall 
-(P: Prop).P))))
-\def
- \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq 
-K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind 
-(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: 
-Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in 
-(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: 
-Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to 
-(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: 
-B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: 
-Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to 
-(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B 
-b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq 
-K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) 
-(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: 
-K).(match e with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) (Bind 
-b) (Bind b0) H1) in (let H3 \def (eq_ind_r B b0 (\lambda (b1: B).((eq B b b1) 
-\to (\forall (P0: Prop).P0))) H0 b H2) in (H3 (refl_equal B b) P))))))) H)))) 
-(\lambda (f: F).(or_intror (eq K (Bind b) (Flat f)) ((eq K (Bind b) (Flat f)) 
-\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Bind b) (Flat f))).(\lambda 
-(P: Prop).(let H0 \def (eq_ind K (Bind b) (\lambda (ee: K).(match ee with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in 
-(False_ind P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda 
-(k: K).(or (eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) 
-(\lambda (b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) 
-\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda 
-(P: Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind b) H) in 
-(False_ind P H0)))))) (\lambda (f0: F).(let H_x \def (terms_props__flat_dec f 
-f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F f f0) \to (\forall (P: 
-Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to 
-(\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind F f (\lambda (f1: 
-F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) \to (\forall (P: 
-Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat f) (Flat f)) \to 
-(\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) (\lambda (H0: (((eq F 
-f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Flat f) (Flat f0)) ((eq 
-K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Flat f) 
-(Flat f0))).(\lambda (P: Prop).(let H2 \def (f_equal K F (\lambda (e: 
-K).(match e with [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat 
-f) (Flat f0) H1) in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) 
-\to (\forall (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) 
-k2))) k1).
-
-lemma term_dec:
- \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall 
-(P: Prop).P))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq 
-T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: 
-T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to 
-(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in 
-(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: 
-Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to 
-(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda 
-(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to 
-(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort 
-n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) 
-(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T 
-(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) 
-(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
-(TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort n0) 
-H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to 
-(\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) 
-(\lambda (n0: nat).(or_intror (eq T (TSort n) (TLRef n0)) ((eq T (TSort n) 
-(TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TSort n) (TLRef 
-n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) 
-(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T 
-(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or 
-(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: 
-Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) 
-(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) 
-(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) I (THead k t t0) H1) in (False_ind 
-P H2)))))))))) t2))) (\lambda (n: nat).(\lambda (t2: T).(T_ind (\lambda (t: 
-T).(or (eq T (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))) 
-(\lambda (n0: nat).(or_intror (eq T (TLRef n) (TSort n0)) ((eq T (TLRef n) 
-(TSort n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TLRef n) (TSort 
-n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) 
-(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind 
-(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) 
-(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda 
-(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) 
-(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) 
-(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to 
-(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq 
-nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) 
-((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T 
-(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow n | (TLRef n1) 
-\Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H1) in 
-(let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to (\forall 
-(P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) (\lambda 
-(k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TLRef n) t) ((eq T (TLRef n) 
-t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T 
-(TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: Prop).P)))).(or_intror 
-(eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) (THead k t t0)) \to (\forall 
-(P: Prop).P)) (\lambda (H1: (eq T (TLRef n) (THead k t t0))).(\lambda (P: 
-Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) 
-(\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t 
-t2) ((eq T t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda 
-(H0: ((\forall (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
-Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t 
-t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: 
-nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort 
-n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort 
-n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind P H2)))))) 
-(\lambda (n: nat).(or_intror (eq T (THead k t t0) (TLRef n)) ((eq T (THead k 
-t t0) (TLRef n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t 
-t0) (TLRef n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in 
-(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq 
-T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: 
-Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq 
-T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in 
-(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: 
-Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) 
-(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let 
-H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
-(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t 
-(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t 
-t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in 
-(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: 
-Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) 
-(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let 
-H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
-(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 
-(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t 
-t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def 
-(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq 
-K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) 
-((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda 
-(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) 
-(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: 
-Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t 
-t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) 
-k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror 
-(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) 
-\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t 
-t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
-\Rightarrow k1])) (THead k t t0) (THead k0 t t0) H11) in (let H13 \def 
-(eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to (\forall (P0: Prop).P0))) 
-H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) t4 H7))) (\lambda (H7: 
-(((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror (eq T (THead k t t0) 
-(THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
-Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t t4))).(\lambda (P: 
-Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) 
-(THead k t t0) (THead k0 t t4) H8) in ((let H10 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in 
-(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq 
-T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r 
-T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
-\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) 
-H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: 
-Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k 
-t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead 
-k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t3 
-t4) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) 
-(THead k t t0) (THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 t4) H5) in 
-(\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def (eq_ind_r 
-T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
-\to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r T t3 
-(\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in (let 
-H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
-(THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 
-(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
-
-lemma binder_dec:
- \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
-T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall 
-(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: 
-B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
-u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T 
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind 
-b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) 
-(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
-(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w 
-u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P 
-H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda 
-(w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: 
-B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to 
-(\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) 
-(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T 
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 
-u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead 
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 
-B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind 
-b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead 
-(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda 
-(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) 
-w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 
-t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: 
-B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda 
-(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: 
-B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to 
-(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T 
-(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w 
-u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead 
-(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T 
-(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) 
-(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: 
-T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: 
-Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: 
-T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal 
-T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda 
-(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 
-(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: 
-T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda 
-(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda 
-(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: 
-T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) 
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) 
-(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
-(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead 
-(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 
-t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) 
-in (False_ind P H2))))))))))))) k)) t).
-
-lemma abst_dec:
- \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead 
-(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to 
-(\forall (P: Prop).P)))))
-\def
- \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda 
-(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead 
-(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda 
-(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v 
-t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall 
-(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind 
-Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in 
-(False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(or_intror (ex T 
-(\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t: 
-T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) 
-(\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v 
-t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind 
-P H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall (v: 
-T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall 
-(t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P: 
-Prop).P))))))).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(or (ex T 
-(\lambda (t1: T).(eq T t0 (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T 
-t0 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))))).(\lambda (v: 
-T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H1 \def H_x in 
-(or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) 
-(or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) 
-(\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall 
-(P: Prop).P)))) (\lambda (H2: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) 
-(\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead 
-(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind 
-Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in 
-(let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P: 
-Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead 
-(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead 
-(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H4: (eq T t 
-v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead 
-(Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead 
-(Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P))))) 
-(or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind 
-Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind 
-Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T 
-(THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead 
-(Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) \to (\forall (P: 
-Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) 
-(THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) 
-(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
-T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
-t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) 
-\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in 
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) 
-(THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T 
-t v)).(H4 H8 P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) 
-\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k 
-t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) 
-(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
-T).(\lambda (H3: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: 
-Prop).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead k t t0) (THead (Bind 
-Abst) v t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
-\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: 
-(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) 
-H1))))))))) u).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma
deleted file mode 100644 (file)
index 0eeff13..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/T/defs.ma
+++ /dev/null
@@ -1,45 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/preamble.ma".
-
-inductive B: Type[0] \def
-| Abbr: B
-| Abst: B
-| Void: B.
-
-inductive F: Type[0] \def
-| Appl: F
-| Cast: F.
-
-inductive K: Type[0] \def
-| Bind: B \to K
-| Flat: F \to K.
-
-inductive T: Type[0] \def
-| TSort: nat \to T
-| TLRef: nat \to T
-| THead: K \to (T \to (T \to T)).
-
-rec definition tweight (t: T) on t: nat \def match t with [(TSort _) 
-\Rightarrow (S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow 
-(S (plus (tweight u) (tweight t0)))].
-
-definition tle:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(le (tweight t1) (tweight t2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma
deleted file mode 100644 (file)
index 5e1833c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/T/fwd.ma
+++ /dev/null
@@ -1,64 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-implied rec lemma T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort 
-n)))) (f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall 
-(t: T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t: 
-T) on t: P t \def match t with [(TSort n) \Rightarrow (f n) | (TLRef n) 
-\Rightarrow (f0 n) | (THead k t0 t1) \Rightarrow (f1 k t0 ((T_rect P f f0 f1) 
-t0) t1 ((T_rect P f f0 f1) t1))].
-
-implied lemma T_ind:
- \forall (P: ((T \to Prop))).(((\forall (n: nat).(P (TSort n)))) \to 
-(((\forall (n: nat).(P (TLRef n)))) \to (((\forall (k: K).(\forall (t: T).((P 
-t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) \to (\forall (t: 
-T).(P t)))))
-\def
- \lambda (P: ((T \to Prop))).(T_rect P).
-
-lemma thead_x_y_y:
- \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq 
-T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda 
-(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def 
-(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H) in (False_ind P H0))))) (\lambda (n: nat).(\lambda (H: 
-(eq T (THead k v (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def 
-(eq_ind T (THead k v (TLRef n)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: 
-Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to 
-(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) 
-(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead 
-k1 _ _) \Rightarrow k1])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) 
-in ((let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) 
-(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T 
-T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead k0 t0 t1) | 
-(TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) \Rightarrow t2])) 
-(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in (\lambda (H5: (eq T v 
-t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T v (\lambda (t2: 
-T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) H0 t0 H5) in (let 
-H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 t1) t1) \to (\forall 
-(P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) H2))))))))) t))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/props.ma b/matita/matita/contribs/lambdadelta/basic_1/T/props.ma
deleted file mode 100644 (file)
index 7b62a5a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/T/props.ma
+++ /dev/null
@@ -1,64 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/fwd.ma".
-
-lemma not_abbr_abst:
- not (eq B Abbr Abst)
-\def
- \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
-B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
-\Rightarrow False])) I Abst H) in (False_ind False H0)).
-
-lemma not_void_abst:
- not (eq B Void Abst)
-\def
- \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: 
-B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
-\Rightarrow True])) I Abst H) in (False_ind False H0)).
-
-lemma not_abbr_void:
- not (eq B Abbr Void)
-\def
- \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
-B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
-\Rightarrow False])) I Void H) in (False_ind False H0)).
-
-lemma not_abst_void:
- not (eq B Abst Void)
-\def
- \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: 
-B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void 
-\Rightarrow False])) I Void H) in (False_ind False H0)).
-
-lemma tweight_lt:
- \forall (t: T).(lt O (tweight t))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: 
-nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda 
-(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O 
-(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S 
-O) (tweight t0) (tweight t1) H))))))) t).
-
-lemma tle_r:
- \forall (t: T).(tle t t)
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(le (tweight t0) (tweight t0))) 
-(\lambda (_: nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: 
-K).(\lambda (t0: T).(\lambda (_: (le (tweight t0) (tweight t0))).(\lambda 
-(t1: T).(\lambda (_: (le (tweight t1) (tweight t1))).(le_n (S (plus (tweight 
-t0) (tweight t1))))))))) t).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma
deleted file mode 100644 (file)
index f339331..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma
+++ /dev/null
@@ -1,21 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/asucc/defs.ma".
-
-rec definition aplus (g: G) (a: A) (n: nat) on n: A \def match n with [O 
-\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
deleted file mode 100644 (file)
index 73ec98c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma
+++ /dev/null
@@ -1,240 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/aplus/defs.ma".
-
-include "basic_1/A/fwd.ma".
-
-include "basic_1/next_plus/props.ma".
-
-lemma aplus_reg_r:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall 
-(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A 
-(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda 
-(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 
-(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n 
-h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n 
-h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
-
-lemma aplus_assoc:
- \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
-(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: 
-nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n 
-h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: 
-nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus 
-g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A 
-(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) 
-(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g 
-(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O 
-n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) 
-n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda 
-(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g 
-(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) 
-n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) 
-(plus_n_Sm n n0)))) h2)))) h1))).
-
-lemma aplus_asucc:
- \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) 
-h) (asucc g (aplus g a h)))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a 
-(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) 
-(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) 
-(aplus_assoc g a (S O) h)))).
-
-lemma aplus_sort_O_S_simpl:
- \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O 
-n) (S k)) (aplus g (ASort O (next g n)) k))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc 
-g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) 
-(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) 
-k)) (aplus_asucc g k (ASort O n))))).
-
-lemma aplus_sort_S_S_simpl:
- \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A 
-(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind 
-A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g 
-(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g 
-(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
-
-lemma aplus_asort_O_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O 
-n) h) (ASort O (next_plus g n h)))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: 
-nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda 
-(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall 
-(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 
-n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) 
-(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat 
-(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next 
-g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) 
-(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n 
-(ASort O n0)))))) h)).
-
-lemma aplus_asort_le_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h 
-k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: 
-nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort 
-(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O 
-k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) 
-(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: 
-nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A 
-(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: 
-nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A 
-(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda 
-(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat 
-O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) 
-h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S 
-x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) 
-H1) in (False_ind (eq A (asucc g (aplus g (ASort O n) h0)) (ASort (minus O (S 
-h0)) n)) H3))))) (le_gen_S h0 O H0)))) (\lambda (n: nat).(\lambda (_: 
-((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n n0) 
-h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: (le (S 
-h0) (S n))).(eq_ind A (aplus g (asucc g (ASort (S n) n0)) h0) (\lambda (a: 
-A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n n0 (le_S_n h0 n H1)) (asucc 
-g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g h0 (ASort (S n) n0))))))) 
-k)))) h)).
-
-lemma aplus_asort_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A 
-(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: 
-nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus 
-g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) 
-(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) 
-(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) 
-(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus 
-h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a 
-(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O 
-(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k 
-h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A 
-(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) 
-(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h 
-(le_S_n (S k) (S h) (le_S (S (S k)) (S h) (le_n_S (S k) h H)))))) (minus k k) 
-(minus_n_n k)) (aplus g (ASort k n) k) (aplus_asort_le_simpl g k k n (le_n 
-k))) (aplus g (ASort k n) (plus k (minus h k))) (aplus_assoc g (ASort k n) k 
-(minus h k))) h (le_plus_minus k h (le_S_n k h (le_S_n (S k) (S h) (le_S (S 
-(S k)) (S h) (le_n_S (S k) h H))))))) (\lambda (H: (le h k)).(eq_ind_r A 
-(ASort (minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g 
-n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) 
-n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) 
-(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) 
-(aplus_asort_le_simpl g h k n H))))))).
-
-lemma aplus_ahead_simpl:
- \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A 
-(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: 
-A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 
-n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) 
-(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A 
-(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: 
-A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda 
-(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g 
-(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) 
-(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n 
-a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) 
-h)).
-
-lemma aplus_asucc_false:
- \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) 
-h) a) \to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: 
-nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A 
-(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) 
-\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
-(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O 
-(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) 
-(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 
-\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 
-(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) 
-(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat 
-(\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) 
-\Rightarrow (next_plus g (next g n0) (minus h O))])) (ASort (minus O h) 
-(next_plus g (next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def 
-(eq_ind_r nat (minus h O) (\lambda (n1: nat).(eq nat (next_plus g (next g n0) 
-n1) n0)) H2 h (minus_n_O h)) in (le_lt_false n0 n0 (le_n n0) (eq_ind nat 
-(next_plus g (next g n0) h) (\lambda (n1: nat).(lt n0 n1)) (next_plus_lt g h 
-n0) n0 H3) P))))) (\lambda (n1: nat).(\lambda (_: (((eq A (aplus g (match n1 
-with [O \Rightarrow (ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 
-n0)]) h) (ASort n1 n0)) \to P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) 
-h) (ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) 
-(\lambda (a0: A).(eq A a0 (ASort (S n1) n0))) H0 (ASort (minus n1 h) 
-(next_plus g n0 (minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def 
-(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
-(AHead _ _) \Rightarrow (minus n1 h)])) (ASort (minus n1 h) (next_plus g n0 
-(minus h n1))) (ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda 
-(e: A).(match e with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow 
-(next_plus g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h 
-n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S 
-n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) 
-(minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: A).(\lambda 
-(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: 
-Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus 
-g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: 
-nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 
-a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g 
-a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 (AHead a0 (aplus g 
-(asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def 
-(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (aplus g 
-(asucc g a1) h) | (AHead _ a2) \Rightarrow a2])) (AHead a0 (aplus g (asucc g 
-a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
-
-lemma aplus_inj:
- \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A 
-(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: 
-nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
-h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A 
-(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: 
-(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: 
-A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: 
-(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g 
-a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g 
-n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq 
-nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: 
-nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
-h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq 
-A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: 
-A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A 
-(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) 
-n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) 
-(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a 
-n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: 
-(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def 
-(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus 
-g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def 
-(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g 
-a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat 
-nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma
deleted file mode 100644 (file)
index 32ec12f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/app/defs.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-rec definition cbk (c: C) on c: nat \def match c with [(CSort m) \Rightarrow 
-m | (CHead c0 _ _) \Rightarrow (cbk c0)].
-
-rec definition app1 (c: C) on c: T \to T \def \lambda (t: T).(match c with 
-[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u 
-t))]).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma
deleted file mode 100644 (file)
index e3ea6cc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/A/defs.ma".
-
-inductive aprem: nat \to (A \to (A \to Prop)) \def
-| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
-| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 
-a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
deleted file mode 100644 (file)
index 3f415a4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma
+++ /dev/null
@@ -1,113 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/aprem/defs.ma".
-
-implied rec lemma aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall 
-(a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: 
-A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to 
-(\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) 
-(a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) 
-\Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4 
-((aprem_ind P f f0) i a2 a3 a4) a5)].
-
-lemma aprem_gen_sort:
- \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem 
-i (ASort h n) x) \to False))))
-\def
- \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: 
-nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda 
-(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: 
-(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: 
-A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: 
-A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A 
-(AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False 
-| (AHead _ _) \Rightarrow True])) I (ASort h n) H1) in (False_ind False 
-H2))))) (\lambda (a2: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (_: 
-(aprem i0 a2 a)).(\lambda (_: (((eq A a2 (ASort h n)) \to False))).(\lambda 
-(a1: A).(\lambda (H3: (eq A (AHead a1 a2) (ASort h n))).(let H4 \def (eq_ind 
-A (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow 
-False | (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False 
-H4))))))))) i y x H0))) H))))).
-
-lemma aprem_gen_head_O:
- \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) 
-\to (eq A x a1))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O 
-(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) 
-(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y 
-x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq 
-A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem 
-y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq 
-nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: 
-A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 
-a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) 
-(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) 
-(AHead a1 a2) H3) in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: 
-A).(\lambda (a: A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda 
-(H3: (((eq nat i O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda 
-(a3: A).(\lambda (H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) 
-(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0) 
-(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) 
-(AHead a1 a2) H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 
-(\lambda (a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a 
-a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i a4 
-a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee 
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
-(eq A a a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
-
-lemma aprem_gen_head_S:
- \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem 
-(S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda 
-(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: 
-A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: 
-A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: 
-nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 
-x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: 
-nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead 
-a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda 
-(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let 
-H4 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow 
-a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 
-\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | 
-(AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: 
-(eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def 
-(eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) 
-\Rightarrow False])) I (S i) H2) in (False_ind (aprem i a2 a1) H7)) a0 H6))) 
-H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (H2: 
-(aprem i0 a0 a)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq A a0 (AHead a1 
-a2)) \to (aprem i a2 a))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S i0) (S 
-i))).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def (f_equal 
-A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) 
-\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A 
-A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead _ a4) 
-\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3 
-a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A 
-a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 a2 H7) in (let H10 \def (eq_ind A 
-a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 H7) in (let H11 \def (f_equal nat 
-nat (\lambda (e: nat).(match e with [O \Rightarrow i0 | (S n) \Rightarrow 
-n])) (S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq 
-nat n (S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in 
-(let H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in 
-H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma
deleted file mode 100644 (file)
index 1932956..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma
+++ /dev/null
@@ -1,70 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/aprem/fwd.ma".
-
-include "basic_1/leq/fwd.ma".
-
-lemma aprem_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
-(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g 
-b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall 
-(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
-(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda 
-(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g 
-(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: 
-A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 
-i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g 
-b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda 
-(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall 
-(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g 
-b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda 
-(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall 
-(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
-(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda 
-(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead 
-a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem 
-n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y 
-\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A 
-(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) 
-(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead 
-a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda 
-(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) 
-(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) 
-(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let 
-H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: 
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: 
-A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda 
-(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 
-A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 
-a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 
-H)))).
-
-lemma aprem_asucc:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i 
-a1 a2) \to (aprem i (asucc g a1) a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda 
-(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda 
-(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: 
-A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda 
-(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) 
-a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 
-H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma
deleted file mode 100644 (file)
index 4426607..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/aprem.ma
+++ /dev/null
@@ -1,257 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/props.ma".
-
-include "basic_1/arity/cimp.ma".
-
-include "basic_1/aprem/props.ma".
-
-lemma arity_aprem:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
-a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
-b)))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: 
-A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
-b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda 
-(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b 
-i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: 
-(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem 
-i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: 
-A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def 
-H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop 
-(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 
-\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def 
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda 
-(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
-(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop 
-(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus 
-i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) 
-H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: 
-((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T 
-nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 
-d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 
-(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem 
-i0 a0 b)).(let H_y \def (H2 i0 b) in (let H4 \def (H_y (aprem_asucc g a0 b i0 
-H3)) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop 
-(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x 
-\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abst c0 u i H0) in (let H7 \def 
-H_x in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: 
-C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
-(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop 
-(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus 
-i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) 
-H9))))) H7)))))))) H4)))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
-g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) 
-\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: 
-((\forall (i: nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead 
-c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity 
-g d u0 (asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda 
-(H5: (aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in 
-(ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda 
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
-(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop 
-(plus i x2) O x0 (CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc 
-g b0))).(let H9 \def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O 
-x0 c0)) (drop_S b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) 
-in (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g b0))))) x0 x1 (S x2) H9 H8))))))) 
-H6))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (_: ((\forall (i: 
-nat).(\forall (b: A).((aprem i (asucc g a1) b) \to (ex2_3 C T nat (\lambda 
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
-(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead 
-c0 (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (i: nat).(\forall (b: 
-A).((aprem i a2 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind Abst) u))))) 
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i (AHead 
-a1 a2) b)).(nat_ind (\lambda (n: nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C 
-T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d 
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
-(asucc g b)))))))) (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H_y \def 
-(aprem_gen_head_O a1 a2 b H5) in (eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d 
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
-(asucc g a0))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a1))))) c0 u O (drop_refl 
-c0) H0) b H_y))) (\lambda (i0: nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) 
-\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i0 j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d u0 (asucc g b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 
-a2) b)).(let H_y \def (aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 
-i0 b H_y) in (let H6 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) 
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind 
-Abst) u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d 
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
-(asucc g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) 
-H6))))))) i H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall 
-(b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: 
-((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
-c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
-(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem 
-i a2 b)).(let H_y \def (H3 (S i) b) in (let H5 \def (H_y (aprem_succ a2 b i 
-H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda 
-(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
-(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (S 
-(plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(C_ind 
-(\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1 x1 (asucc g 
-b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n: nat).(\lambda (H8: 
-(drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity g (CSort n) x1 
-(asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i x2)) O) (eq 
-nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0 (CSort 
-n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O 
-O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee 
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d 
-u0 (asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) 
-(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g 
-d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: 
-K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) 
-c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda 
-(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i 
-x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: 
-(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
-(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda 
-(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) 
-(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: 
-nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d 
-c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda 
-(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
-(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g 
-(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in 
-(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: 
-nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda 
-(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop 
-(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g 
-b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 
-(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) 
-H5))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda 
-(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: 
-A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: 
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall 
-(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: 
-nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) 
-in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
-(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
-b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: 
-(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g 
-b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
-nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
-(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) 
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
-t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to 
-(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
-i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d 
-u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
-a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x 
-\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A 
-(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T 
-nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
-c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc 
-g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i 
-a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat 
-(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
-x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
-(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 
-x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: 
-T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g 
-x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) 
-H4))))))))))))) c t a H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma
deleted file mode 100644 (file)
index e9222f5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/cimp.ma
+++ /dev/null
@@ -1,98 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/fwd.ma".
-
-include "basic_1/cimp/props.ma".
-
-lemma arity_cimp_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
-t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
-A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: 
-C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g 
-c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d 
-c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c 
-c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) 
-a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let 
-H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in 
-(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead 
-d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: 
-(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def 
-(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead 
-x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind 
-Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow x | (CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind 
-Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 
-(CHead x0 (Bind Abbr) u) H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: 
-C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 x H10) in (let H12 \def (eq_ind_r 
-C x0 (\lambda (c0: C).(cimp d c0)) H7 x H10) in (arity_abbr g c2 x u i H11 a0 
-(H2 x H12))))))))) H6))))) H4))))))))))))) (\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
-Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda 
-(H2: ((\forall (c2: C).((cimp d c2) \to (arity g c2 u (asucc g 
-a0)))))).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(let H_x \def (H3 Abst d 
-u i H0) in (let H4 \def H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead 
-d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: 
-(getl i c2 (CHead x (Bind Abst) u))).(let H_x0 \def (cimp_getl_conf c c2 H3 
-Abst d u i H0) in (let H6 \def H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d 
-d2)) (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 
-(TLRef i) a0) (\lambda (x0: C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl 
-i c2 (CHead x0 (Bind Abst) u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) 
-u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 
-(CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow x | (CHead 
-c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind Abst) u) 
-(getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) H8)) in 
-(let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind 
-Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d 
-c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) 
-H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
-(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
-a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u 
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
-(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) 
-u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c 
-c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) 
-(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
-((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
-a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to 
-(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c 
-c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) 
-(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
-(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: 
-A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: 
-C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: 
-C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 
-H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: 
-(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to 
-(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 
-a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 
-a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 
-c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda 
-(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: 
-C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: 
-(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 
-t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma
deleted file mode 100644 (file)
index b193fac..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/defs.ma
+++ /dev/null
@@ -1,45 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/leq/defs.ma".
-
-include "basic_1/getl/defs.ma".
-
-inductive arity (g: G): C \to (T \to (A \to Prop)) \def
-| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O 
-n)))
-| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) 
-\to (arity g c (TLRef i) a)))))))
-| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u 
-(asucc g a)) \to (arity g c (TLRef i) a)))))))
-| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: 
-T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c 
-(THead (Bind b) u t) a2)))))))))
-| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u 
-(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind 
-Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
-| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u 
-a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to 
-(arity g c (THead (Flat Appl) u t) a2)))))))
-| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u 
-(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat 
-Cast) u t) a))))))
-| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t 
-a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma
deleted file mode 100644 (file)
index 7821190..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/fwd.ma
+++ /dev/null
@@ -1,1291 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/defs.ma".
-
-include "basic_1/leq/asucc.ma".
-
-include "basic_1/getl/drop.ma".
-
-implied rec lemma arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f: 
-(\forall (c: C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0: 
-(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a) 
-\to (P c (TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall 
-(u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: 
-A).((arity g d u (asucc g a)) \to ((P d u (asucc g a)) \to (P c (TLRef i) 
-a)))))))))) (f2: (\forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to 
-(\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to ((P 
-(CHead c (Bind b) u) t a2) \to (P c (THead (Bind b) u t) a2))))))))))))) (f3: 
-(\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1)) 
-\to ((P c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g 
-(CHead c (Bind Abst) u) t a2) \to ((P (CHead c (Bind Abst) u) t a2) \to (P c 
-(THead (Bind Abst) u t) (AHead a1 a2)))))))))))) (f4: (\forall (c: 
-C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to 
-(\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to ((P c t 
-(AHead a1 a2)) \to (P c (THead (Flat Appl) u t) a2))))))))))) (f5: (\forall 
-(c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to ((P c u 
-(asucc g a)) \to (\forall (t: T).((arity g c t a) \to ((P c t a) \to (P c 
-(THead (Flat Cast) u t) a)))))))))) (f6: (\forall (c: C).(\forall (t: 
-T).(\forall (a1: A).((arity g c t a1) \to ((P c t a1) \to (\forall (a2: 
-A).((leq g a1 a2) \to (P c t a2))))))))) (c: C) (t: T) (a: A) (a0: arity g c 
-t a) on a0: P c t a \def match a0 with [(arity_sort c0 n) \Rightarrow (f c0 
-n) | (arity_abbr c0 d u i g0 a1 a2) \Rightarrow (f0 c0 d u i g0 a1 a2 
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) d u a1 a2)) | (arity_abst c0 d u i g0 
-a1 a2) \Rightarrow (f1 c0 d u i g0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 
-f6) d u (asucc g a1) a2)) | (arity_bind b n c0 u a1 a2 t0 a3 a4) \Rightarrow 
-(f2 b n c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u a1 a2) t0 a3 
-a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) (CHead c0 (Bind b) u) t0 a3 a4)) | 
-(arity_head c0 u a1 a2 t0 a3 a4) \Rightarrow (f3 c0 u a1 a2 ((arity_ind g P f 
-f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 a4 ((arity_ind g P f f0 f1 
-f2 f3 f4 f5 f6) (CHead c0 (Bind Abst) u) t0 a3 a4)) | (arity_appl c0 u a1 a2 
-t0 a3 a4) \Rightarrow (f4 c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) 
-c0 u a1 a2) t0 a3 a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 (AHead a1 
-a3) a4)) | (arity_cast c0 u a1 a2 t0 a3) \Rightarrow (f5 c0 u a1 a2 
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 
-((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 a1 a3)) | (arity_repl c0 t0 a1 
-a2 a3 l) \Rightarrow (f6 c0 t0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) 
-c0 t0 a1 a2) a3 l)].
-
-lemma arity_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c 
-(TSort n) a) \to (leq g a (ASort O n))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda 
-(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g 
-c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: 
-(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: 
-A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: 
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
-(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
-n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort O n1) (ASort O 
-n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((eq 
-T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H4: (eq T (TLRef i) 
-(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) 
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) 
-\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort 
-n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5))))))))))) 
-(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda 
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T 
-u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t 
-(TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) 
-u t) (TSort n))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind (leq g a2 
-(ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TSort 
-n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T 
-t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H5: (eq T (THead (Bind 
-Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
-(False_ind (leq g (AHead a1 a2) (ASort O n)) H6)))))))))))) (\lambda (c0: 
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda 
-(_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: 
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: 
-(((eq T t (TSort n)) \to (leq g (AHead a1 a2) (ASort O n))))).(\lambda (H5: 
-(eq T (THead (Flat Appl) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat 
-Appl) u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: 
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g 
-a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O 
-n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t 
-(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat 
-Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
-(False_ind (leq g a0 (ASort O n)) H6))))))))))) (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t a1)).(\lambda (H2: (((eq T t 
-(TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (a2: A).(\lambda (H3: (leq 
-g a1 a2)).(\lambda (H4: (eq T t (TSort n))).(let H5 \def (f_equal T T 
-(\lambda (e: T).e) t (TSort n) H4) in (let H6 \def (eq_ind T t (\lambda (t0: 
-T).((eq T t0 (TSort n)) \to (leq g a1 (ASort O n)))) H2 (TSort n) H5) in (let 
-H7 \def (eq_ind T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TSort n) H5) in 
-(leq_trans g a2 a1 (leq_sym g a1 a2 H3) (ASort O n) (H6 (refl_equal T (TSort 
-n))))))))))))))) c y a H0))) H))))).
-
-lemma arity_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c 
-(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c 
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) 
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) 
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda 
-(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g 
-c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl 
-i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind 
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) 
-(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T 
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: 
-C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d 
-(Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u a0)).(\lambda 
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: 
-T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i 
-d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 
-u0 (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef 
-i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d 
-(Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda 
-(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: 
-T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda 
-(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d 
-(Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g 
-a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda 
-(H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead 
-_ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat 
-i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in 
-(or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) 
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) 
-u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) 
-(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
-Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) 
-d u H6 H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
-g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t 
-(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead 
-c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i 
-(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda 
-(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind 
-b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2 
-C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) 
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda 
-(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g 
-a1))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t: 
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t 
-a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abbr) 
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T 
-(\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d 
-(Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g 
-a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) u t) (TLRef i))).(let H6 
-\def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) 
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
-a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: 
-A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef 
-i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d 
-(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 
-a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind 
-Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead 
-a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let 
-H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda 
-(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 (asucc g a2)))))) H6)))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda 
-(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: 
-T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 (asucc g a0))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: 
-T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
-T).(arity g d u0 (asucc g (asucc g a0)))))))))).(\lambda (t: T).(\lambda (_: 
-(arity g c0 t a0)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T 
-(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) 
-(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
-C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))))).(\lambda (H5: (eq T 
-(THead (Flat Cast) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Flat Cast) 
-u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in 
-(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead 
-d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) 
-(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) 
-u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) 
-H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: 
-(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T 
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: 
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 
-\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind 
-T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
-(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind 
-T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 
-(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind 
-Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) 
-(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d 
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d 
-(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or 
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) 
-u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda 
-(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: 
-(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) 
-x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C 
-T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T 
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T 
-(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
-(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
-C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: 
-(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda 
-(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
-T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
-(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
-(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) 
-(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
-
-lemma arity_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: 
-C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind 
-b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: 
-A).(arity g (CHead c (Bind b) u) t a2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda 
-(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity 
-g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
-T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u 
-a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: 
-T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda 
-(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda 
-(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
-a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) 
-(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H2) in (False_ind 
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 
-(Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq 
-T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) 
-(\lambda (_: A).(arity g (CHead d (Bind b) u) t a)))))).(\lambda (H5: (eq T 
-(TLRef i) (THead (Bind b) u t))).(let H6 \def (eq_ind T (TLRef i) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in 
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity 
-g (CHead c0 (Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda 
-(_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u 
-a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g 
-a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def 
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind b) u t) H5) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u 
-a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))) H6))))))))))) 
-(\lambda (b0: B).(\lambda (H2: (not (eq B b0 Abst))).(\lambda (c0: 
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: (arity g c0 u0 
-a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
-a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g (CHead c0 
-(Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u t)) \to (ex2 
-A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) (\lambda (_: 
-A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t a0)))))).(\lambda 
-(H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u t))).(let H8 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef 
-_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead 
-(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
-\Rightarrow t1])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in ((let 
-H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
-| (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind 
-b0) u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda 
-(H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind 
-b0) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind 
-b) u) t a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: 
-T).(arity g (CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def 
-(eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A 
-(\lambda (a3: A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: 
-A).(arity g (CHead (CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) 
-in (let H16 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) 
-t1) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T 
-t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) 
-(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let 
-H18 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let 
-H19 \def (eq_ind B b0 (\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to 
-(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: 
-A).(arity g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in 
-(let H20 \def (eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t 
-a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 
-Abst))) H2 b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) 
-(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) 
-H9)) H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
-A).(\lambda (H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
-(_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: 
-T).(\lambda (a0: A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 
-a0)).(\lambda (H5: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
-A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead 
-(CHead c0 (Bind Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead 
-(Bind Abst) u0 t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst 
-| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow Abst])])) (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) 
-in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) 
-(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) 
-\Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0) 
-(THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda (H11: (eq B 
-Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
-b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u 
-a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind b) u) t 
-a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g 
-(CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let H14 \def (eq_ind T u0 
-(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
-A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda (_: A).(arity g (CHead 
-(CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u H10) in (let H15 \def 
-(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a0)) H13 u 
-H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) 
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g 
-(CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u H10) in (let H17 \def (eq_ind 
-T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H2 u H10) in (let H18 
-\def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u t)) \to (ex2 
-A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) u a3)) (\lambda (_: 
-A).(arity g (CHead (CHead c0 (Bind Abst) u) (Bind b0) u) t a0))))) H14 Abst 
-H11) in (let H19 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind 
-b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: 
-A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) H16 Abst H11) in (let 
-H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H11) in 
-(eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) 
-(\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (AHead a1 a0))))) (let H21 
-\def (match (H20 (refl_equal B Abst)) in False with []) in H21) b 
-H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda (u0: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
-(_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: T).(\lambda 
-(a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: (((eq T t0 
-(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
-(_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 a0))))))).(\lambda (H6: (eq 
-T (THead (Flat Appl) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T 
-(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u 
-a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) H7)))))))))))) 
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0 
-(asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A 
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind 
-b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 
-a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
-a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) u 
-t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A 
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind 
-b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: 
-A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) 
-u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g 
-(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 
-a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T 
-(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 
-(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
-A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
-a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda 
-(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 
-(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g 
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A 
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind 
-b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: 
-(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g 
-c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 
-(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y 
-a2 H1))) H0)))))))).
-
-lemma arity_gen_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
-A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: 
-A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
-A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
-(CHead c (Bind Abst) u) t a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: 
-A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead 
-(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A 
-A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: 
-A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: 
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
-A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: 
-A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
-A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
-(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: 
-nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1: 
-A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 
-a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda 
-(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda 
-(_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
-(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead 
-(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ 
-_ _) \Rightarrow False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 
-A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl 
-i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 
-(asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A 
-A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) 
-(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: 
-A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda 
-(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef 
-i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u 
-t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 
-(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g 
-a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t 
-a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
-Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2: 
-(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to 
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) 
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0 
-a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda 
-(_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) 
-t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u 
-t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
-with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) 
-u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
-(THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0) (THead (Bind Abst) u 
-t) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
-(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 
-u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: 
-T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
-A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda 
-(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t 
-H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind 
-b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T 
-t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead 
-c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let 
-H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) 
-H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead 
-(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g 
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t 
-a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 
-t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t 
-(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq 
-A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 
-(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let 
-H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) 
-H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 
-Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in 
-False with []) in H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0: 
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g 
-a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
-(\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 a3)))) 
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0 
-a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
-(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
-A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3)))) 
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0) 
-(Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
-(THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
-\Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in 
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
-(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T 
-u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
-Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead 
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) 
-u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 
-(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T 
-t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in 
-(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) 
-\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) 
-(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc 
-g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind 
-Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 
-(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let 
-H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to 
-(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 
-a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) 
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) 
-H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc 
-g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A 
-(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u 
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind 
-Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) 
-H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to 
-(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) 
-(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
-A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda 
-(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: 
-A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda 
-(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T 
-(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T 
-(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead c0 (Bind Abst) u) t a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda 
-(u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda 
-(_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: 
-A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (t0: 
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind 
-Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead 
-a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) 
-(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t 
-a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u 
-t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A 
-(\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
-(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda 
-(c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 
-a1)).(\lambda (H2: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
-(\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: 
-A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda 
-(a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (a2: 
-A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u 
-t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind Abst) u t) 
-H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
-Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead 
-a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) 
-(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) 
-H2 (THead (Bind Abst) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: 
-T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) u t) H5) in (let H8 \def (H6 
-(refl_equal T (THead (Bind Abst) u t))) in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
-A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u 
-(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind 
-Abst) u) t a4)))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H9: (eq A a1 
-(AHead x0 x1))).(\lambda (H10: (arity g c0 u (asucc g x0))).(\lambda (H11: 
-(arity g (CHead c0 (Bind Abst) u) t x1)).(let H12 \def (eq_ind A a1 (\lambda 
-(a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in (let H13 \def (eq_ind A a1 
-(\lambda (a0: A).(arity g c0 (THead (Bind Abst) u t) a0)) H7 (AHead x0 x1) 
-H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) in (let H14 \def H_x in 
-(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g x0 a3))) (\lambda (_: 
-A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
-a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g 
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t 
-a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda (H15: (leq g x0 
-x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 (AHead x2 
-x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 x3) H17) in 
-(eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: A).(\lambda 
-(a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
-A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
-(CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead x2 x3)) 
-(arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 H15)) 
-(arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) 
-H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
-
-lemma arity_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: 
-A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity 
-g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: 
-A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead 
-(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A 
-(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 
-a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda 
-(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) 
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t 
-(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T 
-(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H1) in 
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
-g c0 t (AHead a1 (ASort O n))))) H2))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq 
-T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) 
-(\lambda (a1: A).(arity g d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef 
-i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in 
-(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
-g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda 
-(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g 
-d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda 
-(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef 
-i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u 
-t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: 
-A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (b: B).(\lambda (_: 
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
-A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: 
-A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a0)).(\lambda (_: (((eq T 
-t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 
-(Bind b) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) t (AHead 
-a3 a0))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Flat Appl) u 
-t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) u t) H6) in (False_ind (ex2 A 
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
-a0)))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead 
-(Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
-(a3: A).(arity g c0 t (AHead a3 (asucc g a1)))))))).(\lambda (t0: T).(\lambda 
-(a0: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (_: 
-(((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g 
-(CHead c0 (Bind Abst) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind 
-Abst) u0) t (AHead a3 a0))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
-(THead (Flat Appl) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) 
-H5) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
-A).(arity g c0 t (AHead a3 (AHead a1 a0))))) H6)))))))))))) (\lambda (c0: 
-C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 
-a1)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda 
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
-a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 
-(AHead a1 a0))).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A 
-(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
-(AHead a1 a0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead 
-(Flat Appl) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
-\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in 
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
-(THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in (\lambda (H8: (eq T 
-u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat 
-Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
-A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let H10 \def 
-(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t H7) in (let 
-H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to 
-(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t 
-(AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: 
-T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: A).(arity g c0 
-u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 H10))))))) 
-H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: 
-(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) 
-\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t 
-(AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 
-a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 
-a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Appl) u 
-t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) 
-H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
-(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t)) 
-\to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t 
-(AHead a3 a1))))))).(\lambda (a0: A).(\lambda (H3: (leq g a1 a0)).(\lambda 
-(H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 \def (f_equal T T (\lambda 
-(e: T).e) t0 (THead (Flat Appl) u t) H4) in (let H6 \def (eq_ind T t0 
-(\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: 
-A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1)))))) H2 
-(THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: 
-T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in (let H8 \def (H6 
-(refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A (\lambda (a3: A).(arity 
-g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1))) (ex2 A (\lambda 
-(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) 
-(\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda (H10: (arity g c0 t 
-(AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
-A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10 
-(AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c 
-y a2 H0))) H)))))).
-
-lemma arity_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
-A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) 
-(arity g c t a)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: 
-A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead 
-(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land 
-(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: 
-(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
-A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) 
-(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T 
-(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u t) H1) in 
-(False_ind (land (arity g c0 u (asucc g (ASort O n))) (arity g c0 t (ASort O 
-n))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) 
-u t)) \to (land (arity g d u (asucc g a0)) (arity g d t a0))))).(\lambda (H4: 
-(eq T (TLRef i) (THead (Flat Cast) u t))).(let H5 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u 
-t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) 
-H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead 
-(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t 
-(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u 
-t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Cast) u t) H4) in (False_ind (land (arity g c0 u 
-(asucc g a0)) (arity g c0 t a0)) H5))))))))))) (\lambda (b: B).(\lambda (_: 
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t 
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 
-(Bind b) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to 
-(land (arity g (CHead c0 (Bind b) u0) u (asucc g a2)) (arity g (CHead c0 
-(Bind b) u0) t a2))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead 
-(Flat Cast) u t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H6) in (False_ind 
-(land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) H7)))))))))))))) 
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
-u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land 
-(arity g c0 u (asucc g (asucc g a1))) (arity g c0 t (asucc g 
-a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 
-(Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to 
-(land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g (CHead c0 
-(Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
-(THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) 
-H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t 
-(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda 
-(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
-Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t 
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead 
-a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g 
-c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: 
-(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def 
-(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u t) H5) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g 
-c0 t a2)) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: 
-A).(\lambda (H1: (arity g c0 u0 (asucc g a0))).(\lambda (H2: (((eq T u0 
-(THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) 
-(arity g c0 t (asucc g a0)))))).(\lambda (t0: T).(\lambda (H3: (arity g c0 t0 
-a0)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 
-u (asucc g a0)) (arity g c0 t a0))))).(\lambda (H5: (eq T (THead (Flat Cast) 
-u0 t0) (THead (Flat Cast) u t))).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
-(THead _ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) 
-u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow 
-t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in (\lambda (H8: 
-(eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead 
-(Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)))) 
-H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a0)) 
-H3 t H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead 
-(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) (arity g c0 
-t (asucc g a0))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: 
-T).(arity g c0 t1 (asucc g a0))) H1 u H8) in (conj (arity g c0 u (asucc g 
-a0)) (arity g c0 t a0) H12 H10))))))) H6))))))))))) (\lambda (c0: C).(\lambda 
-(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: 
-(((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1)) 
-(arity g c0 t a1))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda 
-(H4: (eq T t0 (THead (Flat Cast) u t))).(let H5 \def (f_equal T T (\lambda 
-(e: T).e) t0 (THead (Flat Cast) u t) H4) in (let H6 \def (eq_ind T t0 
-(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u 
-(asucc g a1)) (arity g c0 t a1)))) H2 (THead (Flat Cast) u t) H5) in (let H7 
-\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) 
-u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in 
-(land_ind (arity g c0 u (asucc g a1)) (arity g c0 t a1) (land (arity g c0 u 
-(asucc g a2)) (arity g c0 t a2)) (\lambda (H9: (arity g c0 u (asucc g 
-a1))).(\lambda (H10: (arity g c0 t a1)).(conj (arity g c0 u (asucc g a2)) 
-(arity g c0 t a2) (arity_repl g c0 u (asucc g a1) H9 (asucc g a2) (asucc_repl 
-g a1 a2 H3)) (arity_repl g c0 t a1 H10 a2 H3)))) H8))))))))))))) c y a H0))) 
-H)))))).
-
-lemma arity_gen_appls:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall 
-(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: 
-A).(arity g c t a))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads 
-(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda 
-(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c 
-t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall 
-(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: 
-A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g 
-c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g 
-c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 
-a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: 
-(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x 
-a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A 
-(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) 
-(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: 
-A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
-
-lemma arity_gen_lift:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: 
-nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: 
-C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T 
-(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall 
-(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda 
-(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) 
-\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat 
-d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: 
-C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: 
-C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq 
-T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
-a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: 
-T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda 
-(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 
-(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) 
-(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: 
-(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u 
-(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) 
-(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def 
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq 
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) 
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef 
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: 
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda 
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: 
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) 
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S 
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) 
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: 
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 
-(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 
-a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u 
-(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in 
-(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h 
-(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h 
-(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) 
-(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef 
-(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda 
-(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda 
-(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) 
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 
-H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) 
-u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda 
-(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall 
-(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda 
-(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x 
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def 
-(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq 
-T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) 
-(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef 
-i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: 
-(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda 
-(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: 
-nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) 
-in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S 
-i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) 
-(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: 
-(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 
-(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 
-\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
-t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 
-(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def 
-(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus 
-x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 
-(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt 
-Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: 
-(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x 
-h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le 
-(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T 
-(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 
-u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 
-H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: 
-(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall 
-(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to 
-(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: 
-(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: 
-nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h 
-x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: 
-nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x 
-x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda 
-(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: 
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u 
-(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T 
-(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def 
-(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to 
-(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T 
-t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) 
-x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c 
-(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def 
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift 
-h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind 
-b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 
-\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
-t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 
-a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: 
-T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 
-(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal 
-T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b 
-x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) 
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u 
-(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u 
-(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
-(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g 
-(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall 
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c 
-(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda 
-(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda 
-(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: 
-T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: 
-T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S 
-x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 
-t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: 
-nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h 
-x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) 
-x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c 
-(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u 
-(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 
-(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: 
-nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall 
-(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) 
-H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall 
-(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: 
-C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) 
-H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g 
-a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T 
-(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) 
-(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 
-H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda 
-(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall 
-(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: 
-((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall 
-(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda 
-(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift 
-h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T 
-(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) 
-(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 
-x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x 
-x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 
-a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall 
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
-(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def 
-(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) 
-H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall 
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
-(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u 
-(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 
-x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 
-(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 
-x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: 
-A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: 
-nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x 
-c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: 
-(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T 
-t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
-a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead 
-(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c 
-c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
-Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) 
-(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) 
-x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h 
-x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 
-a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall 
-(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
-(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 
-(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def 
-(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 
-(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 
-(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u 
-(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in 
-(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 
-x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast 
-u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall 
-(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to 
-(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 
-a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x 
-x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 
-(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
-
-theorem arity_mono:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c 
-t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: 
-(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: 
-A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: 
-C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) 
-a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u 
-a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a 
-a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 
-\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda 
-(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
-Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) 
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def 
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead 
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind 
-Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind 
-Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 
-(CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
-(CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d 
-x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind 
-Abbr) t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity 
-g x0 t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl 
-i c0 (CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 
-(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) 
-H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
-(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 
-(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
-(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
-(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g 
-a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i 
-c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i 
-H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind 
-Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
-[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 
-(CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind 
-(leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u (asucc g a))).(\lambda 
-(H2: ((\forall (a2: A).((arity g d u a2) \to (leq g (asucc g a) 
-a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 
-\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
-C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda 
-(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
-Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
-a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
-(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) 
-(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 
-(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def 
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead 
-x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind 
-Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (leq g a a2) 
-H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
-T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: 
-T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
-(u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda 
-(u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: 
-(arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C (CHead d (Bind Abst) u) 
-(\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 
-(CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
-c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) 
-(getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in 
-((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u) 
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead 
-x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u 
-H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g 
-a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 
-(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 
-(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 
-(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: 
-T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall 
-(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda 
-(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: 
-((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 
-a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) 
-a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A 
-(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind 
-b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u 
-x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) 
-H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda 
-(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 
-u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: 
-A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: 
-((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 
-a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) 
-a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda 
-(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda 
-(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity 
-g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: 
-(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) 
-t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) 
-(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) 
-a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: 
-A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g 
-c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: 
-(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 
-t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity 
-g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 
-H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity 
-g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g 
-c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x 
-a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: 
-T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: 
-((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda 
-(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: 
-A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: 
-(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u 
-t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g 
-a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 
-a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda 
-(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: 
-A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: 
-(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans 
-g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma
deleted file mode 100644 (file)
index 7267b78..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/lift1.ma
+++ /dev/null
@@ -1,41 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/props.ma".
-
-include "basic_1/drop1/fwd.ma".
-
-lemma arity_lift1:
- \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: 
-PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t 
-a) \to (arity g c1 (lift1 hds t) a))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: 
-PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: 
-T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) 
-(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda 
-(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C 
-c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: 
-C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 
-(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 
-(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def 
-(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 
-(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 
-x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) 
-c1 n n0 H3)))) H2))))))))))) hds)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma
deleted file mode 100644 (file)
index 69dfef6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/pr3.ma
+++ /dev/null
@@ -1,579 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/arity.ma".
-
-include "basic_1/pr3/fwd.ma".
-
-include "basic_1/pr1/fwd.ma".
-
-include "basic_1/wcpr0/getl.ma".
-
-include "basic_1/pr0/props.ma".
-
-include "basic_1/arity/subst0.ma".
-
-lemma arity_sred_wcpr0_pr0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
-c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 
-t2) \to (arity g c2 t2 a)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
-(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
-(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: 
-C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) 
-t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
-(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
-(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda 
-(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to 
-(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
-c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) 
-(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
-(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 
-(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 
-(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u 
-x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 
-H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g 
-a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: 
-T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: 
-C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) 
-t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
-T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 
-d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 
-H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i 
-H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
-(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
-a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 
-u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda 
-(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: 
-C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
-(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c 
-c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq 
-T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g 
-c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: 
-T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) 
-(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def 
-(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T 
-(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 
-c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp 
-c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 
-(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) 
-u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k 
-u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead 
-k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind b) u t) H12) in ((let 
-H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
-| (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
-(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead (Bind b) u t) H12) in 
-(\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K k (Bind b))).(eq_ind_r K 
-(Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H18 \def 
-(eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 
-t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 
-t4)) H10 t H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 
-(THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 u H16) in (let H21 \def 
-(eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) in (arity_bind g b H0 c2 
-u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H5 
-u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) H13)))))))))))) (\lambda (u0: 
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
-(((eq T v1 (THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
-(Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (H12: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Bind b) u t))).(let H13 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind b) u t) H12) in (False_ind (arity g 
-c2 (THead (Bind Abbr) v2 t4) a2) H13)))))))))))) (\lambda (b0: B).(\lambda 
-(_: (not (eq B b0 Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g c2 v2 
-a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (((eq T u1 (THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
-(Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (H15: (eq T (THead (Flat 
-Appl) v1 (THead (Bind b0) u1 t3)) (THead (Bind b) u t))).(let H16 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind b) u t) H15) in (False_ind (arity g 
-c2 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) a2) 
-H16))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 
-u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u t)) \to (arity g c2 u2 
-a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda 
-(H11: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (w: 
-T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda (H13: (eq T (THead (Bind Abbr) 
-u1 t3) (THead (Bind b) u t))).(let H14 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | 
-(THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in 
-((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
-_) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) u1 
-t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T u1 u)).(\lambda (H18: 
-(eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead 
-(Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in (let H20 \def (eq_ind T 
-t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let H21 \def (eq_ind T u1 
-(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 
-u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H17) 
-in (let H23 \def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u 
-t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in (let H24 \def (eq_ind_r B b 
-(\lambda (b0: B).((eq T u (THead (Bind b0) u t)) \to (arity g c2 u2 a2))) H21 
-Abbr H18) in (let H25 \def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: 
-C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t5: T).((pr0 t t5) \to 
-(arity g c3 t5 a2)))))) H4 Abbr H18) in (let H26 \def (eq_ind_r B b (\lambda 
-(b0: B).(arity g (CHead c (Bind b0) u) t a2)) H3 Abbr H18) in (let H27 \def 
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H0 Abbr H18) in 
-(arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w a2 (arity_subst0 g (CHead 
-c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H5 u 
-u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr c2 u2) w 
-H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda (H8: (not 
-(eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 
-t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
-a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S O) 
-O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | 
-(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u 
-t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow 
-t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in 
-((let H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) 
-\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) 
-u t) H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 
-\def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let 
-H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to 
-(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T 
-t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to 
-(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O 
-t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c 
-(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 
-(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u 
-(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 
-(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: 
-(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: 
-T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let 
-H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) 
-H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g 
-c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
-(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: 
-C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
-(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
-c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) 
-t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) 
-(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: 
-(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind 
-Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda 
-(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda 
-(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u 
-t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 
-c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u 
-u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
-(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
-(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: 
-K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 
-\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | 
-(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
-(THead (Bind Abst) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
-(THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) 
-in ((let H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) 
-(THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda (H15: (eq T u1 
-u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) (\lambda 
-(k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 \def (eq_ind T 
-t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 t4 
-(AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: 
-T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq 
-T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2)))) H8 u H15) 
-in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H7 u H15) in 
-(arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead c2 (Bind Abst) u2) 
-(wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k H16)))) H13)) 
-H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity 
-g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
-t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
-(AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead 
-(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda 
-(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
-v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 
-(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
-u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 
-(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
-t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
-(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) 
-u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) 
-(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: 
-(((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 
-a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
-(_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 
-a2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
-(THead (Bind Abbr) u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T 
-(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
-Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g c2 (THead 
-(Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: B).(\lambda 
-(H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
-t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
-(AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0 
-(lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 \def (f_equal T B 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O 
-t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead 
-(Bind Abst) u t) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) 
-| (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead 
-(Bind Abst) u t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b 
-Abst)).(let H16 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
-Abst H15) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead 
-(Bind Abst) u t0)) \to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) 
-H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 
-(CHead c (Bind Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 
-t5 a2)))))) H3 (lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda 
-(t0: T).(arity g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in 
-(let H20 \def (match (H16 (refl_equal B Abst)) in False with []) in 
-H20)))))))) H12)) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity 
-g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat 
-Cast) u0 t3) (THead (Bind Abst) u t))).(let H10 \def (eq_ind T (THead (Flat 
-Cast) u0 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u 
-t) H9) in (False_ind (arity g c2 t4 (AHead a1 a2)) H10)))))))) y t2 H6))) 
-H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c u a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to 
-(\forall (t2: T).((pr0 u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: 
-T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: 
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g 
-c2 t2 (AHead a1 a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
-c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) 
-t2)).(insert_eq T (THead (Flat Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) 
-(\lambda (_: T).(arity g c2 t2 a2)) (\lambda (y: T).(\lambda (H6: (pr0 y 
-t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Flat Appl) 
-u t)) \to (arity g c2 t3 a2)))) (\lambda (t0: T).(\lambda (H7: (eq T t0 
-(THead (Flat Appl) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 
-(THead (Flat Appl) u t) H7) in (eq_ind_r T (THead (Flat Appl) u t) (\lambda 
-(t3: T).(arity g c2 t3 a2)) (arity_appl g c2 u a1 (H1 c2 H4 u (pr0_refl u)) t 
-a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 (THead (Flat Appl) u 
-t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: 
-(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g 
-c2 t4 a2)))).(\lambda (k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead 
-(Flat Appl) u t))).(let H12 \def (f_equal T K (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
-(THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead (Flat Appl) u t) H11) 
-in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r 
-K (Flat Appl) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 
-\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to 
-(arity g c2 t4 a2))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: 
-T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq 
-T t0 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 
-\def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 
-u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) 
-H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(H7: (pr0 v1 v2)).(\lambda (H8: (((eq T v1 (THead (Flat Appl) u t)) \to 
-(arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 
-t4)).(\lambda (H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
-a2)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
-(THead (Flat Appl) u t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead 
-(Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow 
-(THead (Bind Abst) u0 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u t) H11) in (\lambda 
-(H14: (eq T v1 u)).(let H15 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 
-(THead (Flat Appl) u t)) \to (arity g c2 v2 a2))) H8 u H14) in (let H16 \def 
-(eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2)) H7 u H14) in (let H17 \def 
-(eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity 
-g c2 t4 a2))) H10 (THead (Bind Abst) u0 t3) H13) in (let H18 \def (eq_ind_r T 
-t (\lambda (t0: T).((eq T u (THead (Flat Appl) u t0)) \to (arity g c2 v2 
-a2))) H15 (THead (Bind Abst) u0 t3) H13) in (let H19 \def (eq_ind_r T t 
-(\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall (t5: T).((pr0 t0 
-t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3 (THead (Bind Abst) u0 t3) H13) 
-in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) 
-H2 (THead (Bind Abst) u0 t3) H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let 
-H22 \def (H19 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) 
-t3 t4 H9 (Bind Abst))) in (let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 
-a2) H22) in (ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 
-a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g 
-a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 
-a4))) (arity g c2 (THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda 
-(x1: A).(\lambda (H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: 
-(arity g c2 u0 (asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) 
-u0) t4 x1)).(let H27 \def (f_equal A A (\lambda (e: A).(match e with [(ASort 
-_ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 a2) (AHead x0 
-x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e with [(ASort 
-_ _) \Rightarrow a2 | (AHead _ a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 
-x1) H24) in (\lambda (H29: (eq A a1 x0)).(let H30 \def (eq_ind_r A x1 
-(\lambda (a0: A).(arity g (CHead c2 (Bind Abst) u0) t4 a0)) H26 a2 H28) in 
-(let H31 \def (eq_ind_r A x0 (\lambda (a0: A).(arity g c2 u0 (asucc g a0))) 
-H25 a1 H29) in (arity_bind g Abbr not_abbr_abst c2 v2 a1 H21 t4 a2 
-(csuba_arity g (CHead c2 (Bind Abst) u0) t4 a2 H30 (CHead c2 (Bind Abbr) v2) 
-(csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 H31 v2 H21))))))) H27))))))) 
-H23)))))))))))) H12)))))))))))) (\lambda (b: B).(\lambda (H7: (not (eq B b 
-Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H8: (pr0 v1 v2)).(\lambda 
-(H9: (((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (((eq T 
-u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H12: (pr0 t3 t4)).(\lambda (H13: (((eq T t3 
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H14: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) u t))).(let 
-H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 
-| (TLRef _) \Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat 
-Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
-(Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ 
-t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) u t) H14) in (\lambda (H17: (eq T v1 u)).(let H18 \def (eq_ind T 
-v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 v2 
-a2))) H9 u H17) in (let H19 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2)) 
-H8 u H17) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead 
-(Flat Appl) u t0)) \to (arity g c2 t4 a2))) H13 (THead (Bind b) u1 t3) H16) 
-in (let H21 \def (eq_ind_r T t (\lambda (t0: T).((eq T u1 (THead (Flat Appl) 
-u t0)) \to (arity g c2 u2 a2))) H11 (THead (Bind b) u1 t3) H16) in (let H22 
-\def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead (Flat Appl) u t0)) \to 
-(arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) H16) in (let H23 \def 
-(eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall 
-(t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3 (THead (Bind 
-b) u1 t3) H16) in (let H24 \def (eq_ind_r T t (\lambda (t0: T).(arity g c t0 
-(AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) in (let H25 \def (H1 c2 H4 v2 
-H19) in (let H26 \def (H23 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H10 
-t3 t4 H12 (Bind b))) in (let H27 \def (arity_gen_bind b H7 g c2 u2 t4 (AHead 
-a1 a2) H26) in (ex2_ind A (\lambda (a3: A).(arity g c2 u2 a3)) (\lambda (_: 
-A).(arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))) (arity g c2 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) a2) (\lambda (x: 
-A).(\lambda (H28: (arity g c2 u2 x)).(\lambda (H29: (arity g (CHead c2 (Bind 
-b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 c2 u2 x H28 (THead (Flat Appl) 
-(lift (S O) O v2) t4) a2 (arity_appl g (CHead c2 (Bind b) u2) (lift (S O) O 
-v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 (Bind b) u2) (S O) O (drop_drop 
-(Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 H29))))) H27))))))))))))) 
-H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
-u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 
-a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
-(_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda 
-(w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind 
-Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T (THead (Bind 
-Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u 
-t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) 
-H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
-(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda 
-(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u 
-t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | 
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind 
-(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity 
-g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) 
-(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
-False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H9) in 
-(False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) 
-(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u 
-(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall 
-(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: 
-T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c 
-c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda 
-(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 
-(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda 
-(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: 
-T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq 
-T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: 
-T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T 
-(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat 
-Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 
-H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
-(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat 
-Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T 
-(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
-(THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat Cast) u t) H11) 
-in ((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) 
-(THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead 
-(Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k 
-(Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 (THead k0 
-u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead 
-(Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def 
-(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind 
-T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 
-a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) 
-H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 
-H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead 
-(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
-\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u t) H11) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a0) 
-H12)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
-(THead (Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) 
-u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity 
-g c2 t4 a0)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) (THead (Flat Cast) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
-Cast) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead 
-(Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
-\to (arity g c2 t4 a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 
-w)).(\lambda (H12: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Cast) u 
-t))).(let H13 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Cast) u t) H12) in (False_ind (arity 
-g c2 (THead (Bind Abbr) u2 w) a0) H13))))))))))))) (\lambda (b: B).(\lambda 
-(_: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
-t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 
-a0)))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0 (lift (S O) O 
-t3)) (THead (Flat Cast) u t))).(let H11 \def (eq_ind T (THead (Bind b) u0 
-(lift (S O) O t3)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Cast) u t) H10) in (False_ind (arity g c2 t4 a0) H11)))))))))) (\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 
-(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda 
-(H9: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Cast) u t))).(let H10 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
-_) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Cast) u0 
-t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Flat Cast) u0 t3) (THead (Flat Cast) 
-u t) H9) in (\lambda (_: (eq T u0 u)).(let H13 \def (eq_ind T t3 (\lambda 
-(t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) 
-in (let H14 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 
-c2 H4 t4 H14))))) H10)))))))) y t2 H6))) H5))))))))))))) (\lambda (c: 
-C).(\lambda (t: T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda 
-(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
-(arity g c2 t2 a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
-a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda 
-(H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 H3 t2 H4) a2 H2)))))))))))) c1 
-t1 a H))))).
-
-lemma arity_sred_wcpr0_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
-(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 
-c2) \to (arity g c2 t2 a)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: 
-A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 
-a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: 
-A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 
-c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda 
-(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: 
-A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 
-a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: 
-(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a 
-(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl 
-c2)))))))))))))) t1 t2 H))).
-
-lemma arity_sred_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
-G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda 
-(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda 
-(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 
-g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
-G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a 
-(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
-H2)))))))))))))) c t1 t2 H)))).
-
-lemma arity_sred_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
-(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: 
-A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: 
-G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c 
-t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: 
-(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 
-H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma
deleted file mode 100644 (file)
index fb8379a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/props.ma
+++ /dev/null
@@ -1,266 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/fwd.ma".
-
-lemma node_inh:
- \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: 
-C).(\lambda (t: T).(arity g c t (ASort k n)))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: 
-nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) 
-(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) 
-(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda 
-(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 
-n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: 
-T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: 
-T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: 
-C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) 
-(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 
-x1) (ASort (S n0) n) H1))))) H0)))) k))).
-
-lemma arity_lift:
- \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 
-t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 
-c2) \to (arity g c1 (lift h d t) a)))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
-A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to 
-(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: 
-nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop 
-h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O 
-n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) 
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: 
-(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall 
-(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda 
-(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 
-c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i 
-d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def 
-(drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S d0) (le_S (S (S i)) (S 
-d0) (le_n_S (S i) d0 H4)))) c1 c h H3 (CHead d (Bind Abbr) u) H0) in 
-(ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda 
-(e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: 
-C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity g c1 (TLRef 
-i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O c1 
-x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
-(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
-nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C 
-(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) 
-u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) 
-a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h 
-(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x 
-d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead 
-x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S 
-i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
-H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: 
-T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i 
-c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) 
-(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
-Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g 
-a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: 
-nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g 
-a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda 
-(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) 
-(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 
-t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S 
-d0) (le_S (S (S i)) (S d0) (le_n_S (S i) d0 H4)))) c1 c h H3 (CHead d (Bind 
-Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 
-e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) 
-(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) u)))) (arity 
-g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O 
-c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
-(CHead d (Bind Abst) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
-nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abst d u H8) in (ex2_ind C 
-(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abst) (lift h (minus d0 (S i)) 
-u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) 
-a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift h 
-(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x 
-d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead 
-x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S 
-i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
-H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: 
-T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) (drop_getl_trans_ge i 
-c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) 
-(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: B).(\lambda (H0: (not 
-(eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: 
-(arity g c u a1)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall 
-(d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) a1))))))).(\lambda 
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 
-a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 (lift h d t0) 
-a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H5: 
-(drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) 
-d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind g b H0 c1 (lift h d 
-u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 (CHead c1 (Bind b) 
-(lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 b u))) (lift h d 
-(THead (Bind b) u t0)) (lift_head (Bind b) u t0 h d))))))))))))))))) (\lambda 
-(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g 
-a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g 
-a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
-(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 
-(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d 
-u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 
-a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) 
-d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) 
-(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) 
-(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
-(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 
-(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity 
-g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead 
-a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s 
-(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 
-(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s 
-(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat 
-Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: 
-A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: 
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift 
-h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 
-a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: 
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 
-c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) 
-t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 
-(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) 
-(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h 
-d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) 
-a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: 
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 
-c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a 
-H))))).
-
-lemma arity_repellent:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: 
-A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c 
-(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: 
-Prop).P)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: 
-A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: 
-A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g 
-a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) 
-w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind 
-A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
-A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: 
-A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: 
-A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g 
-x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def 
-(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y 
-(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind 
-Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))).
-
-theorem arity_appls_cast:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: 
-TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to 
-((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) 
-vs (THead (Flat Cast) u t)) a))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads 
-(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) 
-\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) 
-(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity 
-g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) 
-(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c 
-(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: 
-A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) 
-(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat 
-Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 
-t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: 
-A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda 
-(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat 
-Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat 
-Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) 
-(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g 
-a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat 
-Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda 
-(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g 
-a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) 
-a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g 
-a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 
-(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc 
-g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g 
-(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
-
-lemma arity_appls_abbr:
- \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall 
-(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c 
-(THeads (Flat Appl) vs (TLRef i)) a)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: 
-TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads 
-(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t 
-(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) 
-a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop 
-Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: 
-((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to 
-(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda 
-(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
-v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S 
-i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: 
-A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity 
-g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: 
-A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat 
-Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat 
-Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
-
-theorem arity_appls_bind:
- \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
-C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: 
-T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) 
-(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat 
-Appl) vs (THead (Bind b) v t)) a2)))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda 
-(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v 
-a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
-TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind 
-b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) 
-t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads 
-(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 
-(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead 
-c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts 
-(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) 
-(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in 
-(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) 
-a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda 
-(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity 
-g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x 
-a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) 
-O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 
-(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma
deleted file mode 100644 (file)
index 334505b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/arity/subst0.ma
+++ /dev/null
@@ -1,1115 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/props.ma".
-
-include "basic_1/fsubst0/fwd.ma".
-
-include "basic_1/csubst0/getl.ma".
-
-include "basic_1/subst0/dec.ma".
-
-include "basic_1/subst0/fwd.ma".
-
-include "basic_1/getl/getl.ma".
-
-lemma arity_gen_cvoid_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to 
-(\forall (P: Prop).P))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
-A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d 
-(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to 
-(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d 
-(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w 
-(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: 
-(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall 
-(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall 
-(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: 
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 
-(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w 
-(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) 
-O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O 
-w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 
-(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) 
-(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 
-(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def 
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind 
-Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) 
-u0) H7)) in (False_ind P H9)))))) (subst0_gen_lref w v i0 i 
-H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: 
-((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d0 
-(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) \to 
-(\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda 
-(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: 
-T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: 
-Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq 
-nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat 
-i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let 
-H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 
-(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead 
-d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d 
-(Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) 
-(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (b: B).(\lambda (_: 
-(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: ((\forall (d: C).(\forall 
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
-(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: 
-Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g 
-(CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: C).(\forall (u0: 
-T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d (Bind Void) u0)) 
-\to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
-(H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
-T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) v)).(\lambda (P: 
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) 
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
-(Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
-(t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T (\lambda 
-(u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i w u 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda 
-(u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead 
-(Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 P)))) 
-H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u t2))) 
-(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda (t2: 
-T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w 
-t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u 
-x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) 
-(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d 
-(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u 
-x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 
-P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) 
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
-(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: 
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 
-a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl 
-i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: 
-T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
-T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: 
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) 
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
-(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: 
-(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: 
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind 
-Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda 
-(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d 
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v 
-(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) 
-(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: 
-T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s 
-(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind 
-Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 
-P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v 
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda 
-(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda 
-(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 
-d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i 
-H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall 
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
-(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: 
-Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 
-t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall 
-(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall 
-(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind 
-Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead 
-(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq 
-T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T 
-(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 
-(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w 
-u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 
-t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 
-t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T 
-v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda 
-(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 
-i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: 
-T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) 
-i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) 
-(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: 
-T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s 
-(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
-(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w 
-u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 
-P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) 
-(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u 
-(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: 
-T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: 
-T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall 
-(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
-(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
-Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
-(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
-T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: 
-Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) 
-(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
-(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: 
-(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: 
-T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat 
-Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda 
-(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d 
-u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v 
-(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) 
-(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: 
-T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s 
-(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
-(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w 
-u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 
-P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) 
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
-t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: 
-T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: 
-A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: 
-T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d 
-u i H3 w v H4 P)))))))))))))))) c t a H))))).
-
-lemma arity_gen_cvoid:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
-a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
-(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in 
-(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i 
-v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i 
-v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T 
-t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift 
-(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: 
-(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u 
-(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) 
-(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda 
-(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i 
-x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) 
-(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x 
-(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
-
-lemma arity_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
-c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u 
-c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
-(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead 
-d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 
-t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: 
-nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i 
-c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: 
-(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort 
-n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u 
-(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i 
-u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: 
-(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i 
-u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c 
-c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: 
-C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 
-(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 
-i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 
-(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c 
-c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
-(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) 
-t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c 
-c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) 
-t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g 
-c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
-Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: 
-((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 
-t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
-T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) 
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef 
-i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in 
-(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) 
-(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) 
-t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) 
-(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) 
-t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 
-(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq 
-nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat 
-i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O 
-u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda 
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def 
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind 
-Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
-Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 
-(CHead d1 (Bind Abbr) u0) H11)) in ((let H14 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
-Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d 
-d1)).(let H16 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind 
-Abbr) t))) H12 u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O 
-t) a0)) (let H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 
-(Bind Abbr) u))) H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop 
-Abbr c d u i H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i 
-H8)) c2 H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c 
-c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) 
-(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c 
-c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 
-(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def 
-(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abbr) u) H0) in (or4_ind 
-(getl i c2 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) 
-(\lambda (H11: (getl i c2 (CHead d (Bind Abbr) u))).(let H12 \def (eq_ind nat 
-(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 
-(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d 
-(Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) 
-(le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in 
-(arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda 
-(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind 
-x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def 
-(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) 
-(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 
-(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S 
-i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i 
-H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in 
-(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def 
-(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) 
-in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind 
-x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i 
-c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 
-(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead 
-d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) 
-(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda 
-(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda 
-(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq 
-C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 
-(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 
-x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
-d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
-Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
-(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
-(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
-(Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let 
-H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) 
-H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus 
-i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u 
-i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d 
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind 
-Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) 
-(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind 
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: 
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) 
-(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) 
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 
-(le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0 
-H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal 
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in 
-((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
-Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr 
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C 
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let 
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) 
-H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) 
-(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u 
-(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: 
-(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead 
-d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) 
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) 
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 
-(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
-T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: 
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda 
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def 
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind 
-Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
-Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 
-(CHead d1 (Bind Abbr) u0) H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
-Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d 
-d1)).(let H17 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind 
-Abbr) t))) H13 u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: 
-T).(csubst0 i t c c2)) H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 
-(lift (S i) O t) a0)) (let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c 
-(CHead c0 (Bind Abbr) u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O 
-(getl_drop Abbr c2 d u i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d 
-(Bind Abbr) u) H19)))) u0 H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 
-i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
-Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g 
-a0))).(\lambda (H2: ((\forall (d1: C).(\forall (u0: T).(\forall (i0: 
-nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall 
-(t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g c2 t2 (asucc g 
-a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda 
-(H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: 
-T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x \def 
-(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in (or3_ind 
-(land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) 
-(csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) 
-(arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) 
-t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) 
-(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind 
-C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 
-(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
-T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n 
-c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d 
-(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) 
-(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in 
-(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
-I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
-d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c (lift (S i) O u0) a0) 
-H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda 
-(H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (eq T (TLRef 
-i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) 
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: 
-T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: 
-(lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind 
-Abst) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) 
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
-(arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d (Bind Abst) 
-u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
-(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
-(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: 
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda 
-(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C 
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 
-(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 
-x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
-(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
-(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
-(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
-x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let 
-H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) 
-H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead 
-c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d 
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind 
-Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) 
-(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda 
-(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq 
-C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 
-(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 
-x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
-d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
-Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
-(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
-(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
-(Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
-x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let 
-H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) 
-H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus 
-i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u 
-i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d 
-(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind 
-Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) 
-(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind 
-Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind 
-x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: 
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) 
-(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) 
-(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 
-(le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0 
-H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal 
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in 
-((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead 
-x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
-Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst 
-x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: 
-T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C 
-x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let 
-H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) 
-H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) 
-(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u 
-(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 
-d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: 
-(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead 
-d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) 
-(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) 
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 
-(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
-(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
-T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: 
-nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda 
-(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def 
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
-d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind 
-Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d 
-(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in (False_ind (arity g c2 
-(lift (S i) O u0) a0) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) 
-H6)) H5)))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
-Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity 
-g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: 
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t 
-a2)).(\lambda (H4: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) u0)) \to (\forall 
-(c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) u) t c2 t2) \to 
-(arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i: 
-nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: 
-C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead (Bind b) u t) c2 
-t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u t) t2 u0 i H6) in 
-(let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 (THead (Bind b) u 
-t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (land 
-(subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 a2) 
-(\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind b) u t) 
-t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) (arity g c2 
-t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 (THead (Bind b) 
-u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
-u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
-(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
-u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: 
-T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 
-u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) 
-(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t 
-a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b 
-c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S 
-i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u 
-x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 
-(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
-(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: 
-T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s 
-(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity 
-g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) 
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
-(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c 
-(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda 
-(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t 
-x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) 
-(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) 
-x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind 
-b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both 
-(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) 
-(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head 
-(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead 
-(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) 
-t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind 
-b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u 
-t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 
-i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) 
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
-(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c 
-(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) 
-t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) 
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) 
-(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead 
-(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
-u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
-(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 
-T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: 
-T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 
-u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) 
-(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 
-c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u 
-(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t 
-(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) 
-(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda 
-(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: 
-T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
-(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) 
-(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u 
-x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind 
-b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 
-d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) 
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
-(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c 
-(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 
-u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) 
-i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) 
-(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 
-(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: 
-(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda 
-(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 
-(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind 
-b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) 
-(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 
-H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 
-H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) 
-H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: 
-C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) 
-\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g 
-c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: 
-(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: 
-C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead 
-d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 
-(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda 
-(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 
-(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i 
-u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 
-(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq 
-C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind 
-Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u 
-t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land 
-(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2) 
-(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) 
-(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) 
-t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind 
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: 
-T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u 
-t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 
-(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind 
-Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
-T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) 
-(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
-(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind 
-Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x 
-a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) 
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) 
-u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i 
-u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq 
-T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 
-t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 
-a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u 
-x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead 
-(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g 
-c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) 
-c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind 
-Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 
-H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity 
-g c t2 (AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T 
-t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda 
-(H13: (subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 
-x1) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 
-d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) 
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S 
-i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0) 
-(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10)) 
-(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7: 
-(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T 
-(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) 
-(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0 
-c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0 
-(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c 
-u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) 
-c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind 
-Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind 
-Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda 
-(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c 
-c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) 
-(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u 
-t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: 
-T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: 
-T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind 
-Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) 
-(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 (AHead a1 a2)) (\lambda 
-(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12: 
-(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0: 
-T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x 
-(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind 
-Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) 
-u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind 
-Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2 
-H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 
-(THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 
-a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u 
-x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead 
-(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head 
-g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S 
-i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S 
-i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u) 
-(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c2 t2 
-(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 
-(THead (Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: 
-(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) 
-(\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x0 a1 (H1 d1 
-u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) 
-(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
-(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S 
-i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0) 
-(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) 
-(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) 
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u 
-a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: 
-T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: 
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 
-t2) \to (arity g c2 t2 (AHead a1 a2))))))))))).(\lambda (d1: C).(\lambda (u0: 
-T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
-(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
-Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
-(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) 
-t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) 
-(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) 
-(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 
-(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c 
-c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c 
-(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T 
-t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T 
-(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 
-(s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t 
-t3)))) (arity g c t2 a2) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 
-(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
-(Flat Appl) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat 
-Appl) x t) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 
-i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda 
-(H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda 
-(t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq 
-T t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 
-t t3)) (arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat 
-Appl) u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T 
-(THead (Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u 
-a1 H0 x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity 
-g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead 
-(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: 
-(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
-(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 
-(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c 
-t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) 
-c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0 
-i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2) 
-(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda 
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0: 
-T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst 
-i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2) 
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2) 
-(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead 
-(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10: 
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: 
-T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat 
-Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) 
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda 
-(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0: 
-T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x 
-(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i 
-u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T 
-t2 (THead (Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2) 
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda 
-(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) 
-(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 
-u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c 
-t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat 
-Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: 
-T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 
-(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 
-(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head 
-(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g 
-a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g 
-a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: 
-((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 
-(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 
-t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
-T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
-(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
-Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
-(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) 
-t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) 
-(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) 
-(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 
-(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c 
-c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c 
-(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T 
-t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T 
-(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 
-(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
-t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 
-(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
-(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat 
-Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 
-i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
-(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
-t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat 
-Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T 
-(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u 
-a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity 
-g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead 
-(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: 
-(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) 
-(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 
-(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t 
-x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 
-H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i 
-u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) 
-(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda 
-(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: 
-T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst 
-i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 
-H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) 
-(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) 
-(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead 
-(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 
-i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: 
-(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: 
-T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat 
-Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) 
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda 
-(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: 
-T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x 
-(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 
-c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 
-(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) 
-(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda 
-(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) 
-(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 
-u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t 
-x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 
-x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat 
-Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: 
-T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 
-(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i 
-u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t 
-t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall 
-(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) 
-u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity 
-g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda 
-(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 
-(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u 
-c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def 
-H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) 
-(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 
-t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c 
-c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda 
-(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) 
-(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) 
-c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind 
-(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t 
-t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g 
-c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 
-H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u 
-c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) 
-(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c 
-c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 
-c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
-
-lemma arity_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c 
-t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead 
-d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 
-a)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: 
-(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u 
-c t1 t2 H1)))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma
deleted file mode 100644 (file)
index 520f8e3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/A/defs.ma".
-
-include "basic_1/G/defs.ma".
-
-rec definition asucc (g: G) (l: A) on l: A \def match l with [(ASort n0 n) 
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) 
-\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g 
-a2))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma
deleted file mode 100644 (file)
index e2edc78..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma
+++ /dev/null
@@ -1,92 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/asucc/defs.ma".
-
-include "basic_1/A/fwd.ma".
-
-lemma asucc_gen_sort:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A 
-(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: 
-nat).(eq A a (ASort h0 n0)))))))))
-\def
- \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind 
-(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda 
-(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: 
-nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 
-n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 
-with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 
-n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A 
-(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) 
-(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat 
-nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 
-n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) 
-\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 
-n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 
-\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (asucc g (AHead a0 a1)) 
-H1) in (False_ind (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A 
-(AHead a0 a1) (ASort h0 n0))))) H2))))))) a)))).
-
-lemma asucc_gen_head:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A 
-(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind 
-(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: 
-A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc 
-g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g 
-(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead 
-a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda 
-(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort O (next g n0)) H0) in (False_ind (ex2 A (\lambda 
-(a0: A).(eq A (ASort O n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g 
-a0)))) H1))) (\lambda (n1: nat).(\lambda (_: (((eq A (AHead a1 a2) (asucc g 
-(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 
-a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H0: (eq A (AHead 
-a1 a2) (asucc g (ASort (S n1) n0)))).(let H1 \def (eq_ind A (AHead a1 a2) 
-(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: 
-A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g 
-a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) 
-(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda 
-(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A 
-(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 
-a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead 
-a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: 
-A).(match e with [(ASort _ _) \Rightarrow a1 | (AHead a4 _) \Rightarrow a4])) 
-(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in ((let H3 \def (f_equal A A 
-(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) 
-\Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: 
-(eq A a1 a0)).(let H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 
-a2) (asucc g a4)) \to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) 
-(\lambda (a5: A).(eq A a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda 
-(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda 
-(a5: A).(eq A a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: 
-A).((eq A (AHead a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 
-(AHead a1 a5))) (\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) 
-H3) in (let H7 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc 
-g a1)) \to (ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: 
-A).(eq A a4 (asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) 
-(\lambda (a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) 
-(\lambda (a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq 
-A (AHead a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g 
-a4))) a3 (refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) 
-a0 H4)))) H2))))))) a)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma
deleted file mode 100644 (file)
index 5d53fcf..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/cimp/defs.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/defs.ma".
-
-definition cimp:
- C \to (C \to Prop)
-\def
- \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall 
-(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C 
-(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma
deleted file mode 100644 (file)
index d26ec13..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/cimp/props.ma
+++ /dev/null
@@ -1,125 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/cimp/defs.ma".
-
-include "basic_1/getl/getl.ma".
-
-lemma cimp_flat_sx:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) 
-c)))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: 
-C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) 
-v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat 
-f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 
-(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind 
-b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 
-(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c 
-(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) 
-w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) 
-(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind 
-b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind 
-b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) 
-d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
-
-lemma cimp_flat_dx:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) 
-v))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: 
-C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind 
-b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 
-(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
-
-lemma cimp_bind:
- \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
-(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: 
-C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to 
-(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda 
-(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: 
-T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 
-(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) 
-(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) 
-v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) 
-(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 
-(Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) 
-w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let 
-H3 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 
-| (CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat 
-_) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) 
-(clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) 
-v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) 
-(CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 
-(Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) 
-in (\lambda (H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v 
-(\lambda (t: T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead 
-d2 (Bind b0) t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: 
-C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C 
-(\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 
-(getl_refl b c2 v)) b0 H5) w H4)))) H3)) H2))) (\lambda (h0: nat).(\lambda 
-(_: (((getl h0 (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w)) \to (ex C 
-(\lambda (d2: C).(getl h0 (CHead c2 (Bind b) v) (CHead d2 (Bind b0) 
-w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind b) v) (CHead d1 (Bind b0) 
-w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) (getl_gen_S (Bind b) c1 (CHead 
-d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x in (ex_ind C (\lambda (d2: 
-C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C (\lambda (d2: C).(getl (S h0) 
-(CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: 
-(getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) 
-(CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) h0 c2 
-(CHead x (Bind b0) w) H3 v)))) H2)))))) h H0)))))))))).
-
-lemma cimp_getl_conf:
- \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
-(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) 
-\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead 
-d2 (Bind b) w)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: 
-C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to 
-(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda 
-(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def 
-H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C 
-(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall 
-(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: 
-C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x 
-(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: 
-C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) 
-\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: 
-B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h 
-d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 
-(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 
-(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in 
-(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 
-(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind 
-b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 
-(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind 
-b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat 
-(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: 
-nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 
-(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def 
-(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind 
-b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro 
-C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind 
-b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma
deleted file mode 100644 (file)
index 5428dde..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-inductive clear: C \to (C \to Prop) \def
-| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e 
-(Bind b) u) (CHead e (Bind b) u))))
-| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: 
-F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma
deleted file mode 100644 (file)
index 4383e26..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/drop.ma
+++ /dev/null
@@ -1,168 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/clear/fwd.ma".
-
-include "basic_1/drop/fwd.ma".
-
-lemma drop_clear:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to 
-(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead 
-e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
-c2))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: 
-nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda 
-(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda 
-(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind 
-(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) 
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda 
-(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat 
-O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (ex2_3 B 
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e 
-(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
-c2))))) H3))))) (drop_gen_sort n (S i) O c2 H)))))) (\lambda (c: C).(\lambda 
-(H: ((\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T 
-(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) 
-v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
-c2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (i: 
-nat).(\lambda (H0: (drop (S i) O (CHead c k t) c2)).(K_ind (\lambda (k0: 
-K).((drop (r k0 i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear (CHead c k0 t) (CHead e (Bind b) v))))) (\lambda 
-(_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))) (\lambda (b: 
-B).(\lambda (H1: (drop (r (Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda 
-(b0: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e 
-(Bind b0) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
-c2)))) b c t (clear_bind b c t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r 
-(Flat f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda 
-(b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) 
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C 
-T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) 
-(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: 
-T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O 
-x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: 
-T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c 
-(CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) k (drop_gen_drop k c c2 t i 
-H0))))))))) c1).
-
-lemma drop_clear_O:
- \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c 
-(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 
-e2) \to (drop (S i) O c e2))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: 
-C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: 
-C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) 
-(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort 
-n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: 
-(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O 
-(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: 
-C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: 
-C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 
-e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: 
-T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: 
-C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: 
-K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 
-k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) 
-(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) (CHead e1 
-(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) 
-u) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
-_) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) u) (CHead c0 
-(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H5 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) 
-(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b 
-b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: 
-C).(drop i O c1 e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O 
-(CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) 
-H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 
-(Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead 
-e1 (Bind b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
-
-lemma drop_clear_S:
- \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop 
-h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear 
-x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 
-(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
-\def
- \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: 
-C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: 
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
-c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda 
-(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) 
-u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: 
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
-c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: 
-C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: 
-C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
-c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k 
-t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear 
-(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 
-(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C 
-(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: 
-C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift 
-h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k 
-(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead 
-c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind 
-(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h 
-(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) 
-t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) 
-(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind 
-b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind 
-b0 c (CHead c2 (Bind b) u) t H4)) in ((let H7 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match 
-k0 with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 
-(Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) 
-t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead 
-c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda 
-(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: 
-T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) 
-t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) 
-(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind 
-b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda 
-(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda 
-(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind 
-b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda 
-(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind 
-b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x 
-(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: 
-F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda 
-(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u 
-(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: 
-C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
-c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) 
-t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) 
-(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d 
-u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear 
-(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d 
-u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) 
-(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 
-H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
deleted file mode 100644 (file)
index 1b05b45..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/fwd.ma
+++ /dev/null
@@ -1,192 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/clear/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-implied rec lemma clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: 
-B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) 
-u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to 
-(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) 
-(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) 
-\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 
-((clear_ind P f f0) e c2 c3) f1 u)].
-
-lemma clear_gen_sort:
- \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: 
-Prop).P)))
-\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda 
-(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: 
-C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: 
-C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let 
-H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
-H1) in (False_ind P H2)))))) (\lambda (e: C).(\lambda (c: C).(\lambda (_: 
-(clear e c)).(\lambda (_: (((eq C e (CSort n)) \to P))).(\lambda (f: 
-F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e (Flat f) u) (CSort n))).(let 
-H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
-H3) in (False_ind P H4))))))))) y x H0))) H)))).
-
-lemma clear_gen_bind:
- \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
-(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
-(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda 
-(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: 
-(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e 
-(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) 
-u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) 
-\Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0) 
-(CHead e (Bind b) u) H1) in ((let H3 \def (f_equal C B (\lambda (e1: 
-C).(match e1 with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead 
-e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def (f_equal C T 
-(\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in (\lambda 
-(H5: (eq B b0 b)).(\lambda (H6: (eq C e0 e)).(eq_ind_r T u (\lambda (t: 
-T).(eq C (CHead e0 (Bind b0) t) (CHead e0 (Bind b0) t))) (eq_ind_r C e 
-(\lambda (c: C).(eq C (CHead c (Bind b0) u) (CHead c (Bind b0) u))) (eq_ind_r 
-B b (\lambda (b1: B).(eq C (CHead e (Bind b1) u) (CHead e (Bind b1) u))) 
-(refl_equal C (CHead e (Bind b) u)) b0 H5) e0 H6) u0 H4)))) H3)) H2)))))) 
-(\lambda (e0: C).(\lambda (c: C).(\lambda (_: (clear e0 c)).(\lambda (_: 
-(((eq C e0 (CHead e (Bind b) u)) \to (eq C c e0)))).(\lambda (f: F).(\lambda 
-(u0: T).(\lambda (H3: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) 
-u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind 
-b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0))) 
-H))))).
-
-lemma clear_gen_flat:
- \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
-(CHead e (Flat f) u) x) \to (clear e x)))))
-\def
- \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
-(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda 
-(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda 
-(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead 
-e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) 
-u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat 
-f) u) H1) in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda 
-(e0: C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 
-(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: 
-T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 
-\def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow e0 | 
-(CHead c0 _ _) \Rightarrow c0])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) 
-H3) in ((let H5 \def (f_equal C F (\lambda (e1: C).(match e1 with [(CSort _) 
-\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead 
-e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: C).(match e1 
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 
-(Flat f0) u0) (CHead e (Flat f) u) H3) in (\lambda (_: (eq F f0 f)).(\lambda 
-(H8: (eq C e0 e)).(let H9 \def (eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead 
-e (Flat f) u)) \to (clear e c))) H2 e H8) in (let H10 \def (eq_ind C e0 
-(\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x 
-H0))) H))))).
-
-lemma clear_gen_flat_r:
- \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x 
-(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
-\def
- \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: 
-(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e 
-(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: 
-C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: 
-C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat 
-f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e 
-(Flat f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: 
-C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) 
-\to P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat 
-f) u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) 
-u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda 
-(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C 
-(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
-
-lemma clear_gen_all:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind 
-(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: 
-B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 
-(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: 
-C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) 
-u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) 
-u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c 
-(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c 
-(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C 
-(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B 
-C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind 
-x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) 
-x2))) c H3)))))) H2)))))))) c1 c2 H))).
-
-theorem clear_mono:
- \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c 
-c2) \to (eq C c1 c2)))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to 
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: 
-nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: 
-C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 
-c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to 
-(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) 
-c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind 
-(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) 
-\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) 
-t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 
-(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) 
-(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 
-(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t 
-H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) 
-c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f 
-c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
-
-lemma clear_cle:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
-(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda 
-(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight 
-c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
-C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear 
-(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) 
-(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C 
-(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) 
-(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c 
-c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) 
-c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 
-(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma b/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma
deleted file mode 100644 (file)
index 07e59e0..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clear/props.ma
+++ /dev/null
@@ -1,96 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/clear/fwd.ma".
-
-lemma clear_clear:
- \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
-(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear 
-(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: 
-C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 
-c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear 
-(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to 
-(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) 
-c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) 
-(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: 
-F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c 
-c2 t H1)))) k H0))))))) c1).
-
-theorem clear_trans:
- \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c 
-c2) \to (clear c1 c2)))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to 
-(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: 
-nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: 
-C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) 
-c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to 
-(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) 
-c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: 
-K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: 
-B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 
-(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c 
-c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead 
-c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) 
-(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c 
-c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
-
-lemma clear_ctail:
- \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 
-(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k 
-u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
-\def
- \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
-C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: 
-K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) 
-u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: 
-(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: 
-T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 
-c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n 
-H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind 
-b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear 
-(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) 
-k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: 
-T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: 
-T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda 
-(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear 
-(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: 
-T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to 
-(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) 
-(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind 
-b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) 
-(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in 
-((let H3 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c 
-(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) 
-(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b 
-b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead 
-(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r 
-C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail 
-k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 
-u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail 
-k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: 
-(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 
-c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead 
-c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma
deleted file mode 100644 (file)
index 64209a6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clen/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-include "basic_1/s/defs.ma".
-
-rec definition clen (c: C) on c: nat \def match c with [(CSort _) \Rightarrow 
-O | (CHead c0 k _) \Rightarrow (s k (clen c0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma
deleted file mode 100644 (file)
index 55bad4a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/clen/getl.ma
+++ /dev/null
@@ -1,351 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/clen/defs.ma".
-
-include "basic_1/getl/props.ma".
-
-lemma getl_ctail_clen:
- \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: 
-nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
-\def
- \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex 
-nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) 
-(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O 
-(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b 
-(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl 
-(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: 
-K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl 
-(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat 
-(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) 
-(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen 
-c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: 
-K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t 
-c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat 
-(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) 
-t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail 
-(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: 
-F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t 
-c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) 
-t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
-
-lemma getl_gen_tail:
- \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall 
-(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 
-(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
-(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: 
-nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
-\def
- \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i 
-(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C 
-c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 
-nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind 
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort 
-n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: 
-nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C 
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort 
-n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) 
-(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead 
-(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear 
-(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
-C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e 
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: 
-nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
-nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead 
-(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c _ _) 
-\Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
-(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
-_ k0 _) \Rightarrow (match k0 with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
-(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
-(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: 
-(eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C (CSort n) 
-(\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) 
-(\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda 
-(_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda 
-(_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T 
-u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind 
-b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 
-nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind 
-b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort 
-n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C 
-(CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e 
-(Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: 
-nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda 
-(n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: 
-C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) 
-(CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda 
-(_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda 
-(n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq 
-nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq 
-T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat 
-O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b 
-H4) u2 H3) c2 H5)))) H2)) H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead 
-(CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 
-(Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or 
-(ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl 
-O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) 
-(\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) 
-(\lambda (n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) 
-k u1) (CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 
-(CHead (CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
-C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e 
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: 
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: 
-nat).(eq C c2 (CSort n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k 
-u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) 
-(getl_gen_S k (CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda 
-(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n1: nat).(eq C c2 (CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: 
-((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or 
-(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda 
-(i: nat).(nat_ind (\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) 
-(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
-e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat 
-(\lambda (_: nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k 
-(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort 
-n0))))))) (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind 
-b) u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead 
-c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
-(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat 
-(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind 
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort 
-n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind 
-b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 
-(CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead _ k1 _) 
-\Rightarrow (match k1 with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) 
-(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) 
-(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) 
-in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind 
-T u2 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
-(\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 
-nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_: 
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
-C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e: 
-C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1) 
-(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 
-(\lambda (c0: C).(\forall (i0: nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind 
-b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: 
-C).(getl i0 c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 
-(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in 
-(eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C 
-c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e 
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C 
-(CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) 
-(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C 
-(\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O 
-(CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c)) 
-(getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2)))) (\lambda (f: 
-F).(\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) 
-u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) 
-(CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) 
-(CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 
-(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat 
-(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) 
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or 
-(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O 
-(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq 
-nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda 
-(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: 
-(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c 
-(CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 
-e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda 
-(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) 
-(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4: 
-(eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b) 
-u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: 
-C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) 
-(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) 
-(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: 
-C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c 
-(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
-(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n)))) 
-(ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda 
-(e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C 
-(CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3)) 
-(\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: 
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
-C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c))) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
-e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) 
-(ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: 
-nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
-C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen 
-c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda 
-(H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C 
-(\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c 
-(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
-(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
-nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 
-(\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) 
-(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 
-nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq 
-K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort 
-x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda 
-(e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c 
-(Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
-(Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: 
-nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) 
-(or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) 
-(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 
-nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq 
-K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C 
-(CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat 
-f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: 
-nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 
-(refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) 
-u2 H6) c2 H7)))))) H3)) H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) 
-(CHead c2 (Bind b) u2) H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead 
-(CTail k u1 c) k0 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
-C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e 
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n0: nat).(eq C c2 (CSort n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail 
-k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S 
-k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in 
-(or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: 
-C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq 
-nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
-nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or (ex2 C 
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
-c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
-k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
-u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H3: (ex2 C 
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c 
-(CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 
-e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C 
-(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
-c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
-k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
-u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda 
-(H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl (r k0 n) c (CHead x (Bind 
-b) u2))).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k 
-u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind 
-b) u2) t n H1) (CTail k u1 x) H4) in (let H7 \def (eq_ind C c2 (\lambda (c0: 
-C).((getl n (CHead (CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 
-C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c 
-k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 
-(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
-u2)) (\lambda (n0: nat).(eq C c0 (CSort n0))))))) H0 (CTail k u1 x) H4) in 
-(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C 
-c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind 
-b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda 
-(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
-nat).(eq C c0 (CSort n0)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail 
-k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e 
-(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) 
-(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
-(n0: nat).(eq C (CTail k u1 x) (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq 
-C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) 
-(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_head k0 n c 
-(CHead x (Bind b) u2) H5 t))) c2 H4)))))) H3)) (\lambda (H3: (ex4 nat 
-(\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind 
-b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort 
-n0))))).(ex4_ind nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda 
-(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
-nat).(eq C c2 (CSort n0))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
-e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 
-nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K 
-k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 
-(CSort n0))))) (\lambda (x0: nat).(\lambda (H4: (eq nat (r k0 n) (clen 
-c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda 
-(H7: (eq C c2 (CSort x0))).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(getl 
-(r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 
-c) (CHead c2 (Bind b) u2) t n H1) (CSort x0) H7) in (let H9 \def (eq_ind C c2 
-(\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) 
-\to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: 
-C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: 
-nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) 
-(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0))))))) 
-H0 (CSort x0) H7) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C 
-(\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
-c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
-k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
-u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H10 \def (eq_ind_r T 
-u2 (\lambda (t0: T).((getl n (CHead (CTail k u1 c) k0 t) (CHead (CSort x0) 
-(Bind b) t0)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 
-e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat 
-(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind 
-b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) 
-(CSort n0))))))) H9 u1 H6) in (let H11 \def (eq_ind_r T u2 (\lambda (t0: 
-T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t0))) H8 u1 H6) 
-in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) 
-(CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
-t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda 
-(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: 
-nat).(eq C (CSort x0) (CSort n0)))))) (let H12 \def (eq_ind K k (\lambda (k1: 
-K).((getl n (CHead (CTail k1 u1 c) k0 t) (CHead (CSort x0) (Bind b) u1)) \to 
-(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: 
-C).(getl n (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: 
-nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) 
-(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort 
-n0))))))) H10 (Bind b) H5) in (let H13 \def (eq_ind K k (\lambda (k1: 
-K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) (Bind b) u1))) H11 (Bind 
-b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: 
-C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 
-t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 
-(clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 
-u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n) 
-(\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind 
-b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
-u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_: 
-nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: 
-nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0: 
-nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) 
-(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat 
-(\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind 
-b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) 
-(CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail 
-(Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
-u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq 
-K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq 
-C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S 
-n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 
-u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S 
-n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s 
-k0 (r k0 n)) (s_r k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) 
-H2)))))) i)))))) c1)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma
deleted file mode 100644 (file)
index 149d125..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/cnt/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-inductive cnt: T \to Prop \def
-| cnt_sort: \forall (n: nat).(cnt (TSort n))
-| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt 
-(THead k v t))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma
deleted file mode 100644 (file)
index 9227376..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/cnt/fwd.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/cnt/defs.ma".
-
-implied rec lemma cnt_ind (P: (T \to Prop)) (f: (\forall (n: nat).(P (TSort 
-n)))) (f0: (\forall (t: T).((cnt t) \to ((P t) \to (\forall (k: K).(\forall 
-(v: T).(P (THead k v t)))))))) (t: T) (c: cnt t) on c: P t \def match c with 
-[(cnt_sort n) \Rightarrow (f n) | (cnt_head t0 c0 k v) \Rightarrow (f0 t0 c0 
-((cnt_ind P f f0) t0 c0) k v)].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma
deleted file mode 100644 (file)
index e18a478..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/cnt/props.ma
+++ /dev/null
@@ -1,34 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/cnt/fwd.ma".
-
-include "basic_1/lift/props.ma".
-
-lemma cnt_lift:
- \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i 
-d t)))))
-\def
- \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: 
-nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: 
-nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) 
-(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: 
-T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: 
-nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) 
-(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k 
-(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma
deleted file mode 100644 (file)
index 7b35a43..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/arity.ma
+++ /dev/null
@@ -1,319 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/getl.ma".
-
-include "basic_1/csuba/props.ma".
-
-include "basic_1/arity/fwd.ma".
-
-include "basic_1/csubv/getl.ma".
-
-lemma csuba_arity:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
-t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
-A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: 
-C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c 
-c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall 
-(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda 
-(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in 
-(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda 
-(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d 
-x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc 
-g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u 
-(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def 
-(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc 
-g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
-a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: 
-(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst 
-g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
-a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 
-(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u 
-(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) 
-x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u 
-(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall 
-(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda 
-(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: 
-((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 
-a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 
-u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind 
-b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: 
-C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
-a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) 
-\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c 
-c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) 
-(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda 
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: 
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: 
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 
-a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 
-(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: 
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g 
-a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
-((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: 
-C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 
-H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
-(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity 
-g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: 
-C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
-H2)))))))))) c1 t a H))))).
-
-lemma csuba_arity_rev:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
-t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 
-t a))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
-A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 
-a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: 
-(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda 
-(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u 
-a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to 
-(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 
-c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i 
-H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
-(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d 
-u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity 
-g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) 
-(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x 
-(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf 
-c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda 
-(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 
-(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
-(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let 
-H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 
-(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 
-(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
-(Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i 
-H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) 
-i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def (f_equal C T (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow 
-t0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d 
-(Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B 
-Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda 
-(t0: T).(getl i c (CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def 
-(eq_ind_r C x1 (\lambda (c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) 
-in (let H20 \def (eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in 
-(let H21 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) 
-H19 Abbr H16) in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) 
-H13)))))))) H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u 
-a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a1: A).(arity g d u a1)))) (arity g c2 (TLRef i) a0) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H7: (getl i c2 
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (csuba g x0 d)).(\lambda (H9: (arity 
-g x0 x1 (asucc g x2))).(\lambda (H10: (arity g d u x2)).(arity_repl g c2 
-(TLRef i) x2 (arity_abst g c2 x0 x1 i H7 x2 H9) a0 (arity_mono g d u x2 H10 
-a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
-i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let 
-H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in 
-(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: 
-C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef 
-i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda 
-(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d 
-(Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) 
-(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in 
-(let H13 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow 
-False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead 
-x2 (Bind Void) x3) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind 
-Void) x3) H11)) in (False_ind (arity g c2 (TLRef i) a0) H13))))))) H9))))))) 
-H6)) H5)))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: 
-((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to (arity g c2 u (asucc 
-g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(\lambda (H4: 
-(csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (let 
-H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) (\lambda (H6: 
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 (TLRef i) a0) 
-(\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) u))).(\lambda 
-(H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 Abst x u i H7) in 
-(let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda 
-(v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 (TLRef i) a0) 
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H10: (csubv x 
-x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let H12 \def (eq_ind 
-C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x1 (Bind 
-x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) 
-H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) 
-(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 
-(Bind x0) x2) H11)) in ((let H14 \def (f_equal C B (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) 
-u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
-x1 (Bind x0) x2) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
-(Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i 
-H0 (CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda 
-(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c 
-(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda 
-(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def 
-(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def 
-(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) 
-in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) 
-H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
-i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let 
-H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in 
-(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: 
-C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef 
-i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda 
-(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d 
-(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) 
-(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in 
-(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
-I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
-x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 (TLRef i) a0) H13))))))) 
-H9))))))) H6)) H5)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
-Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
-g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) 
-\to (arity g c2 u a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: 
-(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba 
-g c2 (CHead c (Bind b) u)) \to ((csubv c2 (CHead c (Bind b) u)) \to (arity g 
-c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: 
-(csubv c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 
-(Bind b) u) (csuba_head g c2 c H5 (Bind b) u) (csubv_bind_same c2 c H6 b u 
-u))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 
-c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a1))))))).(\lambda (t0: 
-T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
-a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c (Bind Abst) u)) 
-\to ((csubv c2 (CHead c (Bind Abst) u)) \to (arity g c2 t0 a2)))))).(\lambda 
-(c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_head 
-g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 (CHead c2 (Bind Abst) u) (csuba_head g c2 c 
-H4 (Bind Abst) u) (csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda 
-(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
-a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to 
-(arity g c2 u a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity 
-g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((csuba g c2 c) \to 
-((csubv c2 c) \to (arity g c2 t0 (AHead a1 a2))))))).(\lambda (c2: 
-C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_appl g c2 
-u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 H5)))))))))))))) (\lambda (c: C).(\lambda 
-(u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda 
-(H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u 
-(asucc g a0))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda 
-(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 
-a0)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv 
-c2 c)).(arity_cast g c2 u a0 (H1 c2 H4 H5) t0 (H3 c2 H4 H5))))))))))))) 
-(\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c t0 
-a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to 
-(arity g c2 t0 a1)))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
-a2)).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(\lambda (H4: (csubv c2 
-c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 H2))))))))))) c1 t a H))))).
-
-theorem arity_appls_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c 
-v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: 
-T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs 
-(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead 
-(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: 
-(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g 
-a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
-TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) 
-v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead 
-(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead 
-(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr not_abbr_abst g c v 
-t a2 H1) in (let H2 \def H_x in (ex2_ind A (\lambda (a3: A).(arity g c v a3)) 
-(\lambda (_: A).(arity g (CHead c (Bind Abbr) v) t a2)) (arity g c (THead 
-(Flat Appl) v (THead (Bind Abst) u t)) a2) (\lambda (x: A).(\lambda (_: 
-(arity g c v x)).(\lambda (H4: (arity g (CHead c (Bind Abbr) v) t 
-a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) a2 (arity_head g c u a1 
-H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t a2 H4 (CHead c (Bind 
-Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) (csubv_bind c c 
-(csubv_refl c) Abst not_abst_void Abbr u v))))))) H2))))) (\lambda (t0: 
-T).(\lambda (t1: TList).(\lambda (H1: ((\forall (a2: A).((arity g c (THeads 
-(Flat Appl) t1 (THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat 
-Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u t))) a2))))).(\lambda (a2: 
-A).(\lambda (H2: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind Abbr) v t))) a2)).(let H3 \def (arity_gen_appl g c t0 (THeads 
-(Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) in (ex2_ind A (\lambda (a3: 
-A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c (THeads (Flat Appl) t1 
-(THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u t)))) a2) 
-(\lambda (x: A).(\lambda (H4: (arity g c t0 x)).(\lambda (H5: (arity g c 
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead x a2))).(arity_appl g 
-c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u 
-t))) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma
deleted file mode 100644 (file)
index 06b1cc2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/clear.ma
+++ /dev/null
@@ -1,122 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma csuba_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 
-c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
-(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear 
-(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
-b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda 
-(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g 
-c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
-(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
-(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
-e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 
-e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
-C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
-(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
-u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
-(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda 
-(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
-(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: 
-C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba 
-g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
-u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) 
-(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: 
-((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
-(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda 
-(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) 
-e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) 
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) 
-(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) 
-u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 
-(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
-
-lemma csuba_clear_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 
-c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear 
-c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c 
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
-(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) 
-e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear 
-(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind 
-b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda 
-(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g 
-c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) 
-(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def 
-(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
-e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 
-e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: 
-C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C 
-(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) 
-u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
-(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda 
-(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
-(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) 
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: 
-C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 
-(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 
-u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: 
-((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
-(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda 
-(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
-a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) 
-e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) 
-e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) 
-(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) 
-t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 
-(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma
deleted file mode 100644 (file)
index 948ee34..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/defs.ma
+++ /dev/null
@@ -1,30 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/arity/defs.ma".
-
-inductive csuba (g: G): C \to (C \to Prop) \def
-| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
-| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
-(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
-| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g 
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
-(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: 
-T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind 
-Abbr) u))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma
deleted file mode 100644 (file)
index 5a92a8e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/drop.ma
+++ /dev/null
@@ -1,2453 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/fwd.ma".
-
-include "basic_1/drop/fwd.ma".
-
-lemma csuba_drop_abbr:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
-O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
-c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) 
-(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 
-(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: 
-(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in 
-(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba 
-g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 
-(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x 
-(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda 
-(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) 
-H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: 
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S 
-n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) 
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
-O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind 
-Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: 
-(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) 
-(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead 
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))) (drop_gen_sort 
-n0 (S n) O (CHead d1 (Bind Abbr) u) H0))))))))) (\lambda (c: C).(\lambda (H0: 
-((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) 
-\to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: 
-T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abbr) 
-u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) 
-c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) 
-O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (b: 
-B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r 
-(Bind b) n) O c (CHead d1 (Bind Abbr) u))).(B_ind (\lambda (b0: B).((csuba g 
-(CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H5: (csuba g (CHead c 
-(Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind 
-Abbr) u))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in 
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: 
-C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: 
-(eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C 
-(CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) 
-O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 
-\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead 
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead 
-x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal 
-nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
-nat).(drop n0 O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) 
-n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) 
-(\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r 
-(Bind Abst) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g 
-c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 
-(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq 
-C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
-g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 
-(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x 
-(Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def 
-(H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead 
-x0 (Bind Abbr) u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal 
-nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: 
-nat).(drop n0 O x (CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) 
-n x (CHead x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda 
-(H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g 
-c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity 
-g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O 
-c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 
-(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t 
-(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind 
-Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u 
-H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x 
-(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat 
-(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 
-O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 
-(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) 
-(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r 
-(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g 
-c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda 
-(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C 
-c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead 
-x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def 
-(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead 
-x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal 
-nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
-nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n 
-x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 
-H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) 
-c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let 
-H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) 
-O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) 
-x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) 
-(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 
-H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) 
-u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
-d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) 
-u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
-d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) 
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n 
-H1)))))))))))) c1)))) i).
-
-lemma csuba_drop_abst:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
-O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n 
-O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: 
-T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: 
-G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 
-(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl 
-c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 
-H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: 
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
-O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x 
-(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind 
-Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) 
-u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind 
-Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
-C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 
-H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind 
-Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 
-(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind 
-Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) 
-x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
-Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 
-H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 
-(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to 
-(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: 
-C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind 
-Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) 
-c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) 
-(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda 
-(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S 
-n) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) 
-H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: 
-T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall 
-(c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda 
-(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) 
-u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) 
-c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) 
-O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O 
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c 
-(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) 
-u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop 
-(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g 
-(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead 
-d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 
-\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
-(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) 
-t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) 
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: 
-C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
-(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead 
-x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def 
-(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda 
-(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) 
-in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) 
-u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: 
-(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 
-x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 
-x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 
-(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) 
-n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 
-H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda 
-(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def 
-(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda 
-(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g 
-c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity 
-g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
-t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
-(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: 
-C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c 
-x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g 
-x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C 
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: 
-(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) 
-u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) 
-n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: 
-(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 
-x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 
-x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def 
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 
-(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) 
-n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 
-H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind 
-Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g 
-x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) 
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda 
-(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
-(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead 
-x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal 
-nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: 
-nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in 
-(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x 
-(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: 
-(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 
-x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 
-x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def 
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) 
-H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 
-(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 
-H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g 
-(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead 
-d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 
-\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: 
-T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c 
-x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g 
-x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 
-(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda 
-(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O 
-(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
-(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: 
-C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: 
-(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 
-\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) 
-u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
-(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n 
-x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: 
-(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 
-x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 
-x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
-(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) 
-H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 
-(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 
-H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: 
-F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r 
-(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c 
-c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda 
-(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 
-(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) 
-x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: 
-C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda 
-(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: 
-(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 
-x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) 
-H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: 
-(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 
-x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 
-x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) 
-x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind 
-Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
-(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
-
-lemma csuba_drop_abst_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
-O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
-c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n 
-O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: 
-C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind 
-Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 
-c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 
-(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x 
-\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or 
-(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C 
-(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) 
-u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) 
-(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O 
-(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x 
-(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind 
-Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x 
-(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
-(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) 
-x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
-(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O 
-(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind 
-Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 
-(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or 
-(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
-(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) 
-(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
-O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind 
-Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq 
-nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) 
-(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) 
-(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) H0))))))))) (\lambda (c: 
-C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 
-(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop 
-(S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: 
-C).(\lambda (u: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind 
-Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g c2 (CHead 
-c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0 t)) \to ((drop (r 
-k0 n) O c (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(drop (S 
-n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: 
-B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
-(Bind b) n) O c (CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g 
-c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
-Abst) u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 
-(CHead c (Bind Abbr) t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 
-(Bind Abst) u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 
-\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) 
-t))) (\lambda (d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
-(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
-(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) 
-t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) 
-(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x 
-H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind 
-Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
-g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: 
-(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind 
-Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
-g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 
-t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
-(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda 
-(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
-x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop 
-(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
-(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) 
-H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc 
-g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) 
-(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
-A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g 
-x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t 
-x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda 
-(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: 
-(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind 
-Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
-x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) 
-(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) 
-x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) 
-H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: 
-(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r 
-C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: 
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
-H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O 
-x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C 
-(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x 
-d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
-(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda 
-(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
-x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) 
-H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) 
-t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) 
-u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in 
-(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda 
-(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 
-(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
-g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 
-(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x 
-(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H 
-c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) 
-u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) 
-H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead 
-x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
-(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) 
-H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: 
-(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r 
-C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: 
-C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
-H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O 
-x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C 
-(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x 
-d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
-(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda 
-(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
-x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) 
-H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) 
-t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) 
-u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in 
-(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: 
-C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq 
-C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C 
-(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
-n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H 
-c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
-u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) 
-H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
-Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead 
-x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
-(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) 
-H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 
-(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind 
-Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def 
-H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 
-(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) 
-x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 
-H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
-d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat 
-f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C 
-T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) 
-u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
-(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
-O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 
-x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 
-(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
-(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) 
-H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n 
-H1)))))))))))) c1)))) i).
-
-lemma csuba_drop_abbr_rev:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
-O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: 
-G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n 
-O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
-(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: 
-C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 
-(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in 
-(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda 
-(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x 
-d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C 
-(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: 
-C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) 
-u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl 
-(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
-A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g 
-x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 
-u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C 
-(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: 
-C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) 
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 
-(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda 
-(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
-O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C 
-c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C 
-(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop 
-O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) 
-(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
-Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 
-x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) 
-(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall 
-(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall 
-(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
-C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
-(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: 
-nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) 
-(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: 
-(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort 
-n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O 
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) 
-(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let 
-H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H3) in (False_ind (or3 (ex2 C (\lambda 
-(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O 
-(CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall 
-(d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to 
-(\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda 
-(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda (k: K).(\lambda 
-(t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c 
-k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda 
-(H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead 
-c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: 
-B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
-(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g 
-c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
-Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda 
-(H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: (drop (r (Bind Abbr) 
-n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abbr_rev g c c2 t 
-H5) in (let H7 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
-d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: 
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: 
-C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq 
-C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C 
-(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop 
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind 
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) 
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
-(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
-(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda 
-(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 
-(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 
-x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n 
-x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
-x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
-(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
-Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) 
-H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 
-C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind 
-Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc 
-g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) 
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) 
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x 
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) 
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 
-(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: 
-(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: 
-(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 
-(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 
-H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) 
-x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind 
-Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 
-H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) 
-x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x 
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 
-(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: 
-(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: 
-(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 
-H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
-x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind 
-Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) 
-H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda 
-(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
-(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
-g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: 
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: 
-C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
-Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq 
-C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C 
-(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop 
-(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind 
-(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S 
-n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) 
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
-(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
-(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda 
-(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 
-(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 
-x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n 
-x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
-x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
-(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
-Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) 
-H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 
-(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) 
-x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
-(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
-(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
-O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
-C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
-(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x 
-d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
-x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 
-(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: 
-(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: 
-(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
-n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
-(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 
-H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
-n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
-(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
-x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind 
-Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) 
-H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda 
-(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
-(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda 
-(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) 
-(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: 
-C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x 
-c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x 
-H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C 
-(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 
-(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda 
-(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
-x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) 
-(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 
-(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 
-(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 
-x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n 
-x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
-C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) 
-x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
-(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
-Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) 
-H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat 
-f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) 
-u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in 
-(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda 
-(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 
-c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda 
-(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 
-H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda 
-(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) 
-u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
-(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
-(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) 
-x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g 
-x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: 
-C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
-x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
-(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 
-H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
-(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
-(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 
-(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda 
-(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
-x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 
-(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k 
-H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma
deleted file mode 100644 (file)
index b94c9c8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/fwd.ma
+++ /dev/null
@@ -1,1024 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/defs.ma".
-
-implied rec lemma csuba_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
-nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csuba 
-g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
-(CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csuba g c1 
-c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
-T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
-u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to ((P 
-c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to 
-(\forall (u: T).((arity g c2 u a) \to (P (CHead c1 (Bind Abst) t) (CHead c2 
-(Bind Abbr) u)))))))))))) (c: C) (c0: C) (c1: csuba g c c0) on c1: P c c0 
-\def match c1 with [(csuba_sort n) \Rightarrow (f n) | (csuba_head c2 c3 c4 k 
-u) \Rightarrow (f0 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) k u) | 
-(csuba_void c2 c3 c4 b n u1 u2) \Rightarrow (f1 c2 c3 c4 ((csuba_ind g P f f0 
-f1 f2) c2 c3 c4) b n u1 u2) | (csuba_abst c2 c3 c4 t a a0 u a1) \Rightarrow 
-(f2 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) t a a0 u a1)].
-
-lemma csuba_gen_abbr:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
-(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) 
-(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
-C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda 
-(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda 
-(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C 
-c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda 
-(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) 
-u) H1) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H2)))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (k: 
-K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) 
-u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 
-(Bind Abbr) u) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
-u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in (\lambda (H7: (eq K k (Bind 
-Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u (\lambda (t: T).(ex2 C 
-(\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C 
-(\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C 
-c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) H2 d1 H8) in (let H10 
-\def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex_intro2 C 
-(\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abbr) u)) 
-H10))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind 
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B 
-b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 
-(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 
-(Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
-| (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 
-with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
-True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in 
-(False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) (\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C 
-c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: 
-T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: 
-T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) 
-t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H5) in (False_ind (ex2 C 
-(\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) y c H0))) H))))).
-
-lemma csuba_gen_void:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
-(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) 
-u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda 
-(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) 
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) 
-(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: 
-C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T 
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) 
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind 
-Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
-(Bind Void) u1) H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind b) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H2)))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
-(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))).(\lambda (k: 
-K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead d1 (Bind Void) 
-u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 
-(Bind Void) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
-u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in (\lambda (H7: (eq K k (Bind 
-Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_3 B C 
-T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k t) 
-(CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csuba g d1 d2)))))) (eq_ind_r K (Bind Void) (\lambda (k0: K).(ex2_3 B C T 
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) 
-(CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csuba g d1 d2)))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 
-(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H8) in (let 
-H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in 
-(ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C 
-(CHead c2 (Bind Void) u1) (CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda 
-(d2: C).(\lambda (_: T).(csuba g d1 d2)))) Void c2 u1 (refl_equal C (CHead c2 
-(Bind Void) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
-(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))).(\lambda (b: 
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) 
-u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Void) u0) 
-(CHead d1 (Bind Void) u1) H4) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq 
-C c1 d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
-(Bind Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda 
-(u3: T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C 
-c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda 
-(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead 
-d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba 
-g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) 
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
-c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T 
-(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) 
-u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc 
-g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
-(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C 
-(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind 
-Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
-C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) 
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) 
-H6)))))))))))) y c H0))) H))))).
-
-lemma csuba_gen_abst:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
-(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead 
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) 
-u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: 
-C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead 
-d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
-c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind 
-Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) 
-(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) 
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
-(Bind Abst) u1) H1) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CSort n) 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CSort n) (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
-c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda 
-(H3: (eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C 
-C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H5 
-\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) 
-in ((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind 
-Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 
-d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead 
-c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 
-C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k t) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 
-C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) 
-(let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) 
-u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 d1 H8) 
-in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in 
-(or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind 
-Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 
-(refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) 
-H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
-c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
-(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
-g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind 
-Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind 
-Void) u0) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 with 
-[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | 
-(Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abst) u1) H4) in (False_ind 
-(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead d2 
-(Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 
-u3 a)))))) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
-(csuba g c1 c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or 
-(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda 
-(a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda 
-(H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
-(Bind Abst) u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind 
-Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow 
-t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in (\lambda (H8: 
-(eq C c1 d1)).(let H9 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc 
-g a))) H3 u1 H7) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 
-u1 (asucc g a))) H9 d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: 
-C).((eq C c0 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C 
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc 
-g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g 
-c0 c2)) H1 d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind 
-Abbr) u) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead 
-c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity 
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 
-H10 H4)))))))) H6)))))))))))) y c H0))) H))))).
-
-lemma csuba_gen_flat:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
-(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d1 d2)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead 
-d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: 
-(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
-(Flat f) u1) H1) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d1 d2)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
-(csuba g c1 c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))).(\lambda (k: 
-K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead d1 (Flat f) 
-u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 
-(Flat f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) 
-(CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 k u) (CHead d1 (Flat f) u1) H3) in (\lambda (H7: (eq K k (Flat 
-f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k t) (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) (eq_ind_r K (Flat 
-f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
-c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d1 d2))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
-(Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 
-(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 
-c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-(CHead c2 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d1 d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H10))) k 
-H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
-(csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C 
-T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))).(\lambda (b: 
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 (Flat f) 
-u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(CHead d1 (Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u3: T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda 
-(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat 
-f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
-d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
-a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
-(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C 
-(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) 
-H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
-c2 (Bind Abbr) u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d1 d2)))) H6)))))))))))) y c H0))) H)))))).
-
-lemma csuba_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: 
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
-c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
-e1 e2))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 
-c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
-e2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
-(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
-(eq K k (Bind b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: 
-T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
-K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda 
-(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 
-H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) 
-in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
-C (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C 
-(CHead c3 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: 
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
-v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
-(CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) 
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) 
-in (\lambda (H8: (eq B Void b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def 
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
-T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
-e2))))))) H2 e1 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c 
-c3)) H1 e1 H9) in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 
-(CHead e1 (Bind b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 
-(Bind b) u2)) H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
-C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t: T).(\lambda (a: 
-A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (_: 
-(arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 
-(Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind 
-Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead 
-c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
-\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in 
-(\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def 
-(eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let 
-H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) 
-in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) 
-v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
-C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
-(_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 
-(\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 
-(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda 
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) 
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
-e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) 
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) 
-Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) 
-H6)))))))))))) y c2 H0))) H)))))).
-
-lemma csuba_gen_abst_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
-(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) 
-(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: 
-C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: 
-C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: 
-C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C 
-c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: 
-nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def 
-(eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
-True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) u) H1) in 
-(False_ind (or (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(eq C (CSort n) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
-Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq 
-C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H5 
-\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) 
-in ((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 
-(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C 
-c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C 
-(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) 
-(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C 
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
-H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to 
-(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g 
-c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind 
-Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
-(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind 
-Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda 
-(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 
-(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not 
-(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
-c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in ((let H6 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
-_ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in 
-((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead 
-d1 (Bind Abst) u) H4) in (\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 
-d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst 
-H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind 
-Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: 
-T).(eq C c1 (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda 
-(c0: C).(csuba g c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq 
-C (CHead c1 (Bind Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C 
-(CHead c1 (Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: 
-C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) (CHead d2 (Bind Void) 
-u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C 
-(CHead c1 (Bind Void) u1)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 
-(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: T).(\lambda (a: 
-A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: T).(\lambda (_: 
-(arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 
-(Bind Abst) u))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind (or 
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) H6)))))))))))) c y H0))) H))))).
-
-lemma csuba_gen_void_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
-(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
-Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
-(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) 
-(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
-C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda 
-(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda 
-(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C 
-c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda 
-(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) 
-u) H1) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
-Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H2)))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
-(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (k: 
-K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) 
-u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
-(Bind Void) u) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k 
-u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in (\lambda (H7: (eq K k (Bind 
-Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u (\lambda (t: T).(ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 (Bind Void) u))) (\lambda 
-(d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) (\lambda (k0: K).(ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 (Bind Void) u))) (\lambda 
-(d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C 
-c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H8) in (let H10 
-\def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex_intro2 C 
-(\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Void) u)) 
-H10))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
-Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq 
-B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 
-(Bind b) u2) (CHead d1 (Bind Void) u))).(let H5 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in ((let H6 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
-_ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in 
-((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead 
-d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b Void)).(\lambda (H9: (eq C c2 
-d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Void 
-H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind 
-Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 
-(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in (let H13 \def (match (H10 
-(refl_equal B Void)) in False with []) in H13))))))) H6)) H5))))))))))) 
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: 
-(((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (t: 
-T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: 
-T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) 
-u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) 
-u0) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k 
-_) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind (ex2 C 
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) H6)))))))))))) c y H0))) H))))).
-
-lemma csuba_gen_abbr_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) 
-u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda 
-(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda 
-(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
-(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind 
-Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
-(Bind Abbr) u1) H1) in (False_ind (or3 (ex2 C (\lambda (d2: C).(eq C (CSort 
-n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CSort n) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
-c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
-(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k 
-u) (CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) 
-in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r 
-T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: 
-C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C 
-c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
-(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
-g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
-C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba 
-g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 
-(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
-(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
-(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
-(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
-(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: 
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) 
-u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) 
-(CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match 
-k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 
-(Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead _ _ t) 
-\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in 
-(\lambda (H8: (eq B b Abbr)).(\lambda (H9: (eq C c2 d1)).(let H10 \def 
-(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abbr H8) in (let H11 
-\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to 
-(or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u3: 
-T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u3))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) u3)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in (let H12 
-\def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in (or3_intro2 
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) (CHead d2 
-(Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind 
-Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq 
-C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C (CHead c1 (Bind 
-Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
-Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: 
-T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: 
-T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) 
-u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead 
-_ _ t0) \Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) 
-H5) in (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: 
-T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: 
-C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: 
-C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq 
-C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 
-u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 
-d1 H8) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 
-H8) in (or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) 
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
-(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 
-u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) 
-H6)))))))))))) c y H0))) H))))).
-
-lemma csuba_gen_flat_rev:
- \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
-(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))
-\def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead 
-d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: 
-(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 
-(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
-C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
-u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
-(Flat f) u1) H1) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
-(csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Flat f) u1)) \to (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))).(\lambda (k: 
-K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Flat f) 
-u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 
-(Flat f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) 
-(CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c2 k u) (CHead d1 (Flat f) u1) H3) in (\lambda (H7: (eq K k (Flat 
-f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 (Flat f) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (eq_ind_r K (Flat 
-f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
-c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))))) (let H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 
-(Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 
-(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 
-c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H10))) k 
-H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
-(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat f) u1)) \to (ex2_2 C 
-T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))).(\lambda (b: 
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Flat f) u1))).(let 
-H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) 
-u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C 
-(CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat f) 
-u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
-a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
-(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C 
-(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) 
-H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
-c1 (Bind Abst) t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) H6)))))))))))) c y H0))) H)))))).
-
-lemma csuba_gen_bind_rev:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C 
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: 
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 
-y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind 
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
-e2 e1))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 
-c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
-e1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c3 k u) 
-(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k 
-u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
-(eq K k (Bind b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: 
-T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c1 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
-K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c1 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda 
-(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 
-H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) 
-in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
-C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C 
-(CHead c1 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: 
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let 
-H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c3 
-| (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) 
-v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
-_) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 (Bind b) u2) (CHead e1 
-(Bind b1) v1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) 
-u2) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: 
-(eq C c3 e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 
-Void))) H3 b1 H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let 
-H12 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) Void c1 u1 (refl_equal 
-C (CHead c1 (Bind Void) u1)) H12))))))) H6)) H5))))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t: 
-T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: 
-T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
-u) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 
-(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-Abbr])])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) 
-v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 e1)).(let 
-H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) in (let 
-H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) in (let 
-H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to 
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 (\lambda 
-(c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda 
-(b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H13 
-Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda 
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t 
-(refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) H6)))))))))))) c2 y 
-H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma
deleted file mode 100644 (file)
index 0399924..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/getl.ma
+++ /dev/null
@@ -1,1160 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/drop.ma".
-
-include "basic_1/csuba/clear.ma".
-
-include "basic_1/getl/clear.ma".
-
-lemma csuba_getl_abbr:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g 
-c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
-(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: 
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
-Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
-(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda 
-(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
-(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
-d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: 
-(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 
-(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to 
-((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: 
-C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda 
-(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 
-(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
-(CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 
-(Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with 
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind 
-Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
-t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
-(\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
-C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
-T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r 
-B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr H10) in 
-(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind 
-Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 g c2 
-H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 (CHead 
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: 
-C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: 
-(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 
-(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) 
-H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead 
-x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind 
-Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c 
-(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n 
-O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
-x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 
-\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) 
-(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 
-(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) 
-f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead 
-d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 
-(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
-(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) 
-x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u 
-H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
-(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda 
-(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
-c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl 
-O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 
-(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) 
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
-\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda 
-(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O 
-x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 
-c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
-C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
-b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
-x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: 
-C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) 
-x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def 
-(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C 
-(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: 
-C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: 
-(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x 
-\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B 
-C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
-x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda 
-(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: 
-(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 
-(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind 
-C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: 
-(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 
-x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) 
-(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead 
-x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) 
-i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
-lemma csuba_getl_abst:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba 
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) 
-(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear 
-x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to 
-((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) 
-\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda 
-(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) 
-(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 
-(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 
-C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda 
-(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear 
-(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O 
-c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) 
-\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i 
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) 
-t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) 
-u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u1) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
-in ((let H8 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
-in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abst) u1) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
-in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
-C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
-T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r 
-B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in 
-(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind 
-Abst) u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 
-H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
-(H17: (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda 
-(H18: (drop i O c2 (CHead x1 (Bind Abst) u1))).(\lambda (H19: (csuba g d1 
-x1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
-(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 (Bind Abst) u1) (CHead 
-x1 (Bind Abst) u1) H18 (clear_bind Abst x1 u1)) H19))))) H17)) (\lambda (H17: 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) 
-x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 u1 (asucc g 
-x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x1 x2 x3 
-(getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 
-(clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) 
-H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) 
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
-(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
-c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: 
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda 
-(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
-C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) 
-(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def 
-(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) 
-H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) 
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead 
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) 
-x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 
-H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead 
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
-(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 
-(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in 
-(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
-g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba 
-g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity 
-g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 
-(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O 
-c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead 
-x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) 
-H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
-C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
-\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) 
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat 
-f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def 
-(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) 
-(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat 
-f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
-(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 
-(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) 
-x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) 
-(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) 
-x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 
-x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda 
-(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda 
-(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
-d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl 
-(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: 
-C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
-x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: 
-C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
-x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or 
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
-(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda 
-(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
-A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 
-(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C 
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
-C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
-(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
-d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) 
-H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 
-(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
-u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: 
-(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 
-x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 
-x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 
-c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) 
-H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 
-H2)))) H0))))))).
-
-lemma csuba_getl_abst_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g 
-c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) 
-(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: 
-C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
-Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
-(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) 
-(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear 
-(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) 
-n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: 
-C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) 
-\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i 
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: 
-(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: 
-K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind 
-Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: 
-B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear 
-(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
-\Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | 
-(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead 
-_ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in (\lambda (H10: (eq B 
-Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
-g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
-(Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: B).(drop 
-i O c1 (CHead x0 (Bind b0) u))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 
-(\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u))) H14 d1 H11) in (let 
-H16 \def (csuba_drop_abst_rev i c1 d1 u H15 g c2 H12) in (or_ind (ex2 C 
-(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O 
-c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop 
-i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: (csuba g x1 d1)).(or_introl 
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 (Bind Abst) 
-u) (CHead x1 (Bind Abst) u) H18 (clear_bind Abst x1 u)) H19))))) H17)) 
-(\lambda (H17: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead 
-x1 (Bind Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 
-x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 
-(clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda 
-(f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: 
-(clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in 
-(unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 (Flat f) t)) \to 
-(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (nat_ind (\lambda 
-(n: nat).(\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall 
-(c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: 
-(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g 
-c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead 
-x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def 
-(clear_flat x0 (CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind 
-Abst) u) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) 
-t) c2 H10 (CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba 
-g e2 (CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 
-(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst) 
-u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2 
-u H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H15: 
-(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: 
-C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda (H17: (csuba g 
-x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead 
-x3 (Bind Abst) u) H16) in (or_introl (ex2 C (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda 
-(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H18) 
-H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
-x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C 
-x2 (CHead x3 (Bind Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def 
-(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) 
-in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 
-(drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: 
-nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to 
-(\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: 
-C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H10: (csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 
-(Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat f) t))))) (or 
-(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl 
-(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
-g d2 d1))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
-(H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead 
-x0 (Flat f) t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind 
-x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) 
-x4))) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) 
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: 
-C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear 
-c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def 
-H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) 
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x6: 
-B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind 
-x6) x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda 
-(c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 
-H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) 
-c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C 
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) 
-u))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl 
-(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) 
-u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
-n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
-d2 d1))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: 
-(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 
-d1)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
-(u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
-(_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))) x9 x10 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind 
-Void) x10) n H23) H24)))))) H22)) H21)))))))) H17)))))) H14))))))) 
-H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-
-lemma csuba_getl_abbr_rev:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
-(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba 
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
-(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) 
-(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) 
-(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead 
-d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c 
-(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 
-(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: 
-(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) 
-u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba 
-g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: 
-C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) 
-\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
-d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 
-(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
-u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear 
-(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 
-c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: 
-B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear 
-(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
-\Rightarrow c])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H8 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
-_ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
-(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B 
-Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
-g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
-(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: 
-B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def 
-(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1 
-H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in 
-(or3_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: 
-C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) u1))).(\lambda (H19: 
-(csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 (Bind Abbr) u1) (CHead x1 
-(Bind Abbr) u1) H18 (clear_bind Abbr x1 u1)) H19))))) H17)) (\lambda (H17: 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
-(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) 
-x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 x2 (asucc g 
-x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 
-(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 
-(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
-(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind 
-Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
-C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead 
-x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) 
-H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
-t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) 
-u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop 
-n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 
-(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: 
-(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g 
-c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead 
-x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def 
-(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind 
-Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) 
-t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: 
-C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) 
-(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: 
-C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: 
-(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 
-\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead 
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda 
-(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: 
-C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba 
-g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead 
-x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
-(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
-(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 
-(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
-(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba 
-g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity 
-g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 
-(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O 
-c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) 
-x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 
-C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
-(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind 
-Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
-d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
-(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind 
-Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in 
-(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
-(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
-(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
-(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
-(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 
-(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) 
-(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 
-(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n 
-c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 
-(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let 
-H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T 
-(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) 
-v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 
-(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 
-(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 
-\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind 
-C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: 
-C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: 
-(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let 
-H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
-(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S 
-n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: 
-T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g 
-x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead 
-x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C 
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
-(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
-C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S 
-n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
-T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead 
-d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
-(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
-(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: 
-(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
-(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
-C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) 
-u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) 
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
-a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
-A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: 
-A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: 
-(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda 
-(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
-d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
-(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
-u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead 
-d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
-(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 
-(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T 
-(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) 
-(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: 
-C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
-n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
-A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
-T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
-T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: 
-(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 
-d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
-u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
-C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
-a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
-Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro 
-C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) 
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) 
-H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) 
-x H1 H2)))) H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma
deleted file mode 100644 (file)
index 0e60bfe..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csuba/props.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csuba/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-lemma csuba_refl:
- \forall (g: G).(\forall (c: C).(csuba g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) 
-(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma
deleted file mode 100644 (file)
index 6359363..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/arity.ma
+++ /dev/null
@@ -1,36 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/csuba.ma".
-
-lemma csubc_arity_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
-(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
-c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t 
-a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
-
-lemma csubc_arity_trans:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
-((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to 
-(arity g c1 t a))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
-c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda 
-(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 
-H) H0)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma
deleted file mode 100644 (file)
index eaef555..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/clear.ma
+++ /dev/null
@@ -1,167 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma csubc_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall 
-(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda 
-(e2: C).(csubc g e1 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 
-e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c 
-c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 
-e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: 
-C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def 
-(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 
-C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g 
-e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
-(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq 
-C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda 
-(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
-g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
-a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda 
-(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) 
-(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C 
-(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e 
-c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda 
-(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
-C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 
-(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x 
-(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda 
-(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: 
-C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind 
-b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind 
-b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda 
-(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
-(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda 
-(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind 
-b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda 
-(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: 
-(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 
-C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) 
-u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 with [(Bind 
-b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) (Bind b) (Bind Abst) H3) in 
-(eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x0 
-(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda 
-(e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead x0 (Bind Abbr) x1) 
-(clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 H6 x1 H7)) b H8)) c2 
-H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: 
-C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda 
-(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))))).(ex4_3_ind B C T 
-(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind 
-b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) 
-(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B 
-b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e 
-c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g 
-(CHead e (Bind b) u) e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (H3: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H4: (eq K (Bind 
-b) (Bind Void))).(\lambda (H5: (not (eq B x0 Void))).(\lambda (H6: (csubc g e 
-x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: 
-C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))) (let 
-H7 \def (f_equal K B (\lambda (e0: K).(match e0 with [(Bind b0) \Rightarrow 
-b0 | (Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void 
-(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) 
-(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda 
-(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead 
-e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) 
-(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) 
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: 
-((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) 
-(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: 
-T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x 
-\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind 
-(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: 
-C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
-A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
-(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
-(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: 
-C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: 
-C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C 
-(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e 
-c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda 
-(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
-C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) 
-u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda 
-(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c 
-e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda 
-(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: 
-C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) 
-(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c 
-x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda 
-(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) 
-H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
-A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
-(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
-(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda 
-(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
-(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 
-e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: 
-(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda 
-(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C 
-(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 
-e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) 
-(\lambda (ee: K).(match ee with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])) I (Bind Abst) H5) in (False_ind (ex2 C (\lambda (e2: 
-C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g c e2))) 
-H10)) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda (b: B).(\lambda 
-(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))))).(ex4_3_ind B C T 
-(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) 
-(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) 
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 
-(CHead x1 (Bind x0) x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda 
-(_: (not (eq B x0 Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 
-(Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) 
-(\lambda (e2: C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda 
-(ee: K).(match ee with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: C).(clear (CHead 
-x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) c2 H5)))))))) 
-H4)) H3))))))))))) c1 e1 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma
deleted file mode 100644 (file)
index 9e1d301..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/csuba.ma
+++ /dev/null
@@ -1,37 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/fwd.ma".
-
-include "basic_1/sc3/props.ma".
-
-lemma csubc_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
-c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
-(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
-(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: 
-B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: 
-T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: 
-T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g 
-c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma
deleted file mode 100644 (file)
index e1f71a3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/defs.ma
+++ /dev/null
@@ -1,30 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sc3/defs.ma".
-
-inductive csubc (g: G): C \to (C \to Prop) \def
-| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
-| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
-(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
-| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g 
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
-(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g 
-a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) 
-w))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma
deleted file mode 100644 (file)
index a0bb37e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop.ma
+++ /dev/null
@@ -1,466 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/fwd.ma".
-
-include "basic_1/sc3/props.ma".
-
-lemma csubc_drop_conf_O:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h 
-O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
-C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: 
-C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
-\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
-e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: 
-(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) 
-c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda 
-(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: 
-(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O 
-O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 
-e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: 
-C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c 
-e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: 
-C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) 
-(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: 
-C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
-\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
-e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall 
-(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 
-e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c 
-k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind 
-C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) 
-(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O 
-c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 
-(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: 
-(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) 
-c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g 
-e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: 
-C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l 
-g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C 
-c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda 
-(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
-(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) 
-(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 
-e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) 
-(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 
-(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: 
-C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: 
-C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c 
-x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop 
-(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k 
-n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C 
-(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) 
-(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: 
-C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x 
-x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) 
-O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 
-H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c 
-t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 
-C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k 
-(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda 
-(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 
-g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C 
-(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) 
-(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) 
-(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k 
-(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g 
-(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda 
-(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r 
-(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: 
-C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: 
-C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 
-e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g 
-e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 
-x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda 
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c 
-c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 
-e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda 
-(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: 
-(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C 
-(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) 
-(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) 
-(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k 
-(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g 
-(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda 
-(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r 
-(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: 
-C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: 
-C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 
-e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g 
-e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) 
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) 
-H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)).
-
-lemma drop_csubc_trans:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
-C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
-(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
-(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat 
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
-C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g 
-(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def 
-(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C 
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) 
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c 
-c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
-e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h 
-n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
-(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
-(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O 
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) 
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
-O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) 
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
-(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop 
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 
-e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
-(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
-e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda 
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C 
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
-t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) 
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) 
-c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda 
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
-C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
-(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
-(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
-e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) 
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k 
-x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: 
-C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r 
-T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def 
-(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C 
-(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 
-c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
-(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 
-(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
-(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
-a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) 
-(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda 
-(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k 
-x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) 
-(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: 
-(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda 
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r 
-k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def 
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: 
-C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) 
-(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: 
-C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c 
-x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
-(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r 
-k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k 
-n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 
-x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) 
-(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g 
-(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C 
-e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: 
-(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C 
-(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S 
-n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) 
-(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n 
-(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: 
-C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
-e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) 
-H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r 
-k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: 
-K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) 
-(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 
-\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind 
-C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C 
-(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: 
-C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) 
-(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c 
-x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) 
-x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) 
-n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 
-Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g 
-(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g 
-x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda 
-(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: 
-C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: 
-C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) 
-x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: 
-(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind 
-Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 
-x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda 
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r 
-k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: 
-nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to 
-(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: 
-C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) 
-x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda 
-(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind 
-Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 
-(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) 
-c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def 
-H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc 
-g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) 
-(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) 
-c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g 
-c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) 
-x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) 
-n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 
-x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n 
-x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) 
-(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-
-lemma csubc_drop_conf_rev:
- \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
-C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
-(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
-(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat 
-h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
-C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
-(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
-nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 
-(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
-C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def 
-(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C 
-(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) 
-e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 
-c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
-e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h 
-n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
-(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
-(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O 
-(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 
-\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) 
-(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
-O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) 
-H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
-(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop 
-(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 
-e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
-(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
-(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
-e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda 
-(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C 
-(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c 
-k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) 
-H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
-(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
-(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k 
-t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
-e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda 
-(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
-n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
-C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
-x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
-(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
-(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
-(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
-e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) 
-H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
-n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 
-k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc 
-g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h 
-(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) 
-(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def 
-(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C 
-(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
-x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
-(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 
-(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
-(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
-x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq 
-C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: 
-T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda 
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C 
-(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
-x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: 
-C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda 
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: 
-C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x 
-x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: 
-C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k 
-n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in 
-(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g 
-c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: 
-C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 
-c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
-(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r 
-k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k 
-n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
-C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) 
-(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda 
-(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind 
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 
-C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead 
-c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 
-(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g 
-(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 
-(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 
-c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let 
-H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c 
-k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 
-(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda 
-(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) 
-H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 
-(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda 
-(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc 
-g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind 
-Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: 
-C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: 
-C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 
-(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: 
-C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C 
-(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: 
-C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x 
-(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst 
-g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) 
-(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) 
-H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T 
-(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind 
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
-x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: 
-T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: 
-C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) 
-(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 
-(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: 
-(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 
-(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 
-c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let 
-H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c 
-k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 
-(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda 
-(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) 
-H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 
-(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: 
-C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 
-(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h 
-H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 
-x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) 
-c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind 
-x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n 
-x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S 
-n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind 
-x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) 
-(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n 
-x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) 
-H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma
deleted file mode 100644 (file)
index 32574ea..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/drop1.ma
+++ /dev/null
@@ -1,86 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/drop.ma".
-
-lemma drop1_csubc_trans:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 
-e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def 
-(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: 
-C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda 
-(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: 
-C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) 
-\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n 
-n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def 
-(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda 
-(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) 
-(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x 
-e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C 
-(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C 
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 
-c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g 
-x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def 
-H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: 
-C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) 
-(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 
-x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 
-n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
-
-lemma csubc_drop1_conf_rev:
- \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
-C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
-(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 
-e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def 
-(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: 
-C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 
-e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda 
-(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: 
-C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) 
-\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n 
-n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def 
-(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda 
-(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) 
-(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x 
-e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C 
-(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C 
-(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 
-c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g 
-x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 
-\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: 
-C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) 
-(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 
-x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 
-(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 
-n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma
deleted file mode 100644 (file)
index 22f68a2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/fwd.ma
+++ /dev/null
@@ -1,664 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/defs.ma".
-
-implied rec lemma csubc_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
-nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubc 
-g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (v: T).(P (CHead c1 k v) 
-(CHead c2 k v))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubc g c1 
-c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
-T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
-u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to ((P 
-c1 c2) \to (\forall (v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to 
-(\forall (w: T).((sc3 g a c2 w) \to (P (CHead c1 (Bind Abst) v) (CHead c2 
-(Bind Abbr) w)))))))))))) (c: C) (c0: C) (c1: csubc g c c0) on c1: P c c0 
-\def match c1 with [(csubc_sort n) \Rightarrow (f n) | (csubc_head c2 c3 c4 k 
-v) \Rightarrow (f0 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) k v) | 
-(csubc_void c2 c3 c4 b n u1 u2) \Rightarrow (f1 c2 c3 c4 ((csubc_ind g P f f0 
-f1 f2) c2 c3 c4) b n u1 u2) | (csubc_abst c2 c3 c4 v a s0 w s1) \Rightarrow 
-(f2 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) v a s0 w s1)].
-
-lemma csubc_gen_sort_l:
- \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to 
-(eq C x (CSort n)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g 
-(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda 
-(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g 
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) 
-(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
-(f_equal C nat (\lambda (e: C).(match e with [(CSort n1) \Rightarrow n1 | 
-(CHead _ _ _) \Rightarrow n0])) (CSort n0) (CSort n) H1) in (eq_ind_r nat n 
-(\lambda (n1: nat).(eq C (CSort n1) (CSort n1))) (refl_equal C (CSort n)) n0 
-H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 
-c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 c1)))).(\lambda (k: 
-K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) (CSort n))).(let H4 
-\def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
-(False_ind (eq C (CHead c2 k v) (CHead c1 k v)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 
-(CSort n)) \to (eq C c2 c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind 
-Void) u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c2 (Bind b) 
-u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C 
-c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
-v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
-c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) 
-w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
-
-lemma csubc_gen_head_l:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: 
-K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x 
-(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) 
-(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
-(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
-c2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: 
-K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) 
-(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: 
-C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind 
-Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead 
-c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
-(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
-c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda 
-(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda 
-(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C 
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind 
-Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 
-(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: 
-T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) 
-(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
-(CHead c1 k v) H1) in (False_ind (or3 (ex2 C (\lambda (c2: C).(eq C (CSort n) 
-(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2 (Bind Abbr) 
-w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
-(\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B 
-C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C (CSort n) (CHead 
-c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
-(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
-c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
-C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c1 c3))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: 
-(eq C (CHead c0 k0 v0) (CHead c1 k v))).(let H4 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow 
-c])) (CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
-\Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H6 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v0 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: 
-(eq K k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: 
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k0 t) (CHead c3 
-(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
-(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C 
-(CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C 
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead 
-c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc 
-g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g 
-a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
-w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
-(CHead c2 k1 v) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c1 c3))))))) (let H9 \def (eq_ind C c0 (\lambda 
-(c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 
-(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
-(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c 
-c2)) H1 c1 H8) in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
-(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) 
-w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B 
-C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) 
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c1 c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
-(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 
-k v)) H10)))) k0 H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda 
-(c2: C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k 
-v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: 
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
-T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda 
-(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: 
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead c1 k v))).(let H5 
-\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | 
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) 
-in ((let H6 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind 
-Void) u1) (CHead c1 k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind 
-Void) k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: 
-C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead 
-c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
-(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind 
-b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c 
-c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 
-(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) 
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
-T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
-(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
-Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) 
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
-T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
-(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) 
-u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 
-(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) 
-(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind 
-Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C 
-(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda 
-(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
-T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda 
-(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) 
-(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B 
-b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda 
-(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C 
-(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) 
-H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
-c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
-C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
-Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 
-(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: 
-(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 
-w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 
-\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | 
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) 
-in ((let H7 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind 
-Abst) v0) (CHead c1 k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind 
-Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda 
-(t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 
-(\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind 
-C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
-C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
-Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
-c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
-w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
-c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda 
-(c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda 
-(k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 
-(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda 
-(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
-(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) (ex4_3 B C T 
-(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
-v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
-c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or3 
-(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) 
-(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
-T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
-w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead 
-c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: 
-C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: 
-C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
-T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
-w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
-(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) 
-(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead 
-c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda 
-(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) 
-(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g 
-(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 
-g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 
-(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) 
-H)))))).
-
-lemma csubc_gen_sort_r:
- \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to 
-(eq C x (CSort n)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x 
-(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda 
-(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g 
-(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) 
-(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
-(f_equal C nat (\lambda (e: C).(match e with [(CSort n1) \Rightarrow n1 | 
-(CHead _ _ _) \Rightarrow n0])) (CSort n0) (CSort n) H1) in (eq_ind_r nat n 
-(\lambda (n1: nat).(eq C (CSort n1) (CSort n1))) (refl_equal C (CSort n)) n0 
-H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 
-c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 c2)))).(\lambda (k: 
-K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) (CSort n))).(let H4 
-\def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
-(False_ind (eq C (CHead c1 k v) (CHead c2 k v)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 
-(CSort n)) \to (eq C c1 c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind 
-b) u2) (CSort n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 (Bind Void) 
-u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C 
-c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
-v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
-c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) 
-v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
-
-lemma csubc_gen_head_r:
- \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: 
-K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x 
-(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
-C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) 
-(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
-(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
-(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind 
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))))))))
-\def
- \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: 
-K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) 
-(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: 
-C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind 
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead 
-c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
-k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
-c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda 
-(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda 
-(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C 
-T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind 
-Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
-c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead 
-c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
-k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
-c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k 
-w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort 
-_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c2 k w) H1) 
-in (False_ind (or3 (ex2 C (\lambda (c1: C).(eq C (CSort n) (CHead c1 k w))) 
-(\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: 
-T).(\lambda (_: A).(eq C (CSort n) (CHead c1 (Bind Abst) v))))) (\lambda (c1: 
-C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (c1: 
-C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda 
-(_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C (CSort n) (CHead c1 (Bind 
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) 
-(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda 
-(H2: (((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 
-(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
-(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
-(\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind 
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 
-c2))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 
-v) (CHead c2 k w))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 v) 
-(CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
-k0 v) (CHead c2 k w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 
-v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 
-c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead 
-c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) 
-(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 
-g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
-g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: 
-T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k 
-(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 
-k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind 
-Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
-c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
-c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda 
-(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 
-(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
-(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind 
-Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
-b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
-(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 
-H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) 
-in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) 
-(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
-T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
-(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) 
-(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
-(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
-c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k 
-w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) 
-H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: 
-C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) 
-\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: 
-B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead 
-c _ _) \Rightarrow c])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let H6 
-\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind 
-b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) 
-H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind b) u2) (CHead 
-c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c0 
-c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to 
-(or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b0))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H9) in (let 
-H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H9) in (let H12 
-\def (eq_ind_r K k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 
-C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 
-c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 
-(Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 
-(CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
-(asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
-c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq 
-C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda 
-(_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: 
-T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g c3 c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda 
-(k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead 
-c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: 
-C).(\lambda (v: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 
-(Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
-c3 c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) 
-c3 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
-c1 (Bind Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda 
-(_: C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: 
-C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) 
-v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
-(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
-T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind 
-Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead 
-c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K 
-(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not 
-(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
-c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K 
-(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda 
-(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k 
-w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
-A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
-T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
-(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
-B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: 
-T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: 
-T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) 
-w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind 
-Abbr) w0) (CHead c2 k w) H5) in ((let H7 \def (f_equal C K (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) 
-\Rightarrow k0])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow w0 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) 
-in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq C c0 c2)).(let H11 
-\def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) in (let H12 \def 
-(eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in (let H13 \def 
-(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C 
-(\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) 
-(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind 
-Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead 
-c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
-g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
-A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda 
-(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind 
-C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K 
-k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: 
-C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
-(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) 
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind 
-Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
-c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
-c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 
-w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C 
-c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
-(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
-c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: 
-K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 
-w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda 
-(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda 
-(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
-v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
-v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
-c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: 
-C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: 
-C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
-(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
-v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
-(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
-v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
-(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
-c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda 
-(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) 
-(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
-A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
-g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
-A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead 
-c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) 
-H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma
deleted file mode 100644 (file)
index 869c843..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/getl.ma
+++ /dev/null
@@ -1,42 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/drop.ma".
-
-include "basic_1/csubc/clear.ma".
-
-lemma csubc_getl_conf:
- \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i 
-c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
-C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda 
-(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 
-\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) 
-(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 
-x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 
-c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) 
-(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
-c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 
-H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) 
-(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 
-x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 
-e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) 
-H9)))) H7)))))) H4)))))) H1)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma
deleted file mode 100644 (file)
index a419bad..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubc/props.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/defs.ma".
-
-include "basic_1/sc3/props.ma".
-
-lemma csubc_refl:
- \forall (g: G).(\forall (c: C).(csubc g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
-(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma
deleted file mode 100644 (file)
index d70be31..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/clear.ma
+++ /dev/null
@@ -1,1170 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/props.ma".
-
-include "basic_1/csubst0/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma csubst0_clear_O:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
-(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: 
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) 
-c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda 
-(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: 
-C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) 
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(let H2 \def 
-(csubst0_gen_head k c c2 t v O H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear c2 c0) 
-(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k 
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq 
-nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (H6: 
-(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(clear c3 
-c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 
-x1)) \to (clear (CHead c k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear 
-(CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x1))).(let H9 
-\def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S 
-_) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead c (Bind b) 
-x0) c0) H9))))) (\lambda (f: F).(\lambda (H7: (clear (CHead c (Flat f) t) 
-c0)).(\lambda (H8: (eq nat O (s (Flat f) x1))).(let H9 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(subst0 n v t x0)) H6 O H8) in (clear_flat c c0 
-(clear_gen_flat f c c0 t H7) f x0))))) k H1 H4) c2 H5)))))) H3)) (\lambda 
-(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq 
-nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: 
-(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 
-c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 
-x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear 
-(CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x1))).(let H9 
-\def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S 
-_) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead x0 (Bind b) 
-t) c0) H9))))) (\lambda (f: F).(\lambda (H7: (clear (CHead c (Flat f) t) 
-c0)).(\lambda (H8: (eq nat O (s (Flat f) x1))).(let H9 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(csubst0 n v c x0)) H6 O H8) in (clear_flat x0 c0 (H x0 v 
-H9 c0 (clear_gen_flat f c c0 t H7)) f t))))) k H1 H4) c2 H5)))))) H3)) 
-(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear c2 c0) (\lambda (x0: 
-T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat O (s k 
-x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t 
-x0)).(\lambda (H7: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda 
-(c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to 
-((eq nat O (s k0 x2)) \to (clear (CHead x1 k0 x0) c0)))) (\lambda (b: 
-B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H9: (eq nat O (s 
-(Bind b) x2))).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee with 
-[O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H9) in (False_ind 
-(clear (CHead x1 (Bind b) x0) c0) H10))))) (\lambda (f: F).(\lambda (H8: 
-(clear (CHead c (Flat f) t) c0)).(\lambda (H9: (eq nat O (s (Flat f) 
-x2))).(let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 
-O H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) 
-H6 O H9) in (clear_flat x1 c0 (H x1 v H10 c0 (clear_gen_flat f c c0 t H8)) f 
-x0)))))) k H1 H4) c2 H5)))))))) H3)) H2))))))))))) c1).
-
-lemma csubst0_clear_O_back:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
-(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c 
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: 
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 
-c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: 
-C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to 
-(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O 
-v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(let H2 
-\def (csubst0_gen_head k c c2 t v O H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear (CHead c 
-k t) c0) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
-O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) 
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda 
-(x1: nat).(\lambda (H4: (eq nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead c 
-k x0))).(\lambda (H6: (subst0 x1 v t x0)).(let H7 \def (eq_ind C c2 (\lambda 
-(c3: C).(clear c3 c0)) H1 (CHead c k x0) H5) in (K_ind (\lambda (k0: K).((eq 
-nat O (s k0 x1)) \to ((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) 
-c0)))) (\lambda (b: B).(\lambda (H8: (eq nat O (s (Bind b) x1))).(\lambda (_: 
-(clear (CHead c (Bind b) x0) c0)).(let H10 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
-H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda (f: 
-F).(\lambda (H8: (eq nat O (s (Flat f) x1))).(\lambda (H9: (clear (CHead c 
-(Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n 
-v t x0)) H6 O H8) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H9) f t))))) 
-k H4 H7))))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) 
-(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat O (s k 
-x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c 
-x0)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k 
-t) H5) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 
-k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: (eq 
-nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let 
-H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True 
-| (S _) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead c (Bind 
-b) t) c0) H10))))) (\lambda (f: F).(\lambda (H8: (eq nat O (s (Flat f) 
-x1))).(\lambda (H9: (clear (CHead x0 (Flat f) t) c0)).(let H10 \def (eq_ind_r 
-nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H6 O H8) in (clear_flat c c0 (H 
-x0 v H10 c0 (clear_gen_flat f x0 c0 t H9)) f t))))) k H4 H7))))))) H3)) 
-(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear (CHead c k t) c0) (\lambda 
-(x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat O (s k 
-x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t 
-x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda 
-(c3: C).(clear c3 c0)) H1 (CHead x1 k x0) H5) in (K_ind (\lambda (k0: K).((eq 
-nat O (s k0 x2)) \to ((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) 
-c0)))) (\lambda (b: B).(\lambda (H9: (eq nat O (s (Bind b) x2))).(\lambda (_: 
-(clear (CHead x1 (Bind b) x0) c0)).(let H11 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) 
-H9) in (False_ind (clear (CHead c (Bind b) t) c0) H11))))) (\lambda (f: 
-F).(\lambda (H9: (eq nat O (s (Flat f) x2))).(\lambda (H10: (clear (CHead x1 
-(Flat f) x0) c0)).(let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n 
-v c x1)) H7 O H9) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 
-n v t x0)) H6 O H9) in (clear_flat c c0 (H x1 v H11 c0 (clear_gen_flat f x1 
-c0 x0 H10)) f t)))))) k H4 H8))))))))) H3)) H2))))))))))) c1).
-
-lemma csubst0_clear_S:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
-(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 
-B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq 
-C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v e1 e2))))))))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
-T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c 
-c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: 
-C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 
-(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: 
-nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 
-(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) 
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(let H2 \def 
-(csubst0_gen_head k c c2 t v (S i) H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (clear c2 
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H3: (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
-(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 
-(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda 
-(x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat (S i) (s k x1))).(\lambda 
-(H5: (eq C c2 (CHead c k x0))).(\lambda (H6: (subst0 x1 v t x0)).(eq_ind_r C 
-(CHead c k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda 
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-i v e1 e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to 
-((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H7: (clear (CHead c (Bind b) t) 
-c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) x1))).(let H9 \def (f_equal nat 
-nat (\lambda (e: nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) 
-(S i) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 
-n v t x0)) H6 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 
-(clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) 
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
-T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 
-(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro1 
-(clear (CHead c (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda 
-(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind 
-b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) 
-u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind 
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) 
-u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) 
-(ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) 
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H10)) c0 
-(clear_gen_bind b c c0 t H7))))))) (\lambda (f: F).(\lambda (H7: (clear 
-(CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) x1))).(let 
-H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H8) in 
-(let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H6 (S i) 
-H9) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda 
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) 
-(clear_flat c c0 (clear_gen_flat f c c0 t H7) f x0))))))) k H1 H4) c2 
-H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 
-(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C 
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: 
-nat).(\lambda (H4: (eq nat (S i) (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 
-k t))).(\lambda (H6: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda 
-(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat 
-(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda 
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) 
-u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) 
-(\lambda (b: B).(\lambda (H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: 
-(eq nat (S i) (s (Bind b) x1))).(let H9 \def (f_equal nat nat (\lambda (e: 
-nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H8) 
-in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H6 i 
-H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead 
-x0 (Bind b) t) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) 
-t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind 
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro2 (clear (CHead x0 
-(Bind b) t) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e 
-(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) 
-t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 
-(Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) 
-t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2))))) b c x0 t (refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) 
-H10)) c0 (clear_gen_bind b c c0 t H7))))))) (\lambda (f: F).(\lambda (H7: 
-(clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) 
-x1))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) 
-x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c 
-x0)) H6 (S i) H9) in (let H11 \def (H x0 v i H10 c0 (clear_gen_flat f c c0 t 
-H7)) in (or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (H12: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat 
-f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) 
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H12 f t))) 
-(\lambda (H12: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) 
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e 
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: 
-C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C c0 (CHead x3 (Bind 
-x2) x4))).(\lambda (H14: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H15: 
-(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T 
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) 
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) 
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H13 (clear_flat x0 
-(CHead x3 (Bind x2) x5) H14 f t) H15))))))))) H12)) (\lambda (H12: (ex3_4 B C 
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T 
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) 
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (H13: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H14: (clear x0 
-(CHead x4 (Bind x2) x5))).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro2 
-(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C 
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2))))) x2 x3 x4 x5 H13 (clear_flat x0 (CHead x4 (Bind x2) x5) H14 f t) 
-H15))))))))) H12)) (\lambda (H12: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H13: (eq C c0 (CHead x3 (Bind x2) 
-x5))).(\lambda (H14: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H15: 
-(subst0 i v x5 x6)).(\lambda (H16: (csubst0 i v x3 x4)).(or4_intro3 (clear 
-(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H13 (clear_flat x0 
-(CHead x4 (Bind x2) x6) H14 f t) H15 H16))))))))))) H12)) H11))))))) k H1 H4) 
-c2 H5)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda 
-(x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat (S i) (s k x2))).(\lambda 
-(H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda 
-(H7: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 
-(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind 
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to 
-(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda 
-(H8: (clear (CHead c (Bind b) t) c0)).(\lambda (H9: (eq nat (S i) (s (Bind b) 
-x2))).(let H10 \def (f_equal nat nat (\lambda (e: nat).(match e with [O 
-\Rightarrow i | (S n) \Rightarrow n])) (S i) (S x2) H9) in (let H11 \def 
-(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 i H10) in (let H12 
-\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H6 i H10) in 
-(eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x1 (Bind 
-b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) 
-x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind 
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 (clear (CHead x1 
-(Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e 
-(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
-(CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
-C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
-C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) b c x1 t x0 
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) H12 H11)) c0 
-(clear_gen_bind b c c0 t H8)))))))) (\lambda (f: F).(\lambda (H8: (clear 
-(CHead c (Flat f) t) c0)).(\lambda (H9: (eq nat (S i) (s (Flat f) x2))).(let 
-H10 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x2) H9) in 
-(let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 (S i) 
-H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) 
-H6 (S i) H10) in (let H13 \def (H x1 v i H11 c0 (clear_gen_flat f c c0 t H8)) 
-in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (H14: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat 
-f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) 
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H14 f x0))) 
-(\lambda (H14: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind 
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) 
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e 
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: 
-C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C c0 (CHead x4 (Bind 
-x3) x5))).(\lambda (H16: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H17: 
-(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C 
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) 
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) 
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H15 (clear_flat x1 
-(CHead x4 (Bind x3) x6) H16 f x0) H17))))))))) H14)) (\lambda (H14: (ex3_4 B 
-C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
-c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C 
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) 
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
-T).(\lambda (H15: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H16: (clear x1 
-(CHead x5 (Bind x3) x6))).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro2 
-(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C 
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2))))) x3 x4 x5 x6 H15 (clear_flat x1 (CHead x5 (Bind x3) x6) H16 f x0) 
-H17))))))))) H14)) (\lambda (H14: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) 
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H15: (eq C c0 (CHead x4 (Bind x3) 
-x6))).(\lambda (H16: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H17: 
-(subst0 i v x6 x7)).(\lambda (H18: (csubst0 i v x4 x5)).(or4_intro3 (clear 
-(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda 
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H15 (clear_flat x1 
-(CHead x5 (Bind x3) x7) H16 f x0) H17 H18))))))))))) H14)) H13)))))))) k H1 
-H4) c2 H5)))))))) H3)) H2)))))))))))) c1).
-
-lemma csubst0_clear_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
-i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C 
-(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
-T).(\forall (i: nat).((csubst0 i v c c2) \to (\forall (e2: C).((clear c2 e2) 
-\to (or (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda 
-(e1: C).(clear c e1))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CSort n) c2)).(\lambda 
-(e2: C).(\lambda (_: (clear c2 e2)).(csubst0_gen_sort c2 v i n H (or (clear 
-(CSort n) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: 
-C).(clear (CSort n) e1)))))))))))) (\lambda (c: C).(\lambda (H: ((\forall 
-(c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 i v c c2) \to (\forall 
-(e2: C).((clear c2 e2) \to (or (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 
-i v e1 e2)) (\lambda (e1: C).(clear c e1)))))))))))).(\lambda (k: K).(\lambda 
-(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: 
-(csubst0 i v (CHead c k t) c2)).(\lambda (e2: C).(\lambda (H1: (clear c2 
-e2)).(let H2 \def (csubst0_gen_head k c c2 t v i H0) in (or3_ind (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat 
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3))))) (or (clear (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 
-e2)) (\lambda (e1: C).(clear (CHead c k t) e1)))) (\lambda (H3: (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or (clear 
-(CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: 
-C).(clear (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda 
-(H4: (eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda 
-(H6: (subst0 x1 v t x0)).(eq_ind_r nat (s k x1) (\lambda (n: nat).(or (clear 
-(CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 n v e1 e2)) (\lambda (e1: 
-C).(clear (CHead c k t) e1))))) (let H7 \def (eq_ind C c2 (\lambda (c0: 
-C).(clear c0 e2)) H1 (CHead c k x0) H5) in (K_ind (\lambda (k0: K).((clear 
-(CHead c k0 x0) e2) \to (or (clear (CHead c k0 t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (s k0 x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c k0 t) 
-e1)))))) (\lambda (b: B).(\lambda (H8: (clear (CHead c (Bind b) x0) 
-e2)).(eq_ind_r C (CHead c (Bind b) x0) (\lambda (c0: C).(or (clear (CHead c 
-(Bind b) t) c0) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 c0)) 
-(\lambda (e1: C).(clear (CHead c (Bind b) t) e1))))) (or_intror (clear (CHead 
-c (Bind b) t) (CHead c (Bind b) x0)) (ex2 C (\lambda (e1: C).(csubst0 (s 
-(Bind b) x1) v e1 (CHead c (Bind b) x0))) (\lambda (e1: C).(clear (CHead c 
-(Bind b) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 
-(CHead c (Bind b) x0))) (\lambda (e1: C).(clear (CHead c (Bind b) t) e1)) 
-(CHead c (Bind b) t) (csubst0_snd (Bind b) x1 v t x0 H6 c) (clear_bind b c 
-t))) e2 (clear_gen_bind b c e2 x0 H8)))) (\lambda (f: F).(\lambda (H8: (clear 
-(CHead c (Flat f) x0) e2)).(or_introl (clear (CHead c (Flat f) t) e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear 
-(CHead c (Flat f) t) e1))) (clear_flat c e2 (clear_gen_flat f c e2 x0 H8) f 
-t)))) k H7)) i H4)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or (clear (CHead c k t) e2) 
-(ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear (CHead c 
-k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s k 
-x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c 
-x0)).(eq_ind_r nat (s k x1) (\lambda (n: nat).(or (clear (CHead c k t) e2) 
-(ex2 C (\lambda (e1: C).(csubst0 n v e1 e2)) (\lambda (e1: C).(clear (CHead c 
-k t) e1))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(clear c0 e2)) H1 
-(CHead x0 k t) H5) in (K_ind (\lambda (k0: K).((clear (CHead x0 k0 t) e2) \to 
-(or (clear (CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 x1) v e1 
-e2)) (\lambda (e1: C).(clear (CHead c k0 t) e1)))))) (\lambda (b: B).(\lambda 
-(H8: (clear (CHead x0 (Bind b) t) e2)).(eq_ind_r C (CHead x0 (Bind b) t) 
-(\lambda (c0: C).(or (clear (CHead c (Bind b) t) c0) (ex2 C (\lambda (e1: 
-C).(csubst0 (s (Bind b) x1) v e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind 
-b) t) e1))))) (or_intror (clear (CHead c (Bind b) t) (CHead x0 (Bind b) t)) 
-(ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 (CHead x0 (Bind b) t))) 
-(\lambda (e1: C).(clear (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: 
-C).(csubst0 (s (Bind b) x1) v e1 (CHead x0 (Bind b) t))) (\lambda (e1: 
-C).(clear (CHead c (Bind b) t) e1)) (CHead c (Bind b) t) (csubst0_fst (Bind 
-b) x1 c x0 v H6 t) (clear_bind b c t))) e2 (clear_gen_bind b x0 e2 t H8)))) 
-(\lambda (f: F).(\lambda (H8: (clear (CHead x0 (Flat f) t) e2)).(let H_x \def 
-(H x0 v x1 H6 e2 (clear_gen_flat f x0 e2 t H8)) in (let H9 \def H_x in 
-(or_ind (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 x1 v e1 e2)) (\lambda 
-(e1: C).(clear c e1))) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda 
-(e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c 
-(Flat f) t) e1)))) (\lambda (H10: (clear c e2)).(or_introl (clear (CHead c 
-(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) 
-(\lambda (e1: C).(clear (CHead c (Flat f) t) e1))) (clear_flat c e2 H10 f 
-t))) (\lambda (H10: (ex2 C (\lambda (e1: C).(csubst0 x1 v e1 e2)) (\lambda 
-(e1: C).(clear c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 x1 v e1 e2)) 
-(\lambda (e1: C).(clear c e1)) (or (clear (CHead c (Flat f) t) e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear 
-(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H11: (csubst0 x1 v x 
-e2)).(\lambda (H12: (clear c x)).(or_intror (clear (CHead c (Flat f) t) e2) 
-(ex2 C (\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: 
-C).(clear (CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 
-(s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)) x 
-H11 (clear_flat c x H12 f t)))))) H10)) H9))))) k H7)) i H4)))))) H3)) 
-(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (clear (CHead c k t) e2) (ex2 
-C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear (CHead c k t) 
-e1)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq 
-nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: 
-(subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c x1)).(eq_ind_r nat (s k x2) 
-(\lambda (n: nat).(or (clear (CHead c k t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 n v e1 e2)) (\lambda (e1: C).(clear (CHead c k t) e1))))) (let H8 
-\def (eq_ind C c2 (\lambda (c0: C).(clear c0 e2)) H1 (CHead x1 k x0) H5) in 
-(K_ind (\lambda (k0: K).((clear (CHead x1 k0 x0) e2) \to (or (clear (CHead c 
-k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 x2) v e1 e2)) (\lambda (e1: 
-C).(clear (CHead c k0 t) e1)))))) (\lambda (b: B).(\lambda (H9: (clear (CHead 
-x1 (Bind b) x0) e2)).(eq_ind_r C (CHead x1 (Bind b) x0) (\lambda (c0: C).(or 
-(clear (CHead c (Bind b) t) c0) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) 
-x2) v e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind b) t) e1))))) (or_intror 
-(clear (CHead c (Bind b) t) (CHead x1 (Bind b) x0)) (ex2 C (\lambda (e1: 
-C).(csubst0 (s (Bind b) x2) v e1 (CHead x1 (Bind b) x0))) (\lambda (e1: 
-C).(clear (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 
-(s (Bind b) x2) v e1 (CHead x1 (Bind b) x0))) (\lambda (e1: C).(clear (CHead 
-c (Bind b) t) e1)) (CHead c (Bind b) t) (csubst0_both (Bind b) x2 v t x0 H6 c 
-x1 H7) (clear_bind b c t))) e2 (clear_gen_bind b x1 e2 x0 H9)))) (\lambda (f: 
-F).(\lambda (H9: (clear (CHead x1 (Flat f) x0) e2)).(let H_x \def (H x1 v x2 
-H7 e2 (clear_gen_flat f x1 e2 x0 H9)) in (let H10 \def H_x in (or_ind (clear 
-c e2) (ex2 C (\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: C).(clear c 
-e1))) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s 
-(Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)))) 
-(\lambda (H11: (clear c e2)).(or_introl (clear (CHead c (Flat f) t) e2) (ex2 
-C (\lambda (e1: C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear 
-(CHead c (Flat f) t) e1))) (clear_flat c e2 H11 f t))) (\lambda (H11: (ex2 C 
-(\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: C).(clear c 
-e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: 
-C).(clear c e1)) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat 
-f) t) e1)))) (\lambda (x: C).(\lambda (H12: (csubst0 x2 v x e2)).(\lambda 
-(H13: (clear c x)).(or_intror (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda 
-(e1: C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c 
-(Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (s (Flat f) x2) v e1 
-e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)) x H12 (clear_flat c x 
-H13 f t)))))) H11)) H10))))) k H8)) i H4)))))))) H3)) H2)))))))))))) c1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma
deleted file mode 100644 (file)
index eec1e74..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/defs.ma
+++ /dev/null
@@ -1,32 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/defs.ma".
-
-include "basic_1/C/defs.ma".
-
-inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
-| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: 
-T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) 
-v (CHead c k u1) (CHead c k u2))))))))
-| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s 
-k i) v (CHead c1 k u) (CHead c2 k u))))))))
-| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall 
-(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall 
-(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 
-k u2)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma
deleted file mode 100644 (file)
index 2d75924..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/drop.ma
+++ /dev/null
@@ -1,6291 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/fwd.ma".
-
-include "basic_1/drop/fwd.ma".
-
-lemma csubst0_drop_gt:
- \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
-c1 e) \to (drop n O c2 e)))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) 
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
-\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda 
-(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
-O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall 
-(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop 
-n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda 
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
-c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) 
-(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v 
-(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) 
-e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) 
-O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) 
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop 
-(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee 
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
-(drop (S n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e 
-H2)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S 
-n0) O c2 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda 
-(H3: (drop (S n0) O (CHead c k t) e)).(let H4 \def (csubst0_gen_head k c c2 t 
-v i H2) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i 
-(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) 
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda 
-(_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H5: (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S 
-n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k 
-x1))).(\lambda (H7: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t 
-x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let 
-H9 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop 
-(S n0) O c3 e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda 
-(n1: nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop 
-(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) 
-v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) 
-e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda 
-(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
-(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c 
-e H11 x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c 
-e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) 
-v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) 
-(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
-(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 
-O)).(drop_drop (Flat f) n0 c e H11 x0)) (\lambda (H14: (ex2 nat (\lambda (m: 
-nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
-(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H11 x0)))) H14)) 
-(lt_gen_xS x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2 
-H7)))))) H5)) (\lambda (H5: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda 
-(x0: C).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7: 
-(eq C c2 (CHead x0 k t))).(\lambda (H8: (csubst0 x1 v c x0)).(eq_ind_r C 
-(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind 
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
-c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: 
-nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop (r k0 
-n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c 
-c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) 
-e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda 
-(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
-(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0)))))))).(\lambda (H13: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 
-x0 e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H11) t))))) (\lambda (f: 
-F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3: 
-C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt 
-(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq 
-nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat 
-f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8 
-e H11) t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
-(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
-m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) 
-(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
-n0)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8 e H11) t)))) H14)) (lt_gen_xS 
-x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H7)))))) H5)) 
-(\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: 
-T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H6: (eq nat i (s k 
-x2))).(\lambda (H7: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t 
-x0)).(\lambda (H9: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda 
-(c0: C).(drop (S n0) O c0 e)) (let H10 \def (eq_ind nat i (\lambda (n1: 
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H6) 
-in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
-H6) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: 
-C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop 
-(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to 
-(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H12: (drop 
-(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: 
-T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c 
-e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H14: (lt (s (Bind b) x2) (S 
-n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H12) 
-x0))))) (\lambda (f: F).(\lambda (H12: (drop (r (Flat f) n0) O c e)).(\lambda 
-(H13: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) 
-\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
-e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) 
-(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
-(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 
-O)).(drop_drop (Flat f) n0 x1 e (H13 x1 v H9 e H12) x0)) (\lambda (H15: (ex2 
-nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m 
-n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: 
-nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: 
-nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat 
-f) n0 x1 e (H13 x1 v H9 e H12) x0)))) H15)) (lt_gen_xS x2 n0 H14)))))) k 
-(drop_gen_drop k c e t n0 H3) H10 H11))) c2 H7)))))))) H5)) H4))))))))))) 
-c1)))))) n).
-
-lemma csubst0_drop_gt_back:
- \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
-c2 e) \to (drop n O c1 e)))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) 
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
-\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda 
-(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
-O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall 
-(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop 
-n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda 
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
-c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
-(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i 
-v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 
-e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) 
-(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v 
-c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c 
-e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: 
-T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda 
-(H3: (drop (S n0) O c2 e)).(let H4 \def (csubst0_gen_head k c c2 t v i H2) in 
-(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) 
-(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) (\lambda (H5: 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda 
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead c k 
-x0))).(\lambda (_: (subst0 x1 v t x0)).(let H9 \def (eq_ind C c2 (\lambda 
-(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H7) in (let H10 \def (eq_ind 
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
-c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c 
-e0))))))) H1 (s k x1) H6) in (let H11 \def (eq_ind nat i (\lambda (n1: 
-nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).(((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) 
-(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) 
-(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
-(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
-(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda 
-(H14: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H14 t))))) 
-(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
-(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
-(S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(\lambda 
-(H14: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda 
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O 
-(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c 
-e H14 t)) (\lambda (H15: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
-(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
-m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) 
-(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
-n0)).(drop_drop (Flat f) n0 c e H14 t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k 
-H10 H11 (drop_gen_drop k c e x0 n0 H9)))))))))) H5)) (\lambda (H5: (ex3_2 C 
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S 
-n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H6: (eq 
-nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead x0 k t))).(\lambda (H8: 
-(csubst0 x1 v c x0)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) 
-O c0 e)) H3 (CHead x0 k t) H7) in (let H10 \def (eq_ind nat i (\lambda (n1: 
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H6) 
-in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) 
-H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
-(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
-n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to 
-(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt 
-(s (Bind b) x1) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x0 
-e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H14) 
-t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: 
-T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
-e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S 
-n0))).(\lambda (H14: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) 
-(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
-(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop 
-(Flat f) n0 c e (H12 x0 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: 
-nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
-(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
-(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x0 v H8 e H14) 
-t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k H10 H11 (drop_gen_drop k x0 e t n0 
-H9)))))))))) H5)) (\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O 
-(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
-nat).(\lambda (H6: (eq nat i (s k x2))).(\lambda (H7: (eq C c2 (CHead x1 k 
-x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H9: (csubst0 x2 v c 
-x1)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 
-(CHead x1 k x0) H7) in (let H11 \def (eq_ind nat i (\lambda (n1: 
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H6) 
-in (let H12 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
-H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
-(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
-n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to 
-(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt 
-(s (Bind b) x2) (S n0))).(\lambda (H15: (drop (r (Bind b) n0) O x1 
-e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H15) 
-t))))) (\lambda (f: F).(\lambda (H13: ((\forall (c3: C).(\forall (v0: 
-T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
-e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S 
-n0))).(\lambda (H15: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) 
-(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
-(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop 
-(Flat f) n0 c e (H13 x1 v H9 e H15) t)) (\lambda (H16: (ex2 nat (\lambda (m: 
-nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
-(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
-(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S 
-x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H13 x1 v H9 e H15) 
-t)))) H16)) (lt_gen_xS x2 n0 H14)))))) k H11 H12 (drop_gen_drop k x1 e x0 n0 
-H10)))))))))))) H5)) H4))))))))))) c1)))))) n).
-
-lemma csubst0_drop_lt:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
-c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k 
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k 
-n)) v e1 e2))))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) 
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
-\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T 
-T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: 
-nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: 
-T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O 
-c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
-c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: 
-nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
-c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: 
-K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r 
-nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) 
-(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s 
-k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C 
-(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: 
-K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
-T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 
-O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 
-O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda 
-(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) 
-in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s 
-k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead 
-e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 
-(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in 
-(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k 
-u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k 
-i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
-u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
-v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 
-u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k 
-i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
-u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
-v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k 
-u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 
-u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 
-c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 
-O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 
-e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 
-u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r 
-nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) 
-(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k 
-u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s 
-k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k 
-u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u 
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 
-u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 
-H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda 
-(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: 
-C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: 
-C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead 
-e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: 
-(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
-C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c 
-e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: 
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) 
-c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind 
-(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S 
-n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq 
-nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: 
-C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind 
-nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H3) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CSort n1) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 k u)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CSort n1) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (s k (S n0))) v e1 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) 
-O e H1)))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 
-(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 
-(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (s k (S n0))) v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) 
-c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CHead c k t) e)).(let H3 
-\def (csubst0_gen_head k c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
-n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) 
-(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: 
-(eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: 
-(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S 
-n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let 
-H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 
-(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
-(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C 
-T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i 
-(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
-(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) 
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 
-e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c 
-k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead 
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
-(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) 
-x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2))))))) (drop_drop (Bind b) n0 c e H10 x0)))))) (\lambda (f: F).(\lambda 
-(H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall 
-(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) 
-O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
-x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2))))))) (drop_drop (Flat f) n0 c e H10 x0)))))) k (drop_gen_drop k c e t n0 
-H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: 
-(eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: 
-(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop 
-(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let 
-H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
-T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 
-(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
-(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C 
-T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i 
-(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
-(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) 
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 
-e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 
-k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead 
-e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
-(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) 
-x1))).(let H13 \def (IHn x1 (le_S_n (S n0) x1 H12) c x0 v H7 e H10) in 
-(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 
-n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 
-(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x0 
-e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2))))))) (drop_drop (Bind b) n0 x0 e H14 t))) (\lambda (H14: (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 
-n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0 
-(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x4))).(\lambda (H16: 
-(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H17: (subst0 (minus x1 (s x2 n0)) 
-v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
-x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) 
-(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H16 t) (eq_ind_r nat (S (s x2 n0)) 
-(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H17 (s x2 (S 
-n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 
-n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 
-k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0 
-(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 
-C).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
-(drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H17: (csubst0 (minus x1 (s x2 n0)) 
-v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 
-x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H16 t) (eq_ind_r nat (S (s x2 
-n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s 
-x2 (S n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 
-k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T 
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
-(drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H17: (subst0 (minus x1 (s x2 n0)) 
-v x5 x6)).(\lambda (H18: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C 
-(CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) 
-c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
-(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6 
-(refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6) 
-H16 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind 
-b) x1) n1) v x5 x6)) H17 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2 
-n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H18 (s 
-x2 (S n0)) (s_S x2 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f: 
-F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: 
-C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
-x1))).(let H13 \def (H11 x0 v H7 e H10) in (or4_ind (drop (S n0) O x0 e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 
-(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))))) (\lambda (H14: (drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O 
-(CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x0 e H14 
-t))) (\lambda (H14: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x1 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S n0))) v u 
-w))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: 
-C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 
-x4))).(\lambda (H16: (drop (S n0) O x0 (CHead x3 x2 x5))).(\lambda (H17: 
-(subst0 (minus x1 (s x2 (S n0))) v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 x2 
-x4)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x3 x2 x4) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w))))) 
-x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) (drop_drop (Flat f) n0 x0 (CHead 
-x3 x2 x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 
-(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead 
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead 
-x0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 
-C).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
-(drop (S n0) O x0 (CHead x4 x2 x5))).(\lambda (H17: (csubst0 (minus x1 (s x2 
-(S n0))) v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop 
-(S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 
-(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 
-x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H16 t) H17)) e H15)))))))) 
-H14)) (\lambda (H14: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 
-(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 
-(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 K 
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
-(drop (S n0) O x0 (CHead x4 x2 x6))).(\lambda (H17: (subst0 (minus x1 (s x2 
-(S n0))) v x5 x6)).(\lambda (H18: (csubst0 (minus x1 (s x2 (S n0))) v x3 
-x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead 
-x0 (Flat f) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
-(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
-e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f) 
-n0 x0 (CHead x4 x2 x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))) k 
-(drop_gen_drop k c e t n0 H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4: 
-(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s 
-k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
-nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k 
-x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c 
-x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
-(S n0))) v e1 e2))))))))) (let H9 \def (eq_ind nat i (\lambda (n1: 
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S 
-n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
-(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H5) in (let H10 \def 
-(eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r 
-nat (s k x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 
-K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
-(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
-(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall 
-(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) 
-(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 
-u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 
-e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 
-k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) 
-(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 
-K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 
-k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
-(b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b) 
-x2))).(let H14 \def (IHn x2 (le_S_n (S n0) x2 H13) c x1 v H8 e H11) in 
-(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 
-n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4 
-(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H15: (drop n0 O x1 
-e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2))))))) (drop_drop (Bind b) n0 x1 e H15 x0))) (\lambda (H15: (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 
-n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
-k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 
-(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x5))).(\lambda (H17: 
-(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H18: (subst0 (minus x2 (s x3 n0)) 
-v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 
-x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) 
-(drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H17 x0) (eq_ind_r nat (S (s x3 
-n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H18 (s 
-x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 
-n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 
-k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 
-(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
-C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
-(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) 
-v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 
-x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 
-x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H17 x0) (eq_ind_r nat (S (s 
-x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18 
-(s x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead 
-e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T 
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
-u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
-(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H18: (subst0 (minus x2 (s x3 n0)) 
-v x6 x7)).(\lambda (H19: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C 
-(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) 
-c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
-(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 
-x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 
-x7) H17 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s 
-(Bind b) x2) n1) v x6 x7)) H18 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s 
-x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H19 
-(s x3 (S n0)) (s_S x3 n0)))) e H16)))))))))) H15)) H14)))))) (\lambda (f: 
-F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3: 
-C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: 
-C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 
-(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
-x2))).(let H14 \def (H12 x1 v H8 e H11) in (or4_ind (drop (S n0) O x1 e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 
-(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) 
-(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))))) (\lambda (H15: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O 
-(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H15 
-x0))) (\lambda (H15: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead 
-e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u 
-w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda 
-(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: 
-C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 
-x5))).(\lambda (H17: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H18: 
-(subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T 
-T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 
-x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
-(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
-w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1 
-(CHead x4 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C 
-C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 
-(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead 
-e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead 
-x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
-T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 
-(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
-C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
-(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3 
-(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop 
-(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 
-(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) 
-(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K 
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
-(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 
-(CHead x5 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C 
-C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u 
-w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 
-e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 
-(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
-u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
-(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: 
-C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H16: (eq C e 
-(CHead x4 x3 x6))).(\lambda (H17: (drop (S n0) O x1 (CHead x5 x3 
-x7))).(\lambda (H18: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda 
-(H19: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 
-x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K 
-C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 
-k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
-n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 
-x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
-(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
-u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 
-x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 
-x7) H17 x0) H18 H19)) e H16)))))))))) H15)) H14)))))) k (drop_gen_drop k c e 
-t n0 H2) H9 H10) i H5))) c2 H6)))))))) H4)) H3))))))))))) c1)))))) n).
-
-lemma csubst0_drop_eq:
- \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
-n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 
-e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
-c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 
-e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
-c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) 
-(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 
-y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: 
-C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 
-t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: 
-K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: 
-T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 
-(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
-c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) 
-u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s 
-k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) 
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) 
-(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda 
-(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 
-u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat 
-(S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c 
-(Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) 
-v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) 
-(S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 
-e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 
-u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i 
-(\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O (\lambda 
-(n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) u1)) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c 
-(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
-(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c 
-(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) 
-u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat 
-f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
-(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) 
-u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) 
-(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: 
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: 
-C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop 
-i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall 
-(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) 
-(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 
-i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 
-i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) 
-u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 
-e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 
-(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-(CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) 
-u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) 
-u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat 
-f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 
-(Bind b) u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead 
-c4 (Bind b) u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 
-w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: 
-F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
-T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 
-(drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 
-(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i 
-c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda 
-(u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0: 
-nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 
-(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 
-w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H3 O H4) in (let H6 \def 
-(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r 
-nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 
-(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 
-(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C 
-T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) 
-(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 
-(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) 
-u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 
-(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O 
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F 
-C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq 
-C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 
-(Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u 
-(refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4 (Flat f) u)) H6)) i 
-H4)))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: 
-nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) 
-\to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i 
-O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop i i c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T 
-T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
-e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead 
-c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e0 (Flat f) 
-u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 
-i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda 
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) (s k0 
-i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F 
-C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
-e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 
-u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False 
-| (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i) 
-(CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) 
-u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat 
-f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
-i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 
-e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 
-u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 
-c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
-\def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 
-c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 
-u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 
-(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in 
-(let H7 \def (eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) 
-in (let H8 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O 
-H5) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) 
-u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 
-u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F 
-C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 
-(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) 
-u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) 
-u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) 
-u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) 
-(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 
-H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to 
-(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda 
-(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall 
-(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: 
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort 
-n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) 
-e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop 
-(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) 
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 
-(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with 
-[O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (or4 
-(drop (S n0) O c2 (CSort n1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 (Flat f) 
-u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort 
-n1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda 
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop 
-(S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e 
-H0)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 
-(drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: 
-C).(\lambda (H1: (drop (S n0) O (CHead c k t) e)).(let H2 \def 
-(csubst0_gen_head k c c2 t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
-n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
-nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T 
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat 
-(S n0) (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (H6: 
-(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S 
-n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
-n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
-n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: 
-nat).(subst0 n1 v t x0)) H6 n0 H9) in (or4_intro0 (drop (S n0) O (CHead c 
-(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c 
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(drop_drop (Bind b) n0 c e H7 x0))))))) (\lambda (f: F).(\lambda (H7: (drop 
-(r (Flat f) n0) O c e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(let 
-H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in 
-(let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S 
-n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H7 
-x0))))))) k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3: 
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0) 
-(s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 
-v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 
-e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
-n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
-n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: 
-nat).(csubst0 n1 v c x0)) H6 n0 H9) in (let H11 \def (IHn c x0 v H10 e H7) in 
-(or4_ind (drop n0 O x0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
-x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (H12: (drop n0 O x0 e)).(or4_intro0 (drop (S n0) O 
-(CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(drop_drop (Bind b) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 
-(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop 
-n0 O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4 
-x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x0 (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat 
-x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
-x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4)) 
-(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) 
-H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
-x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0 
-(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O 
-x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C 
-(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind 
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat 
-x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) 
-x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) 
-(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H14 t) H15)) e H13)))))))) 
-H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) 
-O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda 
-(H14: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v x5 
-x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
-(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
-x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat 
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat 
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 
-x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) 
-(\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c e)).(\lambda (H8: (eq 
-nat (S n0) (s (Flat f) x1))).(let H9 \def (f_equal nat nat (\lambda (e0: 
-nat).e0) (S n0) (s (Flat f) x1) H8) in (let H10 \def (eq_ind_r nat x1 
-(\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H9) in (let H11 \def (H x0 
-v H10 e H7) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: 
-(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat 
-f) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) 
-O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop 
-(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4 
-x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 
-(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 
-(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15)) 
-e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
-C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda 
-(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop 
-(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 
-x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 
-(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
-(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead 
-x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H14 t) 
-H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 
-(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda 
-(H14: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v 
-x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) 
-x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
-(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
-x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat 
-f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat 
-f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat 
-f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 
-x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) 
-k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3: (ex4_3 T C 
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
-C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F 
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: 
-C).(\lambda (x2: nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5: 
-(eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7: 
-(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop 
-(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
-n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H8: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H9: (eq nat (S n0) (s (Bind b) x2))).(let H10 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
-n1])) (S n0) (S x2) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: 
-nat).(csubst0 n1 v c x1)) H7 n0 H10) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H10) in (let H13 \def (IHn c x1 
-v H11 e H8) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O x1 
-e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H14 
-x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
-x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind 
-b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop n0 O 
-x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C 
-(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind 
-b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat 
-x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) 
-x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) 
-(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H16 x0) H17)) e H15)))))))) 
-H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
-x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 
-(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
-T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O 
-x1 (CHead x5 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C 
-(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind 
-b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
-f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat 
-x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) 
-x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) 
-(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H16 x0) H17)) e H15)))))))) 
-H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) 
-O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda 
-(H16: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H17: (subst0 O v x6 
-x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T 
-T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
-O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
-(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) 
-x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat 
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat 
-f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 
-x1 (CHead x5 (Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) 
-H13)))))))) (\lambda (f: F).(\lambda (H8: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(let H10 \def (f_equal nat 
-nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H9) in (let H11 \def 
-(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H10) in 
-(let H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S 
-n0) H10) in (let H13 \def (H x1 v H11 e H8) in (or4_ind (drop (S n0) O x1 e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop (S n0) O 
-x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14 
-x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) 
-O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda 
-(x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop 
-(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5 
-x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead 
-x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) 
-u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 
-(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H16 x0) 
-H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind 
-F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: 
-C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat 
-x3) x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda 
-(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat 
-f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) 
-u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 
-(CHead x5 (Flat x3) x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
-(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) 
-x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda 
-(H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C 
-(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat 
-f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
-f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat 
-x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
-(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C 
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 
-(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 
-(Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))))) k 
-(drop_gen_drop k c e t n0 H1) H4) c2 H5)))))))) H3)) H2))))))))))) c1)))) n).
-
-lemma csubst0_drop_eq_back:
- \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
-n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 
-e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 
-e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O 
-O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) 
-(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda 
-(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: 
-T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 
-t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda 
-(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall 
-(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s 
-k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: 
-T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 
-u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: 
-T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda 
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 
-u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 
-u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: 
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 
-i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def 
-(eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c 
-(Bind b) u1) (CHead c (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead 
-e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: 
-T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e0 (Flat f) 
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
-T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Bind b) u2) (CHead e2 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
-i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead 
-e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1) 
-(CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: 
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 
-i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind 
-nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O 
-(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u1) (CHead c (Flat f) 
-u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 
-n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 
-u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 
-(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F 
-C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u3))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) 
-(or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C 
-(CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) 
-(CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: 
-T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) 
-u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) 
-u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) 
-(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 
-(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
-T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) 
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
-T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) 
-(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: 
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: 
-C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop 
-i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to 
-(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead 
-c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) 
-u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s 
-k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 
-i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) 
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 
-e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
-(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 
-(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) 
-u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) 
-u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat 
-f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 
-(Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead 
-c3 (Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: 
-F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
-T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 
-(drop i i c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 
-(Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i 
-c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
-e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def 
-(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 
-n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
-C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 
-e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: 
-nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: 
-nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) 
-u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) 
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) 
-u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 
-(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O 
-(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C 
-C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C 
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) 
-(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 
-(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) 
-u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) 
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) 
-(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: 
-K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: 
-T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: 
-C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
-C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop 
-(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 
-u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 
-(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat 
-f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 
-u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 
-u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: 
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 
-i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 
-c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 
-(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: 
-T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
-\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False 
-| (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i) 
-(CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) 
-u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e0 
-(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda 
-(u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead 
-e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
-i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead 
-e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) 
-(CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda 
-(i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: 
-(subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 
-i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F 
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
-C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 
-u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 
-(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 
-(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) 
-u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: 
-T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
-e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda 
-(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e0 
-(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
-T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 
-u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c3 (CHead e1 
-(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 
-(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) 
-u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: 
-T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 
-e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: 
-nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda 
-(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: 
-nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F 
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
-C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 
-(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) 
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C 
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 
-n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 
-n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O 
-(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) 
-u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) 
-u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
-T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
-(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead 
-e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) 
-u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) 
-(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) 
-(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 
-H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to 
-(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind 
-(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to 
-(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: 
-nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort 
-n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort 
-c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat 
-f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) 
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S 
-n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O 
-c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: 
-C).(\lambda (H1: (drop (S n0) O c2 e)).(let H2 \def (csubst0_gen_head k c c2 
-t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
-nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: 
-T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
-n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H3: (ex3_2 T nat 
-(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
-(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 
-(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda 
-(x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda (H5: (eq C c2 
-(CHead c k x0))).(\lambda (H6: (subst0 x1 v t x0)).(let H7 \def (eq_ind C c2 
-(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H5) in (K_ind 
-(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to 
-(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: 
-B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9: (drop (r 
-(Bind b) n0) O c e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match 
-e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H8) in 
-(let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 
-H10) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e 
-H9 t))))))) (\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) 
-x1))).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(let H10 \def (f_equal nat 
-nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def 
-(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H10) in 
-(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e 
-H9 t))))))) k H4 (drop_gen_drop k c e x0 n0 H7)))))))) H3)) (\lambda (H3: 
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
-v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda 
-(H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c x0)).(let H7 
-\def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead x0 k t) 
-H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 
-n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) 
-(\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9: 
-(drop (r (Bind b) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S 
-x1) H8) in (let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c 
-x0)) H6 n0 H10) in (let H12 \def (IHn c x0 v H11 e H9) in (or4_ind (drop n0 O 
-c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c 
-(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (H13: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead 
-c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(drop_drop (Bind b) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
-(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 (Flat x2) 
-x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H16: 
-(subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0 
-c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C (CHead x3 (Flat x2) x5) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
-(CHead c (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat 
-x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
-f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
-x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c 
-(CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda (H13: 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H14: (eq 
-C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) 
-x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let H17 \def (eq_ind C e (\lambda 
-(c0: C).(drop n0 O x0 c0)) H9 (CHead x4 (Flat x2) x5) H14) in (eq_ind_r C 
-(CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind 
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat 
-x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
-(CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead 
-x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H15 t) 
-H16)) e H14))))))))) H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind 
-F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 
-O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda 
-(x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq 
-C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) 
-x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda (H17: (csubst0 O v x3 
-x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0 c0)) H9 (CHead 
-x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind 
-b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) 
-u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 
-c (CHead x3 (Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12))))))) 
-(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H9: 
-(drop (r (Flat f) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0: 
-nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def (eq_ind_r nat x1 
-(\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H10) in (let H12 \def (H x0 
-v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(H13: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
-(Flat f) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
-T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda 
-(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 
-(Flat x2) x5))).(\lambda (H15: (drop (S n0) O c (CHead x3 (Flat x2) 
-x4))).(\lambda (H16: (subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda 
-(c0: C).(drop (S n0) O x0 c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C 
-(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat 
-f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
-f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x3 (Flat 
-x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) 
-u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) 
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) 
-u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat 
-f) n0 c (CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda 
-(H13: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) 
-O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda 
-(x5: T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop 
-(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let 
-H17 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x0 c0)) H9 (CHead x4 
-(Flat x2) x5) H14) in (eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat 
-f) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop 
-(Flat f) n0 c (CHead x3 (Flat x2) x5) H15 t) H16)) e H14))))))))) H13)) 
-(\lambda (H13: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) 
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
-C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop (S 
-n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda 
-(H17: (csubst0 O v x3 x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop 
-(S n0) O x0 c0)) H9 (CHead x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4 
-(Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) 
-(or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x6)) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) u2)))))) (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O 
-(CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C 
-T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
-(CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda 
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 
-(refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Flat f) n0 c (CHead x3 
-(Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12))))))) k H4 
-(drop_gen_drop k x0 e t n0 H7)))))))) H3)) (\lambda (H3: (ex4_3 T C nat 
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) 
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
-C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O (CHead c k t) 
-e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k 
-t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
-nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 
-k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c 
-x1)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 
-(CHead x1 k x0) H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to 
-((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
-C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead 
-e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H9: (eq nat (S n0) (s (Bind b) 
-x2))).(\lambda (H10: (drop (r (Bind b) n0) O x1 e)).(let H11 \def (f_equal 
-nat nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) 
-\Rightarrow n1])) (S n0) (S x2) H9) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n1: nat).(csubst0 n1 v c x1)) H7 n0 H11) in (let H13 \def (eq_ind_r 
-nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H11) in (let H14 \def 
-(IHn c x1 v H12 e H10) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 
-O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H15: (drop n0 O c 
-e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e 
-H15 t))) (\lambda (H15: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) 
-O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead 
-e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda 
-(x6: T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop 
-n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19 
-\def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 c0)) H10 (CHead x4 (Flat x3) 
-x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop 
-(S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead 
-e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 
-(Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead 
-x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H17 t) 
-H18)) e H16))))))))) H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
-C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda 
-(x5: C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) 
-x6))).(\lambda (H17: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H18: 
-(csubst0 O v x4 x5)).(let H19 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 
-c0)) H10 (CHead x5 (Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
-(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
-x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
-f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 
-(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 
-(Flat x3) x6) H17 t) H18)) e H16))))))))) H15)) (\lambda (H15: (ex4_5 F C C T 
-T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
-(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: 
-T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop n0 O 
-c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda (H19: 
-(csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 
-c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5 (Flat x3) x7) 
-(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
-(CHead c (Bind b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
-x3) x7) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
-f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
-f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 
-c (CHead x4 (Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) 
-(\lambda (f: F).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H10: 
-(drop (r (Flat f) n0) O x1 e)).(let H11 \def (f_equal nat nat (\lambda (e0: 
-nat).e0) (S n0) (s (Flat f) x2) H9) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H11) in (let H13 \def 
-(eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H11) in 
-(let H14 \def (H x1 v H12 e H10) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T 
-T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S 
-n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (H15: (drop (S n0) O c e)).(or4_intro0 (drop (S 
-n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Flat f) n0 c e H15 t))) (\lambda (H15: (ex3_4 F 
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq 
-C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead 
-e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat 
-f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop (S 
-n0) O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19 
-\def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x4 (Flat 
-x3) x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 
-(drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat 
-f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) 
-(drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H17 t) H18)) e H16))))))))) 
-H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) 
-O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda 
-(x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H17: (drop 
-(S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (csubst0 O v x4 x5)).(let 
-H19 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x5 
-(Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
-f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat 
-f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 
-(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
-u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
-(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
-(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop 
-(Flat f) n0 c (CHead x4 (Flat x3) x6) H17 t) H18)) e H16))))))))) H15)) 
-(\lambda (H15: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) 
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
-C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: 
-T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop (S 
-n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda 
-(H19: (csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop 
-(S n0) O x1 c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5 
-(Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
-e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) 
-(or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x7)) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) u2)))))) (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O 
-(CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C 
-T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
-(CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda 
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 
-(refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 
-(Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) k H4 
-(drop_gen_drop k x1 e x0 n0 H8)))))))))) H3)) H2))))))))))) c1)))) n).
-
-lemma csubst0_drop_lt_back:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O 
-c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) 
-v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) 
-\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
-\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O 
-c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
-c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda 
-(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 
-c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: 
-nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) 
-(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C 
-(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) 
-(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O 
-c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl 
-c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt 
-n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 
-c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O 
-c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda 
-(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
-c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c 
-e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: 
-C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda 
-(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O 
-(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) 
-(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: 
-C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to 
-(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
-C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda 
-(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(let H3 \def (csubst0_gen_head k 
-c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
-nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) O (CHead c k t) 
-e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: 
-C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H4: (ex3_2 T nat (\lambda 
-(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
-(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or (drop (S n0) O (CHead c k 
-t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k 
-x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda 
-(c0: C).(drop (S n0) O c0 e2)) H2 (CHead c k x0) H6) in (let H9 \def (eq_ind 
-nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
-c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) 
-(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: 
-C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let H10 \def (eq_ind nat 
-i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
-(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
-(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall 
-(v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 
-e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
-k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to 
-((lt (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e2) \to (or (drop (S n0) O 
-(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) 
-v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda 
-(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) 
-x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) 
-O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 
-e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) 
-(s (Bind b) x1))).(\lambda (H13: (drop (r (Bind b) n0) O c e2)).(or_introl 
-(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 
-(minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
-(CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H13 t)))))) (\lambda 
-(f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) 
-x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) 
-O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1 
-e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) 
-(s (Flat f) x1))).(\lambda (H13: (drop (r (Flat f) n0) O c e2)).(or_introl 
-(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 
-(minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
-(CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H13 t)))))) k H9 H10 
-(drop_gen_drop k c e2 x0 n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex3_2 C 
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or (drop 
-(S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) 
-v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: 
-C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C 
-c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(let H8 \def (eq_ind C 
-c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 (CHead x0 k t) H6) in (let H9 
-\def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
-T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or 
-(drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 
-e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let 
-H10 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in 
-(eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) 
-e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda (k0: 
-K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to 
-(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: 
-C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 
-n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) 
-O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: 
-C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: 
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
-(S n0) O c e1))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) x1))).(\lambda 
-(H13: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 
-H12) c x0 v H7 e2 H13) in (let H14 \def H_x in (or_ind (drop n0 O c e2) (ex2 
-C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 
-O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c 
-e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) 
-(\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) 
-(\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
-(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O 
-(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) 
-x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) 
-e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 n0) v x 
-e2)).(\lambda (H17: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind 
-b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v 
-e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 
-C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 
-c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: 
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
-(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda 
-(H13: (drop (r (Flat f) n0) O x0 e2)).(let H_x \def (H11 x0 v H7 e2 H13) in 
-(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c 
-e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c 
-e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) 
-(\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) 
-(\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
-(minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop 
-(S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
-(Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat 
-f) t) e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 (S n0)) v x 
-e2)).(\lambda (H17: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c 
-(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S 
-n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) 
-(ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 
-e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H16 
-(drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 
-(drop_gen_drop k x0 e2 t n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex4_3 T C 
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
-nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k 
-x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c 
-x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 
-(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: 
-nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
-(e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda 
-(e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O 
-c e1)))))))))) H0 (s k x2) H5) in (let H11 \def (eq_ind nat i (\lambda (n1: 
-nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r nat (s k x2) (\lambda (n1: 
-nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 
-(minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) 
-e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
-(s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop 
-(S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v0 
-e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 
-x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) 
-(ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: 
-((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to 
-(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 e3)) (\lambda 
-(e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b) 
-x2))).(\lambda (H14: (drop (r (Bind b) n0) O x1 e2)).(let H_x \def (IHn x2 
-(lt_S_n n0 x2 H13) c x1 v H8 e2 H14) in (let H15 \def H_x in (or_ind (drop n0 
-O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda 
-(e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C 
-(\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H16: (drop n0 O 
-c e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda 
-(e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda (e1: 
-C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H16 
-t))) (\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) 
-(\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
-(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O 
-(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) 
-x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) 
-e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 n0) v x 
-e2)).(\lambda (H18: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind 
-b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v 
-e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 
-C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda 
-(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H17 (drop_drop (Bind b) n0 
-c x H18 t)))))) H16)) H15))))))) (\lambda (f: F).(\lambda (H12: ((\forall 
-(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: 
-C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
-(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda 
-(H14: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H12 x1 v H8 e2 H14) in 
-(let H15 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c 
-e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H16: (drop (S n0) O c 
-e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
-(S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H16 t))) 
-(\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) 
-(\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
-(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop 
-(S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
-(Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat 
-f) t) e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 (S n0)) v x 
-e2)).(\lambda (H18: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c 
-(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S 
-n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) 
-(ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 
-e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H17 
-(drop_drop (Flat f) n0 c x H18 t)))))) H16)) H15))))))) k H10 H11 
-(drop_gen_drop k x1 e2 x0 n0 H9)) i H5))))))))))) H4)) H3))))))))))) c1)))))) 
-n).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
deleted file mode 100644 (file)
index a267513..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/fwd.ma
+++ /dev/null
@@ -1,473 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-implied rec lemma csubst0_ind (P: (nat \to (T \to (C \to (C \to Prop))))) (f: 
-(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(P (s k i) v (CHead c k u1) 
-(CHead c k u2)))))))))) (f0: (\forall (k: K).(\forall (i: nat).(\forall (c1: 
-C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to ((P i v c1 c2) 
-\to (\forall (u: T).(P (s k i) v (CHead c1 k u) (CHead c2 k u))))))))))) (f1: 
-(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i 
-v c1 c2) \to ((P i v c1 c2) \to (P (s k i) v (CHead c1 k u1) (CHead c2 k 
-u2))))))))))))) (n: nat) (t: T) (c: C) (c0: C) (c1: csubst0 n t c c0) on c1: 
-P n t c c0 \def match c1 with [(csubst0_snd k i v u1 u2 s0 c2) \Rightarrow (f 
-k i v u1 u2 s0 c2) | (csubst0_fst k i c2 c3 v c4 u) \Rightarrow (f0 k i c2 c3 
-v c4 ((csubst0_ind P f f0 f1) i v c2 c3 c4) u) | (csubst0_both k i v u1 u2 s0 
-c2 c3 c4) \Rightarrow (f1 k i v u1 u2 s0 c2 c3 c4 ((csubst0_ind P f f0 f1) i 
-v c2 c3 c4))].
-
-lemma csubst0_gen_sort:
- \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 
-i v (CSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) 
-(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: 
-C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda 
-(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) 
-(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq 
-C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
-\Rightarrow True])) I (CSort n) H2) in (False_ind P H3)))))))))) (\lambda (k: 
-K).(\lambda (i0: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v0: 
-T).(\lambda (_: (csubst0 i0 v0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to 
-P))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CSort n))).(let H4 
-\def (eq_ind C (CHead c1 k u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
-(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 
-u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 
-c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead 
-c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort n) H4) in (False_ind P H5))))))))))))) i v y x H0))) H)))))).
-
-lemma csubst0_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
-(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T 
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k 
-u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C 
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
-(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
-c2))))))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) 
-x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda 
-(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda 
-(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: 
-nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j 
-v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: 
-nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
-nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y 
-x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda 
-(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k 
-j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda 
-(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: 
-T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda 
-(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C 
-(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead c k0 u0) (CHead c1 k u1) H2) in ((let H4 \def (f_equal C K (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
-k1])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq K 
-k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 (ex3_2 
-T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda 
-(u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) (\lambda 
-(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda 
-(_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda 
-(j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u3))))) (\lambda 
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (let H8 \def 
-(eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H5) in (eq_ind_r K k 
-(\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
-(s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k1 
-u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 
-u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
-j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k 
-u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C 
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
-j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 
-u2) (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: 
-nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 (ex3_2 T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda 
-(_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda 
-(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C 
-(CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda 
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))) (ex3_2_intro T 
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda 
-(u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda 
-(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 (refl_equal nat (s k 
-i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c H7)))) H4)) H3)))))))))) 
-(\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: C).(\lambda (c2: 
-C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 c2)).(\lambda (H2: (((eq 
-C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: 
-nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
-c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 
-C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda 
-(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda (u: T).(\lambda 
-(H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H5 \def 
-(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
-_ k1 _) \Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) (CHead c1 k u1) H3) in 
-(\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T u1 
-(\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
-(s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k0 t) 
-(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u1)))) 
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat 
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 
-t) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: 
-nat).(subst0 j v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: 
-C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
-(j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
-(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
-(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda 
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H2 c1 H8) 
-in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H1 c1 H8) 
-in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
-(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq 
-C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
-(CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
-(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) 
-(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
-(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) 
-(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 
-k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 
-(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u 
-H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: 
-T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 
-u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 
-c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
-(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: 
-nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j 
-v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda 
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda 
-(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def 
-(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
-_ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in 
-(\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind 
-C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
-(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: 
-nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j 
-v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda 
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) 
-in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) 
-in (let H12 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) 
-in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
-(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq 
-C (CHead c2 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
-nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
-(CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
-(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u3))))) (\lambda 
-(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro2 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
-(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3)))) 
-(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
-(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
-(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k 
-u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
-u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
-c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda 
-(_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) u2 c2 i0 (refl_equal nat (s k 
-i0)) (refl_equal C (CHead c2 k u2)) H12 H11)) k0 H8))))))) H6)) 
-H5))))))))))))) i v y x H0))) H))))))).
-
-lemma csubst0_gen_S_bind_2:
- \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall 
-(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to 
-(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x 
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
-(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: 
-C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: 
-T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 
-(Bind b) v1))))))))))))
-\def
- \lambda (b: B).(\lambda (x: C).(C_ind (\lambda (c: C).(\forall (c2: 
-C).(\forall (v: T).(\forall (v2: T).(\forall (i: nat).((csubst0 (S i) v c 
-(CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
-(\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
-C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C c (CHead c1 (Bind b) v2)))) 
-(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda 
-(c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (n: nat).(\lambda (c2: 
-C).(\lambda (v: T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 
-(S i) v (CSort n) (CHead c2 (Bind b) v2))).(csubst0_gen_sort (CHead c2 (Bind 
-b) v2) v (S i) n H (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda 
-(v1: T).(eq C (CSort n) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
-C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CSort n) (CHead c1 (Bind b) 
-v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) 
-(\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
-C).(\lambda (v1: T).(eq C (CSort n) (CHead c1 (Bind b) v1))))))))))))) 
-(\lambda (c: C).(\lambda (_: ((\forall (c2: C).(\forall (v: T).(\forall (v2: 
-T).(\forall (i: nat).((csubst0 (S i) v c (CHead c2 (Bind b) v2)) \to (or3 
-(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C c (CHead 
-c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: 
-C).(eq C c (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) 
-v1))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: 
-T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c 
-k t) (CHead c2 (Bind b) v2))).(let H1 \def (csubst0_gen_head k c (CHead c2 
-(Bind b) v2) t v (S i) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda 
-(j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C 
-(CHead c2 (Bind b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind 
-b) v2) (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq 
-nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq 
-C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
-C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c c3))))) (or3 (ex2 T (\lambda (v1: 
-T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
-b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
-(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
-(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
-b) v1)))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
-nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind 
-b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t 
-u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k 
-j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead 
-c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or3 (ex2 T 
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) 
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
-(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
-(x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C (CHead 
-c2 (Bind b) v2) (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 
-\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | 
-(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead c k x0) H4) in 
-((let H7 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) 
-v2) (CHead c k x0) H4) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 
-(Bind b) v2) (CHead c k x0) H4) in (\lambda (H9: (eq K (Bind b) k)).(\lambda 
-(H10: (eq C c2 c)).(let H11 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x1 v 
-t t0)) H5 v2 H8) in (eq_ind_r C c (\lambda (c0: C).(or3 (ex2 T (\lambda (v1: 
-T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c0 (Bind 
-b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c0)) (\lambda (c1: C).(eq C 
-(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
-(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c0))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
-b) v1))))))) (let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
-x1))) H3 (Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
-(CHead c (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c)) (\lambda 
-(c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda 
-(_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: 
-T).(csubst0 i v c1 c))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 
-t) (CHead c1 (Bind b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: 
-nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) 
-H12) in (let H14 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t v2)) 
-H11 i H13) in (or3_intro0 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
-(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1)))) (ex2 C 
-(\lambda (c1: C).(csubst0 i v c1 c)) (\lambda (c1: C).(eq C (CHead c (Bind b) 
-t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
-T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c))) 
-(\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind 
-b) v1))))) (ex_intro2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: 
-T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1))) t H14 (refl_equal C 
-(CHead c (Bind b) t)))))) k H9)) c2 H10))))) H7)) H6))))))) H2)) (\lambda 
-(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) 
-(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k 
-t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C 
-nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: 
-C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or3 (ex2 T (\lambda (v1: 
-T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
-b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
-(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
-(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
-b) v1)))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) 
-(s k x1))).(\lambda (H4: (eq C (CHead c2 (Bind b) v2) (CHead x0 k 
-t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H7 \def (f_equal C K 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 
-_) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H8 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v2 | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in 
-(\lambda (H9: (eq K (Bind b) k)).(\lambda (H10: (eq C c2 x0)).(let H11 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(csubst0 x1 v c c0)) H5 c2 H10) in (eq_ind_r 
-T t (\lambda (t0: T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 t0)) 
-(\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind b) v1)))) (ex2 C 
-(\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c k t) 
-(CHead c1 (Bind b) t0)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 
-i v v1 t0))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda 
-(c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind b) v1))))))) 
-(let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 x1))) H3 
-(Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: 
-T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C (CHead c k0 t) (CHead c2 (Bind 
-b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
-(CHead c k0 t) (CHead c1 (Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda 
-(v1: T).(subst0 i v v1 t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
-c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 t) (CHead c1 (Bind 
-b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: nat).(match e with 
-[O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H12) in (let H14 \def 
-(eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c c2)) H11 i H13) in 
-(or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C 
-(CHead c (Bind b) t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
-C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 
-(Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 
-t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
-C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) 
-(ex_intro2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
-(CHead c (Bind b) t) (CHead c1 (Bind b) t))) c H14 (refl_equal C (CHead c 
-(Bind b) t)))))) k H9)) v2 H8))))) H7)) H6))))))) H2)) (\lambda (H2: (ex4_3 T 
-C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 
-(Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
-(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
-C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
-C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) 
-(\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) 
-(\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or3 
-(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k 
-t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
-(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
-(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda 
-(H4: (eq C (CHead c2 (Bind b) v2) (CHead x1 k x0))).(\lambda (H5: (subst0 x2 
-v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) in ((let H8 \def 
-(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) 
-in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) 
-(CHead x1 k x0) H4) in (\lambda (H10: (eq K (Bind b) k)).(\lambda (H11: (eq C 
-c2 x1)).(let H12 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 x2 v c c0)) H6 
-c2 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x2 v t t0)) 
-H5 v2 H9) in (let H14 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
-x2))) H3 (Bind b) H10) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
-(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
-(\lambda (c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c k0 t) (CHead c1 (Bind b) v1))))))) (let H15 \def (f_equal 
-nat nat (\lambda (e: nat).(match e with [O \Rightarrow i | (S n) \Rightarrow 
-n])) (S i) (S x2) H14) in (let H16 \def (eq_ind_r nat x2 (\lambda (n: 
-nat).(csubst0 n v c c2)) H12 i H15) in (let H17 \def (eq_ind_r nat x2 
-(\lambda (n: nat).(subst0 n v t v2)) H13 i H15) in (or3_intro2 (ex2 T 
-(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c (Bind b) 
-t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
-(\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v2)))) (ex3_2 
-C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) (ex3_2_intro C T 
-(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
-C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
-T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1)))) c t H17 H16 
-(refl_equal C (CHead c (Bind b) t))))))) k H10))))))) H8)) H7))))))))) H2)) 
-H1))))))))))) x)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma
deleted file mode 100644 (file)
index 8949329..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/getl.ma
+++ /dev/null
@@ -1,1145 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/clear.ma".
-
-include "basic_1/csubst0/drop.ma".
-
-include "basic_1/getl/fwd.ma".
-
-lemma csubst0_getl_ge:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
-e) \to (getl n c2 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all 
-c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: 
-C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 
-x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: 
-(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) 
-(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: 
-nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: 
-nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) 
-(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: 
-(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat 
-x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: 
-(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
-(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) 
-H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) 
-(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 
-(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x 
-(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) 
-x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda 
-(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e 
-(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e 
-(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F 
-C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O 
-c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O 
-c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: 
-(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) 
-H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) 
-H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e 
-x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
-i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
-
-lemma csubst0_getl_lt:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
-e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all 
-c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: 
-C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: 
-C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def 
-(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O 
-c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B 
-C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) 
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u 
-w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k 
-n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x 
-(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda 
-(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda 
-(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop 
-n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear 
-(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: 
-B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: 
-(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 
-(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 
-(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind 
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 
-(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 
-(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead 
-e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 
-(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) 
-e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n 
-O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v 
-x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n 
-c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 
-(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) 
-x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind 
-K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 
-e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x 
-(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda 
-(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda 
-(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop 
-n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear 
-(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: 
-B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: 
-(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 
-(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 
-(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind 
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 
-(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 
-(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead 
-e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n 
-c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) 
-H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: 
-(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s 
-(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let 
-H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 
-(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 
-x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear 
-x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
-x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C 
-T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 
-B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e 
-H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 
-(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) 
-v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: 
-B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e 
-(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4) 
-x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5 
-(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 
-(getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead 
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
-x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) 
-u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 
-x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind 
-x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 
-f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda 
-(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: 
-(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v 
-x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 
-c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) 
-x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) 
-x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) 
-(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 
-(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) 
-x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f 
-x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind 
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda 
-(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq 
-C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) 
-x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: 
-(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) 
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
-(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead 
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) 
-(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 
-(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) 
-H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k 
-n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k 
-n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n 
-c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda 
-(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq 
-C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda 
-(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s 
-x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
-(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) 
-\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 
-x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 
-(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 
-x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda 
-(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) 
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) 
-(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
-(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
-(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead 
-e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda 
-(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C 
-T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) 
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 
-(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) 
-(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind 
-b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 
-(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 
-x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda 
-(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) 
-(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy 
-i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e 
-(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) 
-v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: 
-(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e 
-(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: 
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 
-(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) 
-(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda 
-(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) 
-x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: 
-(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) 
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 
-(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead 
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
-x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) 
-u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 
-x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind 
-x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 
-f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda 
-(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: 
-(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v 
-x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 
-c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5) 
-x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) 
-x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) 
-(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 
-(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) 
-x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f 
-x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind 
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda 
-(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq 
-C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) 
-x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: 
-(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) 
-(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
-u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
-(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead 
-e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
-C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
-x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
-x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) 
-(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 
-(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) 
-H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
-
-lemma csubst0_getl_ge_back:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 
-e) \to (getl n c1 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
-c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all 
-c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: 
-C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 
-x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: 
-(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) 
-H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: 
-nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: 
-nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) 
-(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O 
-c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 
-e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat 
-x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: 
-(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
-(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) 
-H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) 
-(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x 
-(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) 
-x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda 
-(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e 
-(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v 
-H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i 
-O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
-O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O 
-c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: 
-(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) 
-H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) 
-H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 
-x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
-i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
-
-lemma csubst0_getl_lt_back:
- \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 
-e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
-e2)) (\lambda (e1: C).(getl n c1 e1))))))))))))
-\def
- \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
-c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def 
-(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) 
-(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
-(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x 
-\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in 
-(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) 
-(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
-(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) 
-(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 
-(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda 
-(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or 
-(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) 
-(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 
-(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def 
-(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in 
-(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) 
-(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
-(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) 
-(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: 
-C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind 
-C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 
-e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
-e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: 
-(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl 
-n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: 
-C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
-e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 
-H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma
deleted file mode 100644 (file)
index bb427a6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst0/props.ma
+++ /dev/null
@@ -1,52 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/defs.ma".
-
-lemma csubst0_snd_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c 
-(Bind b) u1) (CHead c (Bind b) u2))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind 
-b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) 
-u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S 
-i))))))))).
-
-lemma csubst0_fst_bind:
- \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall 
-(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 
-(Bind b) u) (CHead c2 (Bind b) u))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
-(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind 
-b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) 
-u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))).
-
-theorem csubst0_both_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i 
-v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) 
-u2))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: 
-nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) 
-(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S 
-i))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma
deleted file mode 100644 (file)
index dcfa784..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/defs.ma
+++ /dev/null
@@ -1,22 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/defs.ma".
-
-inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
-| csubst1_refl: csubst1 i v c1 c1
-| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma
deleted file mode 100644 (file)
index c063401..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/fwd.ma
+++ /dev/null
@@ -1,128 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst1/defs.ma".
-
-include "basic_1/csubst0/fwd.ma".
-
-include "basic_1/subst1/defs.ma".
-
-include "basic_1/s/fwd.ma".
-
-implied lemma csubst1_ind:
- \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (P: ((C \to 
-Prop))).((P c1) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2)))) \to 
-(\forall (c: C).((csubst1 i v c1 c) \to (P c))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (P: ((C \to 
-Prop))).(\lambda (f: (P c1)).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1 
-c2) \to (P c2))))).(\lambda (c: C).(\lambda (c0: (csubst1 i v c1 c)).(match 
-c0 with [csubst1_refl \Rightarrow f | (csubst1_sing x x0) \Rightarrow (f0 x 
-x0)])))))))).
-
-lemma csubst1_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
-(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T 
-C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: 
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: 
-C).(csubst1 i v c1 c2))))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
-(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) 
-x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C 
-(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: 
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: 
-C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: 
-C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 
-c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl 
-i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) 
-c2)).(let H1 \def (csubst0_gen_head k c1 c2 u1 v (s k i) H0) in (or3_ind 
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) 
-(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 
-j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
-C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2: 
-T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
-c3)))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
-(s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k 
-u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T 
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda 
-(u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2: 
-T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
-c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (s k i) (s k 
-x1))).(\lambda (H4: (eq C c2 (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1 
-x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: 
-T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
-c3))))) (let H_y \def (s_inj k i x1 H3) in (let H6 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in (ex3_2_intro T C (\lambda 
-(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda 
-(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: 
-C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single 
-i v u1 x0 H6) (csubst1_refl i v c1)))) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 
-C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda 
-(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: 
-C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 
-j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 
-k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: 
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: 
-nat).(\lambda (H3: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C c2 (CHead 
-x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) 
-(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead 
-c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda 
-(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x1 
-H3) in (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) 
-H5 i H_y) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead 
-x0 k u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 
-u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 
-(refl_equal C (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 
-H6)))) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda 
-(_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: 
-T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
-(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: 
-T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C 
-nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 
-c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k 
-u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: 
-T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: 
-C).(\lambda (x2: nat).(\lambda (H3: (eq nat (s k i) (s k x2))).(\lambda (H4: 
-(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: 
-(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C 
-(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: 
-T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: 
-C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x2 H3) in (let H7 \def 
-(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c1 x1)) H6 i H_y) in (let H8 
-\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in 
-(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) 
-(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) 
-(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C 
-(CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7))))) 
-c2 H4)))))))) H2)) H1)))) x H))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma
deleted file mode 100644 (file)
index 1d37fd7..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/getl.ma
+++ /dev/null
@@ -1,269 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst1/props.ma".
-
-include "basic_1/csubst0/getl.ma".
-
-include "basic_1/subst1/props.ma".
-
-include "basic_1/drop/props.ma".
-
-lemma csubst1_getl_ge:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 
-e) \to (getl n c2 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to 
-(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda 
-(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
-(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
-
-lemma csubst1_getl_lt:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 
-e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: 
-C).(getl n c2 e2)))))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to 
-(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl 
-n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S 
-(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 
-e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 
-(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) 
-(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda 
-(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: 
-nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n 
-c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind 
-(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
-B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: 
-T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n 
-c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: 
-C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) 
-(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S 
-(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl 
-(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind 
-b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u 
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 
-(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S 
-(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 
-(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) 
-x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 
-(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S 
-n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: 
-C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) 
-v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus 
-i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T 
-(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 
-(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: 
-C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 
-(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 
-(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) 
-x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 
-(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S 
-n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: 
-C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) 
-v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus 
-i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T 
-T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda 
-(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 
-(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda 
-(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) 
-(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: 
-C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind 
-x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: 
-(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 
-x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: 
-C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind 
-x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) 
-(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind 
-x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 
-H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
-
-lemma csubst1_getl_ge_back:
- \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 
-e) \to (getl n c1 e)))))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to 
-(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda 
-(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
-(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
-
-lemma getl_csubst1:
- \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c 
-(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: 
-C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 
-a))))))))
-\def
- \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: 
-C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C 
-(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: 
-C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind 
-(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind 
-Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 
-a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda 
-(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) 
-(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 
-C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda 
-(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda 
-(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to 
-(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda 
-(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind 
-(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O 
-(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: 
-C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: 
-C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: 
-T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) 
-t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) 
-(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e 
-(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) 
-H0))) in ((let H2 \def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort 
-_) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e (Bind Abbr) u) 
-(CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t 
-(getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H3 
-\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) 
-t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind 
-b) t) (CHead e (Bind Abbr) u) H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda 
-(_: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(ex2_2 C C (\lambda (a0: 
-C).(\lambda (_: C).(csubst1 O t0 (CHead c0 (Bind b) t) a0))) (\lambda (a0: 
-C).(\lambda (a: C).(drop (S O) O a0 a))))) (eq_ind B Abbr (\lambda (b0: 
-B).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind 
-b0) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) 
-(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 
-(Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) 
-(CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead c0 (Bind Abbr) t)) 
-(drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u H3)))) H2)) 
-H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let 
-H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T (\lambda (t2: 
-T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: 
-C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
-C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: (subst1 O u t (lift 
-(S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead e (Bind Abbr) u) c0 
-(drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) u) t (getl_gen_O 
-(CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in (ex2_2_ind C C 
-(\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: 
-C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda 
-(_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
-C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H4: 
-(csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 x1)).(ex2_2_intro C C 
-(\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead x0 (Flat f) 
-(lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O x) H2 c0 x0 H4) 
-(drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) H1)))))))) k)))) 
-c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall (e: C).(\forall 
-(u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: 
-C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: 
-C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind (\lambda (c0: 
-C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) 
-\to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) (\lambda (n0: 
-nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) (CSort n0) 
-(CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind Abbr) u) H0 
-(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CSort n0) a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) (\lambda 
-(c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead 
-e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S 
-n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 
-a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall 
-(e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 t) (CHead e (Bind Abbr) u)) 
-\to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 k0 
-t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) 
-(\lambda (b: B).(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: 
-(getl (S n) (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H_x \def 
-(subst1_ex u t n) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 n 
-u t (lift (S O) n t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 
-(S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S 
-O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 n u t (lift (S O) n 
-x))).(let H4 \def (H c0 e u (getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t 
-n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 
-a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C 
-(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: 
-C).(\lambda (x1: C).(\lambda (H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S 
-O) n x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) 
-u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S 
-n) a0 a))) (CHead x0 (Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) 
-(csubst1_bind b n u t (lift (S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n 
-x0 x1 H6 b x)))))) H4)))) H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda 
-(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead 
-e (Bind Abbr) u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in 
-(ex_ind T (\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C 
-C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) 
-a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda 
-(x: T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def 
-(H0 e u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in 
-(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C 
-(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) 
-(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: 
-C).(\lambda (x1: C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: 
-(drop (S O) (S n) x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: 
-C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
-C).(drop (S O) (S n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead 
-x1 (Flat f) x) (csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) 
-(drop_skip_flat (S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma
deleted file mode 100644 (file)
index 349485a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubst1/props.ma
+++ /dev/null
@@ -1,66 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst1/fwd.ma".
-
-include "basic_1/subst1/fwd.ma".
-
-theorem csubst1_head:
- \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
-v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2))))))))))
-\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: 
-T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k 
-i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: 
-C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) 
-v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 
-c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i 
-c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 
-t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 
-c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) 
-(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) 
-(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v 
-c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both 
-k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))).
-
-theorem csubst1_bind:
- \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
-v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) 
-u2))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: 
-nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) 
-(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S 
-i))))))))))).
-
-theorem csubst1_flat:
- \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
-(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
-v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) 
-u2))))))))))
-\def
- \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
-(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: 
-nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) 
-(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma
deleted file mode 100644 (file)
index d2dc87d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/clear.ma
+++ /dev/null
@@ -1,71 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma csubt_clear_conf:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to 
-(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
-(\lambda (e2: C).(clear c2 e2))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 
-e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
-e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
-(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 
-e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
-(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
-e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear 
-(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
-b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
-(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda 
-(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g 
-c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
-(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
-(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g 
-e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 
-e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
-C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
-(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
-u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
-C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall 
-(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda 
-(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
-(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
-(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: 
-C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt 
-g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
-u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) 
-(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: 
-((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
-(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda 
-(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda 
-(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) 
-t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: 
-C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 
-(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 
-H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 
-c2 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma
deleted file mode 100644 (file)
index 0de7f11..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/csuba.ma
+++ /dev/null
@@ -1,39 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/arity.ma".
-
-lemma csubt_csuba:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba 
-g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
-(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
-(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
-(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: 
-B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u 
-t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A 
-(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g 
-a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda 
-(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g 
-x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) 
-H4))))))))))) c1 c2 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma
deleted file mode 100644 (file)
index 35ef17f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/defs.ma
+++ /dev/null
@@ -1,29 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/defs.ma".
-
-inductive csubt (g: G): C \to (C \to Prop) \def
-| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
-| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
-(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u))))))
-| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
-(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g 
-(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
-(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g 
-(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma
deleted file mode 100644 (file)
index 8d05e4b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/drop.ma
+++ /dev/null
@@ -1,579 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/fwd.ma".
-
-include "basic_1/drop/fwd.ma".
-
-lemma csubt_drop_flat:
- \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
-(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
-(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
-\def
- \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
-nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
-C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
-(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
-(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let 
-H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C 
-(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g 
-d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O 
-c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x 
-(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) 
-(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) 
-u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda 
-(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) 
-\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) 
-\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 
-(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
-(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall 
-(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
-d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
-T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind 
-(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
-(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort 
-n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 
-\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Flat f) u) 
-H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 
-c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
-(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (k: 
-K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: 
-T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) u0)) \to (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda 
-(d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) 
-(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
-(CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
-x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) 
-u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Flat f) 
-u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: 
-C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) 
-(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) 
-u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
-x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
-x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
-(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
-u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
-n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
-u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
-(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
-(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
-d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
-u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
-t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
-(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) 
-u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 
-O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) 
-u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O 
-c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) 
-u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 
-c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 
-H5)))))))))))))) c1 c2 H0)))))) n))).
-
-lemma csubt_drop_abbr:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
-c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
-Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
-n O c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
-C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: 
-T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
-c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
-(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
-(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
-(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq 
-C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) 
-u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) 
-(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead 
-d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) 
-H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
-C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda 
-(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c 
-(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda 
-(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O 
-(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) 
-u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
-u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: 
-(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) 
-(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H3) in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
-H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Bind Abbr) u) H1)))))) (\lambda 
-(c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (H2: 
-((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) 
-u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
-n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (k: K).(K_ind (\lambda 
-(k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O 
-(CHead c0 k0 u) (CHead d1 (Bind Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind 
-Abbr) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda 
-(u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind 
-Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: 
-(drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 
-(Bind Abbr) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 
-u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) 
-u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda 
-(u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind 
-Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) 
-(CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
-x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Flat f) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead 
-x (Bind Abbr) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 
-(Bind Abbr) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
-T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
-u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
-n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
-(Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda 
-(x: C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x 
-(Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 
-(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u 
-(drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) 
-(\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: 
-((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) 
-u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
-n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: 
-C).(\lambda (u0: T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) 
-(CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind 
-Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g 
-d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 
-(CHead x (Bind Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind 
-Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
-
-lemma csubt_drop_abst:
- \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
-c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
-Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
-C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: 
-T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
-g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g 
-c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 
-(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
-(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in 
-(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: 
-C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) 
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: 
-C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
-g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O 
-O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C 
-(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt 
-g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) 
-(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda 
-(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or 
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead 
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead 
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) 
-(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead 
-d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) 
-(\lambda (H3: (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) 
-(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: 
-T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csubt g d1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead 
-x0 (Bind Abbr) x1))).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (ty3 g d1 
-x1 t)).(\lambda (H7: (ty3 g x0 x1 t)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) 
-(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O 
-O c (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind 
-Abbr) x1) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x0 
-(Bind Abbr) x1) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) 
-(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
-Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H5 (drop_refl (CHead x0 (Bind Abbr) 
-x1)) H6 H7)) c2 H4))))))) H3)) H2))))))))) (\lambda (n0: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
-C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: 
-C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) 
-t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
-(S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 
-(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: 
-nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) 
-(CHead d1 (Bind Abst) t))).(and3_ind (eq C (CHead d1 (Bind Abst) t) (CSort 
-n1)) (eq nat (S n0) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) t) (CSort 
-n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 
-\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow 
-False | (S _) \Rightarrow True])) I O H3) in (False_ind (or (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))) H5))))) (drop_gen_sort n1 (S n0) O 
-(CHead d1 (Bind Abst) t) H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda 
-(H1: (csubt g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop 
-(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: 
-T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 
-(Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
-(_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: 
-C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead 
-d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g 
-d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda 
-(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) 
-(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
-g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: 
-(drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
-(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) 
-(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) 
-O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) 
-n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda 
-(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
-(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H5: (csubt g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind 
-Abbr) x1))).(\lambda (H7: (ty3 g d1 x1 t)).(\lambda (H8: (ty3 g x0 x1 
-t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C 
-T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex4_2_intro C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
-g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H6 
-u) H7 H8)))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead d1 
-(Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: 
-C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead 
-d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda 
-(u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
-n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda 
-(_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: 
-T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead 
-d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: 
-C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csubt 
-g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl 
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
-(CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
-g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t))) x H5 
-(drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda 
-(H4: (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 
-(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 
-t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) 
-u) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead 
-c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csubt g d1 x0)).(\lambda 
-(H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g d1 x1 
-t)).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) 
-(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 
-t))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex4_2_intro C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind 
-Abbr) x1) H6 u) H7 H8)))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead 
-d1 (Bind Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: 
-T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
-Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: 
-T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind 
-Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 
-u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
-n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O 
-(CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O c3 
-(CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind 
-Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g 
-d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 
-(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 
-u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 
-u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
-u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda 
-(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 
-t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
-(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda 
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda 
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
-g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 
-u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead 
-d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: 
-T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
-(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: 
-T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind 
-Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g 
-d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
-(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
-g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
-(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: 
-(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) 
-t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
-(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind 
-Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda 
-(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 
-(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 
-t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda 
-(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
-t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
-n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) 
-(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda 
-(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: 
-(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 
-t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
-(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
-(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
-(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
-g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) 
-H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 
-(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
deleted file mode 100644 (file)
index eb57146..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/fwd.ma
+++ /dev/null
@@ -1,386 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/defs.ma".
-
-implied rec lemma csubt_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
-nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubt 
-g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
-(CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubt g c1 
-c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
-T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
-u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to ((P 
-c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u 
-t) \to (P (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))))) (c: C) 
-(c0: C) (c1: csubt g c c0) on c1: P c c0 \def match c1 with [(csubt_sort n) 
-\Rightarrow (f n) | (csubt_head c2 c3 c4 k u) \Rightarrow (f0 c2 c3 c4 
-((csubt_ind g P f f0 f1 f2) c2 c3 c4) k u) | (csubt_void c2 c3 c4 b n u1 u2) 
-\Rightarrow (f1 c2 c3 c4 ((csubt_ind g P f f0 f1 f2) c2 c3 c4) b n u1 u2) | 
-(csubt_abst c2 c3 c4 u t t0 t1) \Rightarrow (f2 c2 c3 c4 ((csubt_ind g P f f0 
-f1 f2) c2 c3 c4) u t t0 t1)].
-
-lemma csubt_gen_abbr:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
-(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) 
-v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: 
-C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) 
-(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: 
-C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
-(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
-e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind 
-Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 
-(Bind Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) 
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda 
-(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C 
-c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: 
-K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abbr) 
-v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 
-(Bind Abbr) v) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
-u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in (\lambda (H7: (eq K k (Bind 
-Abbr))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C 
-(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda 
-(e2: C).(csubt g e1 e2)))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C 
-(\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda 
-(e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
-(CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in (let H10 
-\def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (ex_intro2 C 
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) 
-(\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abbr) v)) 
-H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: 
-C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind 
-Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) 
-(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B 
-b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 
-(Bind Void) u1) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 
-(Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
-| (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 
-with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
-True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H4) in 
-(False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
-(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
-c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u 
-t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) 
-v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
-with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
-\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
-I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C 
-(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
-e1 e2))) H6))))))))))) y c2 H0))) H))))).
-
-lemma csubt_gen_abst:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
-(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
-e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda 
-(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda 
-(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind 
-Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda 
-(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 
-e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
-Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
-C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
-g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g 
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or 
-(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
-C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: 
-(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or (ex2 C 
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
-C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H2)))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
-(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
-e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
-(CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in (\lambda 
-(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 
-(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
-(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2: 
-C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 
-v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda 
-(c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: 
-C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
-C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
-g e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: 
-C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C 
-(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind 
-Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda 
-(e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abst) v1)) H10)))) 
-k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda 
-(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to 
-(or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda 
-(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
-c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not 
-(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
-c1 (Bind Void) u1) (CHead e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead 
-c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
-False | (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match 
-b0 with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
-True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H4) in 
-(False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
-(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
-(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt 
-g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C 
-(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
-g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) 
-(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: 
-(ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 
-(Bind Abst) t) (CHead e1 (Bind Abst) v1))).(let H6 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow 
-c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow t | (CHead 
-_ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) 
-H5) in (\lambda (H8: (eq C c1 e1)).(let H9 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c: 
-C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: 
-C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C 
-c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T 
-(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) 
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
-(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
-v1))))))) H2 e1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
-c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind 
-Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex4_2_intro C T (\lambda (e2: 
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
-C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
-g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13 H11 
-H9))))))))) H6))))))))))) y c2 H0))) H))))).
-
-lemma csubt_gen_flat:
- \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall 
-(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C 
-c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
-\def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
-(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead 
-e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C 
-(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
-e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda 
-(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda 
-(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 
-e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) 
-v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort 
-_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Flat f) 
-v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat 
-f) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda 
-(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
-(Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) 
-(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: 
-T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Flat f) v))).(let H4 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
-c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 
-\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in 
-((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat 
-f) v) H3) in (\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 
-e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k 
-t) (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K 
-(Flat f) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) 
-(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def 
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C 
-(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
-e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
-c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) 
-(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C 
-(CHead c3 (Flat f) v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 
-(CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat 
-f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: 
-(not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C 
-(CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C 
-(CHead c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
-H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead 
-e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
-(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
-c1 (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
-(Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: T).(\lambda 
-(t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda 
-(H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v))).(let H6 \def 
-(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
-H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) 
-(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 
-H0))) H)))))).
-
-lemma csubt_gen_bind:
- \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
-(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
-\def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
-(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
-(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: 
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
-(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
-T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y 
-c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
-C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
-e1 e2))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 
-c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
-(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
-u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
-(eq K k (Bind b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: 
-T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
-K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda 
-(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
-H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) 
-in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
-C (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C 
-(CHead c3 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
-C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
-(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: 
-B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
-v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
-(CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) 
-(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) 
-in (\lambda (H8: (eq B Void b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def 
-(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
-T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
-e2))))))) H2 e1 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
-c3)) H1 e1 H9) in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 
-(CHead e1 (Bind b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
-C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in 
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
-(CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
-(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 
-(Bind b) u2)) H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
-C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
-b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: 
-(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
-c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) 
-in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) 
-(CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) 
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B 
-Abst b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda 
-(t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: 
-C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: 
-C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
-H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) 
-in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) 
-v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
-C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
-(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: 
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) 
-H7)) H6))))))))))) y c2 H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma
deleted file mode 100644 (file)
index 7156f62..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/getl.ma
+++ /dev/null
@@ -1,417 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/clear.ma".
-
-include "basic_1/csubt/drop.ma".
-
-include "basic_1/getl/clear.ma".
-
-lemma csubt_getl_abbr:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
-(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g 
-c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: 
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
-(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: 
-C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
-Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 
-(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda 
-(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
-(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
-(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
-Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 
-(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
-u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear 
-(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 
-c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 
-(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 
-(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind 
-Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
-t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
-((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u) 
-(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
-(\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
-C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
-T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r 
-B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr H10) in 
-(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c (Bind 
-Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 
-(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n 
-c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 
-u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) 
-(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda 
-(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 
-in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to 
-(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda 
-(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
-(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: 
-C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
-C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
-f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) 
-(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def 
-(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) 
-H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) 
-(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: 
-C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: 
-(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C 
-(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt 
-g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 
-(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def 
-(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) 
-in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) 
-c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda 
-(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
-(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: 
-C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
-C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 
-(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
-C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
-B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) 
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: 
-T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 
-O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 
-(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead 
-x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) 
-x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 
-x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 
-(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 
-(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 
-\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl 
-n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 
-(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) 
-H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 
-H2)))) H0))))))).
-
-lemma csubt_getl_abst:
- \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
-(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g 
-c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda 
-(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def 
-(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: 
-C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) 
-(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) 
-(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead 
-d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c 
-(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 
-(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) 
-t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt 
-g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: 
-C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) 
-\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 
-(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) 
-t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear 
-(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 
-c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
-c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: 
-(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) 
-t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
-(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
-d1 (Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow 
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
-(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
-d1 (Bind Abst) t) t0 H6)) in ((let H9 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) 
-(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
-d1 (Bind Abst) t) t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq 
-C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def 
-(eq_ind_r T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t 
-H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 
-(Bind b0) t))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: 
-C).(drop n O c1 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (H16: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) 
-(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 
-(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H17: 
-(csubt g d1 x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abst) 
-t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda 
-(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) (CHead x1 (Bind Abst) 
-t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda (H16: (ex4_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
-t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O 
-c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: 
-(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 
-H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 
-(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 
-d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 
-(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead 
-d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n 
-O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 
-C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: 
-C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 
-c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl 
-n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda 
-(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: 
-(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) 
-H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in 
-(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 
-(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g 
-(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C 
-(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: 
-C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
-c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
-t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: 
-C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: 
-(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C 
-(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt 
-g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) 
-(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda 
-(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda 
-(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
-T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) 
-(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 
-(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
-Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 
-(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 
-(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl 
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead 
-d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O 
-c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: 
-(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) 
-v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: 
-C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
-g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 
-(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 
-e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: 
-C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind 
-Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 
-t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: 
-C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
-Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
-(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 
-(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) 
-H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 
-(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 
-(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
-d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
-C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: 
-(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: 
-(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) 
-in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 
-(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: 
-T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
-(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 
-(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind 
-x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) 
-e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) 
-x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 
-x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
-T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
-C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
-d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
-(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 
-(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def 
-(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) 
-in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
-T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or 
-(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
-(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
-d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
-(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C 
-(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 
-(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt 
-g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: 
-(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
-C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) 
-t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
-g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
-C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 
-x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C 
-T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
-C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
-d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
-(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
-(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
-(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
-(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g 
-d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: 
-C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 
-x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda 
-(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
-(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
-(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
-(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) 
-(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
-(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
-(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
-g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) 
-x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) 
-H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma
deleted file mode 100644 (file)
index 60a4452..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/pc3.ma
+++ /dev/null
@@ -1,56 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/getl.ma".
-
-include "basic_1/pc3/left.ma".
-
-lemma csubt_pr2:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: 
-C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
-C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 
-c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl 
-i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
-(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
-c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
-
-lemma csubt_pc3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: 
-T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
-T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: 
-T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 
-c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 
-c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 
-(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 
-t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall 
-(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
-(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 
-t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma
deleted file mode 100644 (file)
index 3122146..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/props.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-lemma csubt_refl:
- \forall (g: G).(\forall (c: C).(csubt g c c))
-\def
- \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) 
-(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 
-c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma
deleted file mode 100644 (file)
index 7b3094b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubt/ty3.ma
+++ /dev/null
@@ -1,98 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/pc3.ma".
-
-include "basic_1/csubt/props.ma".
-
-include "basic_1/ty3/fwd.ma".
-
-lemma csubt_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda 
-(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda 
-(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
-C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
-t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 
-c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
-(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c 
-c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
-(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g 
-d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c 
-c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
-(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
-u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: 
-(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g 
-n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
-Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: 
-((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
-C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 
-c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: 
-C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 
-u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) 
-(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) 
-(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda 
-(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
-u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: 
-(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g 
-n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
-c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 
-u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T 
-(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda 
-(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
-T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 
-g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g 
-n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: 
-C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda 
-(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 
-t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 
-c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) 
-u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: 
-(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 
-w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind 
-Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v 
-(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
-C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
-(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
-T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c 
-c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
-c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-
-lemma csubt_ty3_ld:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
-v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
-t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: 
-(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead 
-c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
-c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma
deleted file mode 100644 (file)
index 4f4a012..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/clear.ma
+++ /dev/null
@@ -1,179 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubv/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma csubv_clear_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
-B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to 
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 
-(Bind b2) v2))))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall 
-(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
-B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) 
-v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda 
-(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) 
-v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
-B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) 
-v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
-c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear 
-c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
-C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: 
-T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) 
-v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b1) v0) 
-(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 
-H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow b1 | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) (CHead 
-c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 H2)) in 
-((let H5 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b1) v0) 
-(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 
-H2)) in (\lambda (_: (eq B b1 Void)).(\lambda (H7: (eq C d1 c3)).(eq_ind_r C 
-c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csubv c d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
-(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))))) (ex2_3_intro B C T 
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda 
-(b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) 
-(CHead d2 (Bind b2) v3))))) Void c4 v2 H0 (clear_bind Void c4 v2)) d1 H7)))) 
-H4)) H3)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
-c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear 
-c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
-C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda 
-(b1: B).(\lambda (_: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (v0: 
-T).(\lambda (H3: (clear (CHead c3 (Bind b1) v1) (CHead d1 (Bind b0) 
-v0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) 
-(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) 
-in ((let H5 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow b0 | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) v0) (CHead 
-c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in 
-((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) 
-(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) 
-in (\lambda (_: (eq B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 
-(\lambda (c: C).(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csubv c d2)))) (\lambda (b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
-(CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T 
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda 
-(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) 
-(CHead d2 (Bind b3) v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) 
-H4))))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 
-c4)).(\lambda (H1: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: 
-T).((clear c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: 
-B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
-B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) 
-v2)))))))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda 
-(v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H2: 
-(clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind b1) v0))).(let H_x \def (H1 b1 
-d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind b1) v0) v1 H2)) in (let H3 \def 
-H_x in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
-c4 (CHead d2 (Bind b2) v3))))) (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
-C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) 
-v3)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: 
-(csubv d1 x1)).(\lambda (H5: (clear c4 (CHead x1 (Bind x0) x2))).(ex2_3_intro 
-B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) 
-(\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) 
-v2) (CHead d2 (Bind b2) v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind 
-x0) x2) H5 f2 v2))))))) H3))))))))))))))) c1 c2 H))).
-
-lemma csubv_clear_conf_void:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
-C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T 
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
-(v2: T).(clear c2 (CHead d2 (Bind Void) v2))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c 
-(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 
-(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: 
-T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort 
-(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead 
-d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: 
-(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 
-(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 
-(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: 
-C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 
-(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind 
-Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind 
-Void) v0) v1 H2)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind 
-Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind 
-Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: 
-C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: 
-C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) 
-v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) 
-(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 
-(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: 
-((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to 
-(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: 
-C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda 
-(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear 
-(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
-\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
-(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in ((let H5 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Void | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Void])])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
-(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in ((let H6 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v0 | (CHead 
-_ _ t) \Rightarrow t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
-(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq 
-B Void b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 
-C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda 
-(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 
-\def (eq_ind_r B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let 
-H10 \def (match (H9 (refl_equal B Void)) in False with []) in H10)) d1 H8)))) 
-H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv 
-c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 
-(Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 
-d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) 
-v2)))))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) 
-v1) (CHead d1 (Bind Void) v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 
-(CHead d1 (Bind Void) v0) v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T 
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
-(v3: T).(clear c4 (CHead d2 (Bind Void) v3)))) (ex2_2 C T (\lambda (d2: 
-C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear 
-(CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3))))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H4: (csubv d1 x0)).(\lambda (H5: (clear c4 
-(CHead x0 (Bind Void) x1))).(ex2_2_intro C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat 
-f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 (clear_flat c4 (CHead x0 (Bind 
-Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 c2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma
deleted file mode 100644 (file)
index fe704a1..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/defs.ma
+++ /dev/null
@@ -1,30 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-inductive csubv: C \to (C \to Prop) \def
-| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n))
-| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
-(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind 
-Void) v2))))))
-| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
-(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall 
-(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))
-| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
-(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 
-(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma
deleted file mode 100644 (file)
index 0fb7bbc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/drop.ma
+++ /dev/null
@@ -1,114 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubv/props.ma".
-
-include "basic_1/csubv/fwd.ma".
-
-include "basic_1/drop/fwd.ma".
-
-lemma csubv_drop_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: 
-C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 
-e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h 
-O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O 
-c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda 
-(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq 
-nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O 
-(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h 
-O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C 
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) 
-(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) 
-(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: 
-C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) 
-(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort 
-n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
-c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to 
-(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 
-e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: 
-nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda 
-(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: 
-C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) 
-e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C 
-(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c 
-e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C 
-(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop 
-O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same 
-c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl 
-(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 
-O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) 
-(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: 
-(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind 
-Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in 
-(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 
-e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O 
-(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 
-x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 
-e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 
-(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda 
-(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall 
-(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: 
-C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: 
-B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h 
-O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead 
-c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: 
-C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead 
-c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 
-C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind 
-b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) 
-e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind 
-b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) 
-v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: 
-nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C 
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind 
-b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) 
-e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 
-h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) 
-(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) 
-(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: 
-C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C 
-(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 
-(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h 
-H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
-c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to 
-(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 
-e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat 
-f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) 
-e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O 
-(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) 
-v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda 
-(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) 
-e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) 
-(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) 
-v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 
-(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda 
-(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: 
-C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) 
-e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x 
-\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in 
-(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: 
-C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda 
-(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda 
-(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda 
-(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) 
-v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h 
-H2)))))))))))) c1 c2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma
deleted file mode 100644 (file)
index 23883b4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/fwd.ma
+++ /dev/null
@@ -1,35 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubv/defs.ma".
-
-implied rec lemma csubv_ind (P: (C \to (C \to Prop))) (f: (\forall (n: 
-nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubv 
-c1 c2) \to ((P c1 c2) \to (\forall (v1: T).(\forall (v2: T).(P (CHead c1 
-(Bind Void) v1) (CHead c2 (Bind Void) v2))))))))) (f1: (\forall (c1: 
-C).(\forall (c2: C).((csubv c1 c2) \to ((P c1 c2) \to (\forall (b1: B).((not 
-(eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall (v2: T).(P 
-(CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))))) (f2: (\forall (c1: 
-C).(\forall (c2: C).((csubv c1 c2) \to ((P c1 c2) \to (\forall (f2: 
-F).(\forall (f3: F).(\forall (v1: T).(\forall (v2: T).(P (CHead c1 (Flat f2) 
-v1) (CHead c2 (Flat f3) v2))))))))))) (c: C) (c0: C) (c1: csubv c c0) on c1: 
-P c c0 \def match c1 with [(csubv_sort n) \Rightarrow (f n) | (csubv_void c2 
-c3 c4 v1 v2) \Rightarrow (f0 c2 c3 c4 ((csubv_ind P f f0 f1 f2) c2 c3 c4) v1 
-v2) | (csubv_bind c2 c3 c4 b1 n b2 v1 v2) \Rightarrow (f1 c2 c3 c4 
-((csubv_ind P f f0 f1 f2) c2 c3 c4) b1 n b2 v1 v2) | (csubv_flat c2 c3 c4 f3 
-f4 v1 v2) \Rightarrow (f2 c2 c3 c4 ((csubv_ind P f f0 f1 f2) c2 c3 c4) f3 f4 
-v1 v2)].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma
deleted file mode 100644 (file)
index f648b23..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/getl.ma
+++ /dev/null
@@ -1,84 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubv/clear.ma".
-
-include "basic_1/csubv/drop.ma".
-
-include "basic_1/getl/fwd.ma".
-
-lemma csubv_getl_conf:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
-B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 
-(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl 
-i c2 (CHead d2 (Bind b2) v2)))))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: 
-B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i 
-c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind 
-b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
-C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda 
-(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
-C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: 
-C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) 
-v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in 
-(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) 
-(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 
-d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead 
-d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda 
-(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) 
-in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: 
-C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
-C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T 
-(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda 
-(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) 
-v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: 
-(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro 
-B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) 
-(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind 
-b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 
-H9))))))) H7)))))) H4)))))) H1))))))))).
-
-lemma csubv_getl_conf_void:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
-C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) 
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: 
-C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: 
-C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 
-(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i 
-H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e 
-(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 
-(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda 
-(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 
-c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) 
-(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 
-(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda 
-(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 
-H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: 
-T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 
-(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 
-d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) 
-v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 
-x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T 
-(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
-(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 
-(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma b/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma
deleted file mode 100644 (file)
index eb8448a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/csubv/props.ma
+++ /dev/null
@@ -1,43 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubv/defs.ma".
-
-include "basic_1/C/fwd.ma".
-
-include "basic_1/T/props.ma".
-
-lemma csubv_bind_same:
- \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall 
-(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) 
-v2)))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: 
-B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 
-(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: 
-T).(csubv_bind c1 c2 H Abbr not_abbr_void Abbr v1 v2))) (\lambda (v1: 
-T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst not_abst_void Abst v1 v2))) 
-(\lambda (v1: T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))).
-
-lemma csubv_refl:
- \forall (c: C).(csubv c c)
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: 
-nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda 
-(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead 
-c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) 
-(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/definitions.ma b/matita/matita/contribs/lambdadelta/basic_1/definitions.ma
deleted file mode 100644 (file)
index 010b9d2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/definitions.ma
+++ /dev/null
@@ -1,68 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/tlt/defs.ma".
-
-include "basic_1/iso/defs.ma".
-
-include "basic_1/clen/defs.ma".
-
-include "basic_1/flt/defs.ma".
-
-include "basic_1/app/defs.ma".
-
-include "basic_1/cnt/defs.ma".
-
-include "basic_1/cimp/defs.ma".
-
-include "basic_1/subst1/defs.ma".
-
-include "basic_1/subst/defs.ma".
-
-include "basic_1/csubst1/defs.ma".
-
-include "basic_1/fsubst0/defs.ma".
-
-include "basic_1/next_plus/defs.ma".
-
-include "basic_1/sty1/defs.ma".
-
-include "basic_1/llt/defs.ma".
-
-include "basic_1/aprem/defs.ma".
-
-include "basic_1/ex0/defs.ma".
-
-include "basic_1/wcpr0/defs.ma".
-
-include "basic_1/csubv/defs.ma".
-
-include "basic_1/csuba/defs.ma".
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/ex2/defs.ma".
-
-include "basic_1/csubc/defs.ma".
-
-include "basic_1/pc1/defs.ma".
-
-include "basic_1/ex1/defs.ma".
-
-include "basic_1/csubt/defs.ma".
-
-include "basic_1/wf3/defs.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma
deleted file mode 100644 (file)
index 426f370..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/defs.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-include "basic_1/lift/defs.ma".
-
-include "basic_1/r/defs.ma".
-
-inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
-| drop_refl: \forall (c: C).(drop O O c c)
-| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: 
-C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) 
-e))))))
-| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: 
-C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) 
-(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma
deleted file mode 100644 (file)
index 0b7876e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/fwd.ma
+++ /dev/null
@@ -1,475 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop/defs.ma".
-
-include "basic_1/lift/fwd.ma".
-
-include "basic_1/r/props.ma".
-
-include "basic_1/C/fwd.ma".
-
-implied rec lemma drop_ind (P: (nat \to (nat \to (C \to (C \to Prop))))) (f: 
-(\forall (c: C).(P O O c c))) (f0: (\forall (k: K).(\forall (h: nat).(\forall 
-(c: C).(\forall (e: C).((drop (r k h) O c e) \to ((P (r k h) O c e) \to 
-(\forall (u: T).(P (S h) O (CHead c k u) e))))))))) (f1: (\forall (k: 
-K).(\forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop 
-h (r k d) c e) \to ((P h (r k d) c e) \to (\forall (u: T).(P h (S d) (CHead c 
-k (lift h (r k d) u)) (CHead e k u))))))))))) (n: nat) (n0: nat) (c: C) (c0: 
-C) (d: drop n n0 c c0) on d: P n n0 c c0 \def match d with [(drop_refl c1) 
-\Rightarrow (f c1) | (drop_drop k h c1 e d0 u) \Rightarrow (f0 k h c1 e d0 
-((drop_ind P f f0 f1) (r k h) O c1 e d0) u) | (drop_skip k h d0 c1 e d1 u) 
-\Rightarrow (f1 k h d0 c1 e d1 ((drop_ind P f f0 f1) h (r k d0) c1 e d1) u)].
-
-lemma drop_gen_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop 
-h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: 
-C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: 
-C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d 
-O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: 
-nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) 
-\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: 
-C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
-C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C 
-c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq 
-nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal 
-nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda 
-(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) 
-\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: 
-T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C 
-(CHead c k u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (and3 (eq C e 
-(CHead c k u)) (eq nat (S h0) O) (eq nat O O)) H4)))))))))) (\lambda (k: 
-K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c: C).(\lambda (e: 
-C).(\lambda (_: (drop h0 (r k d0) c e)).(\lambda (_: (((eq C c (CSort n)) \to 
-(and3 (eq C e c) (eq nat h0 O) (eq nat (r k d0) O))))).(\lambda (u: 
-T).(\lambda (H3: (eq C (CHead c k (lift h0 (r k d0) u)) (CSort n))).(let H4 
-\def (eq_ind C (CHead c k (lift h0 (r k d0) u)) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
-(CSort n) H3) in (False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r 
-k d0) u))) (eq nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) 
-H))))).
-
-lemma drop_gen_refl:
- \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
-\def
- \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O 
-(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: 
-nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y 
-n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: 
-nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to 
-(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq 
-nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: 
-C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq 
-nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda 
-(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S 
-h) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H5) in (False_ind (eq C (CHead c k u) e0) H6))))))))))) (\lambda 
-(k: K).(\lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e0: 
-C).(\lambda (H2: (drop h (r k d) c e0)).(\lambda (H3: (((eq nat (r k d) O) 
-\to ((eq nat h (r k d)) \to (eq C c e0))))).(\lambda (u: T).(\lambda (H4: (eq 
-nat (S d) O)).(\lambda (H5: (eq nat h (S d))).(let H6 \def (f_equal nat nat 
-(\lambda (e1: nat).e1) h (S d) H5) in (let H7 \def (eq_ind nat h (\lambda (n: 
-nat).((eq nat (r k d) O) \to ((eq nat n (r k d)) \to (eq C c e0)))) H3 (S d) 
-H6) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n (r k d) c e0)) H2 
-(S d) H6) in (eq_ind_r nat (S d) (\lambda (n: nat).(eq C (CHead c k (lift n 
-(r k d) u)) (CHead e0 k u))) (let H9 \def (eq_ind nat (S d) (\lambda (ee: 
-nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) 
-in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h 
-H6)))))))))))))) y y0 x e H1))) H0))) H))).
-
-lemma drop_gen_drop:
- \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
-nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: 
-nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) 
-(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) 
-(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda 
-(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to 
-(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y 
-x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: 
-nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) 
-(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) 
-\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c 
-c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq 
-nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) 
-(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda 
-(ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I 
-(S h) H3) in (False_ind (drop (r k h) O c (CHead c k u)) H6)) c0 H5))))) 
-(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda 
-(H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: (((eq nat (r k0 h0) (S h)) \to 
-((eq nat O O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c 
-e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S h))).(\lambda (_: 
-(eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H8 
-\def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | 
-(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let 
-H9 \def (f_equal C K (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
-k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c k u) H7) in 
-((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k 
-u) H7) in (\lambda (H11: (eq K k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 
-\def (eq_ind C c0 (\lambda (c1: C).((eq nat (r k0 h0) (S h)) \to ((eq nat O 
-O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) O c e))))) H4 c H12) in 
-(let H14 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) H3 c 
-H12) in (let H15 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 h0) (S h)) 
-\to ((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) 
-H13 k H11) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c 
-e)) H14 k H11) in (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 
-with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in (let H18 
-\def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to ((eq nat O 
-O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h H17) in 
-(let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) H16 h 
-H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda (h0: 
-nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: (drop 
-h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 d) O) 
-\to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e)))))).(\lambda 
-(u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d) 
-O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k 
-u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 (lift n 
-(r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind nat h0 
-(\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq C c0 
-(CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let H10 
-\def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) H5) in 
-(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 (lift (S h) (r 
-k0 d) u0)) (CHead c k u) H8) in ((let H12 \def (f_equal C K (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
-k1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 
-\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
-(lref_map (\lambda (x0: nat).(plus x0 (S h))) (r k0 d) u0) | (CHead _ _ t) 
-\Rightarrow t])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in 
-(\lambda (H14: (eq K k0 k)).(\lambda (H15: (eq C c0 c)).(let H16 \def (eq_ind 
-C c0 (\lambda (c1: C).((eq nat (S h) (S h)) \to ((eq nat (r k0 d) O) \to ((eq 
-C c1 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H9 c H15) in (let H17 
-\def (eq_ind C c0 (\lambda (c1: C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in 
-(let H18 \def (eq_ind K k0 (\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) 
-u)) H13 k H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) 
-(S h)) \to ((eq nat (r k1 d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) 
-(r k1 d) c e))))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: 
-K).(drop (S h) (r k1 d) c e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: 
-K).(drop (r k h) (S d) c (CHead e k1 u0))) (let H21 \def (eq_ind_r T u 
-(\lambda (t: T).((eq nat (S h) (S h)) \to ((eq nat (r k d) O) \to ((eq C c 
-(CHead c k t)) \to (drop (r k h) (r k d) c e))))) H19 (lift (S h) (r k d) u0) 
-H18) in (let H22 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r 
-k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) 
-y1 y0 y x H2))) H1))) H0))) H)))))).
-
-lemma drop_gen_skip_r:
- \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
-(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda 
-(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k 
-d) e c)))))))))
-\def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k 
-u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) 
-(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) 
-u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: 
-(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) 
-(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x 
-(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) 
-(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) 
-\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k 
-(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda 
-(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k 
-u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C 
-c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) 
-(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow 
-True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind (ex2 C (\lambda 
-(e: C).(eq C (CHead c k u) (CHead e k (lift O (r k d) u)))) (\lambda (e: 
-C).(drop O (r k d) e c))) H4)) c0 H3)))) (\lambda (k0: K).(\lambda (h0: 
-nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r k0 h0) O c0 
-e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e (CHead c k u)) \to (ex2 C 
-(\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 h0) (r k d) u)))) (\lambda 
-(e0: C).(drop (r k0 h0) (r k d) e0 c))))))).(\lambda (u0: T).(\lambda (H4: 
-(eq nat O (S d))).(\lambda (H5: (eq C e (CHead c k u))).(let H6 \def (eq_ind 
-C e (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to (ex2 
-C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 h0) (r k d) u)))) 
-(\lambda (e0: C).(drop (r k0 h0) (r k d) e0 c)))))) H3 (CHead c k u) H5) in 
-(let H7 \def (eq_ind C e (\lambda (c1: C).(drop (r k0 h0) O c0 c1)) H2 (CHead 
-c k u) H5) in (let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 
-C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) 
-u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda 
-(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: 
-C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) 
-(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 
-(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 
-c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: 
-(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) 
-(CHead e k0 u0) (CHead c k u) H5) in ((let H7 \def (f_equal C K (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
-k1])) (CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
-\Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda (H9: (eq K k0 
-k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: T).(ex2 C (\lambda 
-(e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead e0 k (lift h0 (r k 
-d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let H11 \def (eq_ind C e 
-(\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) \to 
-(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda 
-(e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 \def (eq_ind C e 
-(\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in (let H13 \def 
-(eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to ((eq C c (CHead c 
-k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) 
-(\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) in (let H14 \def 
-(eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 k H9) in 
-(eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 
-(lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: 
-C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) 
-H4) in (let H16 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat (r k n) (S d)) 
-\to ((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k 
-(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H13 d 
-H15) in (let H17 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c0 
-c)) H14 d H15) in (eq_ind_r nat d (\lambda (n: nat).(ex2 C (\lambda (e0: 
-C).(eq C (CHead c0 k (lift h0 (r k n) u)) (CHead e0 k (lift h0 (r k d) u)))) 
-(\lambda (e0: C).(drop h0 (r k d) e0 c)))) (ex_intro2 C (\lambda (e0: C).(eq 
-C (CHead c0 k (lift h0 (r k d) u)) (CHead e0 k (lift h0 (r k d) u)))) 
-(\lambda (e0: C).(drop h0 (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift 
-h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) u0 H8)))) H7)) H6)))))))))))) h y0 
-x y H1))) H0))) H))))))).
-
-lemma drop_gen_skip_l:
- \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
-(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T 
-(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: 
-C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: 
-T).(drop h (r k d) c e))))))))))
-\def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) 
-x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda 
-(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: 
-C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: 
-(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) 
-(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: 
-C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
-d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq 
-nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: 
-C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k 
-d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda 
-(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: 
-C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: 
-C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O 
-(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow 
-False])) I (S d) H2) in (False_ind (ex3_2 C T (\lambda (e: C).(\lambda (v: 
-T).(eq C (CHead c k u) (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T 
-u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k d) c 
-e)))) H4)) c0 H3)))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: 
-C).(\lambda (e: C).(\lambda (H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq 
-nat O (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: 
-C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T u (lift (r k0 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: 
-T).(drop (r k0 h0) (r k d) c e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq 
-nat O (S d))).(\lambda (H5: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H6 
-\def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | 
-(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let 
-H7 \def (f_equal C K (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
-k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c k u) H5) in 
-((let H8 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k 
-u) H5) in (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 
-\def (eq_ind C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k 
-u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) 
-(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c 
-H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) 
-H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) 
-\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: 
-T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r 
-k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) 
-c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r 
-k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) 
-H4) in (False_ind (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead 
-e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S h0) (r k d) v)))) 
-(\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c e0)))) H15))))))))) 
-H7)) H6))))))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (d0: 
-nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop h0 (r k0 d0) c0 
-e)).(\lambda (H3: (((eq nat (r k0 d0) (S d)) \to ((eq C c0 (CHead c k u)) \to 
-(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: 
-C).(\lambda (_: T).(drop h0 (r k d) c e0)))))))).(\lambda (u0: T).(\lambda 
-(H4: (eq nat (S d0) (S d))).(\lambda (H5: (eq C (CHead c0 k0 (lift h0 (r k0 
-d0) u0)) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: C).(match e0 
-with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 
-k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
-\Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in 
-((let H8 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow (lref_map (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead 
-_ _ t) \Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) 
-H5) in (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def 
-(eq_ind C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead 
-c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k 
-v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda 
-(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 
-\def (eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in 
-(let H13 \def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) 
-H8 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S 
-d)) \to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: 
-T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 
-(r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) 
-H11 k H9) in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c 
-e)) H12 k H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: 
-C).(\lambda (v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: 
-C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda 
-(_: T).(drop h0 (r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: 
-T).((eq nat (r k d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T 
-(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: 
-C).(\lambda (v: T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda 
-(_: T).(drop h0 (r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind 
-T (lift h0 (r k d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda 
-(v: T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 
-(r k d) c e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 
-with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H4) in (let H18 
-\def (eq_ind nat d0 (\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c 
-(CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T (\lambda (e0: C).(\lambda 
-(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift 
-h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop 
-h0 (r k d) c e0))))))) H16 d H17) in (let H19 \def (eq_ind nat d0 (\lambda 
-(n: nat).(drop h0 (r k n) c e)) H15 d H17) in (eq_ind_r nat d (\lambda (n: 
-nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead 
-e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift 
-h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))) 
-(ex3_2_intro C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead 
-e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h0 (r k d) u0) (lift 
-h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))) e 
-u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h0 (r k d) u0)) H19) d0 
-H17)))) u H13)) k0 H9))))))))) H7)) H6)))))))))))) h y0 y x H1))) H0))) 
-H))))))).
-
-lemma drop_S:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
-nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
-C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to 
-(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) 
-u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O 
-O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort 
-n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O 
-(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead 
-e (Bind b) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ _ _) \Rightarrow True])) I (CSort n) H0) in (False_ind (drop (S O) O 
-(CSort n) e) H3)) h H1)))) (drop_gen_sort n h O (CHead e (Bind b) u) H))))))) 
-(\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).(\forall (h: 
-nat).((drop h O c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 
-e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) 
-(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: 
-(drop O O (CHead c0 k t) (CHead e (Bind b) u))).(let H1 \def (f_equal C C 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead 
-c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
-(CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e 
-(Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 
-with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k 
-t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) 
-H0)) in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C 
-c0 (\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) 
-(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 
-c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: 
-(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 
-k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) 
-u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: 
-nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) 
-u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
-
-theorem drop_mono:
- \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h 
-d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: 
-nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 
-x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: 
-C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq 
-nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort 
-n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C 
-x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 
-(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r 
-C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h 
-(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d 
-(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda 
-(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) 
-(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda 
-(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 
-x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) 
-x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) 
-(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) 
-\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) 
-(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: 
-(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C 
-x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) 
-(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 
-(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: 
-(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) 
-x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) 
-x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O 
-(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n 
-H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
-(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq 
-C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) 
-x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) 
-x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: 
-C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: 
-C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: 
-(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 
-x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: 
-C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: 
-C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: 
-(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r 
-C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 
-(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to 
-(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 
-(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 
-(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
-nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop 
-h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k 
-n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r 
-k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda 
-(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) 
-(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) 
-t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) 
-H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k 
-x3))) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (sym_eq C (CHead x4 k x3) 
-(CHead x0 k x3) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C 
-CHead x0 x4 k k x3 x3 (H x0 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal 
-T x3))))) x5 (lift_inj x5 x3 h (r k n) H11))))) x1 H6)) x2 H3)))))) 
-(drop_gen_skip_l c0 x1 t h n k H1))))))) (drop_gen_skip_l c0 x2 t h n k 
-H2)))))))) d))))))) c).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
deleted file mode 100644 (file)
index 1d662e8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
+++ /dev/null
@@ -1,594 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop/fwd.ma".
-
-lemma drop_skip_bind:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
-d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) 
-(lift h d u)) (CHead e (Bind b) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
-(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) 
-d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e 
-(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
-
-lemma drop_skip_flat:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
-(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat 
-f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
-(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat 
-f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead 
-e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S 
-d))))))))).
-
-lemma drop_ctail:
- \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
-h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) 
-(CTail k u c2))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: 
-nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: 
-T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda 
-(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) 
-c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h 
-O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: 
-(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d 
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) 
-(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u 
-(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O 
-(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 
-H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: 
-C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k 
-u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: 
-C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n 
-(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u 
-(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: 
-nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop 
-n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O 
-(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) 
-(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) 
-(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 
-H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to 
-(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail 
-k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: 
-K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O 
-(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: 
-nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to 
-(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail 
-k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) 
-c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda 
-(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
-(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) 
-(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: 
-(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k 
-t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 
-k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k 
-x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u 
-c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n 
-(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop 
-h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 
-(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: 
-T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) 
-(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) 
-x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) 
-c1).
-
-theorem drop_conf_lt:
- \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall 
-(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
-(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
-h (r k d) c0 e0)))))))))))))
-\def
- \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: 
-T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall 
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
-(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: 
-T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k 
-u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop 
-h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S 
-(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in 
-(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) 
-(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
-T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda 
-(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) 
-x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k 
-x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
-h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T 
-(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: 
-T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
-(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda 
-(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda 
-(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: 
-T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k 
-d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 
-e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall 
-(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) 
-c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda 
-(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) 
-O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S 
-i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind 
-(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
-T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort 
-n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) 
-O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee 
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
-(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) H5))))) (drop_gen_sort n h 
-(S (plus (S i0) d)) e H1)))))))) (\lambda (c1: C).(\lambda (H0: (((drop (S 
-i0) O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop h (r k d) c0 e0))))))))))).(\lambda (k0: K).(K_ind (\lambda (k1: 
-K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 t) (CHead c0 k u)) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) 
-d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e 
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
-e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
-(CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 
-(Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e 
-(CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r 
-(Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
-(Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: 
-C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S 
-i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
-e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 
-(Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) (plus (S i0) d)) 
-x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
-(CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
-(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
-(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
-d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) c1 (CHead c0 k 
-u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq 
-T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O x0 
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
-(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead 
-e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) 
-(\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: (eq T u (lift h (r k d) 
-x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k x2))).(\lambda (H9: (drop h (r 
-k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind 
-b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
-e0))) x2 x3 H7 (drop_drop (Bind b) i0 x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) 
-e H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Bind b) H2))))))))) 
-(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c1 (Flat 
-f) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 (Flat f) t) 
-e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat 
-f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus (S 
-i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus (S 
-i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 (Flat f) 
-x1))).(\lambda (_: (eq T t (lift h (r (Flat f) (plus (S i0) d)) 
-x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
-(CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
-(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
-(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
-d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O x0 (CHead e0 k 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
-T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) (CHead e0 k v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (x2: 
-T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) x2))).(\lambda (H7: 
-(drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h (r k d) c0 
-x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) 
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
-x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) H8)))))) (H0 
-(drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e H3)))))) 
-(drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) k0)))) c)))))) 
-i)).
-
-theorem drop_conf_ge:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
-(plus d h) i) \to (drop (minus i h) O e a)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: 
-C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e 
-a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c 
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
-d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda 
-(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def 
-(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus 
-O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 
-\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def 
-(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O 
-(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: 
-C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h 
-H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: 
-nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
-(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: 
-C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall 
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d 
-h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: 
-nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: 
-(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d 
-O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda 
-(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) 
-(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: 
-(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O 
-O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 
-O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S 
-i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) 
-O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 
-a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) 
-c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop 
-(minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4)))))) 
-(drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1))))))))) 
-(\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e: 
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) 
-(S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind 
-(\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0 
-t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))) 
-(\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind 
-b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
-(drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S 
-i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le 
-(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h 
-O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind 
-(\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S 
-i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 
-(Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 
-(Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind 
-b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead 
-c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead 
-c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O 
-e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7: 
-(le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e 
-h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h 
-H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) 
-e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
-a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5: 
-(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
-T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
-(lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
-(Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t 
-(lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 
-x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) 
-h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead 
-x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 
-(drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) 
-x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus 
-d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d 
-H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
-(CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le 
-(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) 
-t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) 
-(\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O 
-h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) 
-\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda 
-(H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S 
-i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) 
-O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) 
-e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda 
-(_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to 
-(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 
-(Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop 
-(Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) 
-H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 
-(Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
-a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5: 
-(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
-T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
-(lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
-(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t 
-(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 
-x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) 
-h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: 
-nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) 
-H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n 
-(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop 
-n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) 
-(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 
-h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 
-H3))))))))) k)))) c))))) i).
-
-theorem drop_conf_rev:
- \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
-(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: 
-C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
-\def
- \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: 
-C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) 
-\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 
-e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 
-e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let 
-H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl 
-e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: 
-C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda 
-(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: 
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O 
-c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind 
-(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: 
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S 
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: 
-nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda 
-(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 
-(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S 
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: 
-(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O 
-O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) 
-H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C 
-(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
-(CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2: 
-C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2: 
-C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S 
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k 
-t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 
-e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda 
-(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 
-k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let 
-H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C 
-(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
-(CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x 
-c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S 
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) 
-(CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 
-(lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2))))) 
-(\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def 
-(IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop 
-(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: 
-C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat 
-f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: 
-(drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) 
-(\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) 
-(lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat 
-f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))) k (drop_gen_drop k 
-e2 e3 t j0 H))))))))))) e1)))) j).
-
-theorem drop_trans_le:
- \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O 
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h 
-(minus d i) e1 e2)))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to 
-(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to 
-(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 
-e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: 
-nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: 
-nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O 
-c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 
-(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda 
-(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C 
-(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 
-(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: 
-nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: 
-C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
-C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda 
-(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind 
-(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: 
-C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O 
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
-C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) 
-O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h 
-O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat 
-(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C 
-(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S 
-i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: 
-(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S x) H2) in (False_ind (ex2 
-C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O 
-(S i0)) e1 e2))) H4))))) (le_gen_S i0 O H))))))))) (\lambda (d0: 
-nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2: 
-C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O 
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
-C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S 
-d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
-nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to 
-(ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus 
-(S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: 
-nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda 
-(H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq 
-nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
-(e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort 
-n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def 
-(eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in 
-(and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda 
-(e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) 
-(S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S 
-i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 
-C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h 
-(minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee: 
-nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) 
-in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
-(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) 
-(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) 
-(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h 
-(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda 
-(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) 
-e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
-T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to 
-(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S 
-i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
-e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: 
-nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: 
-C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda 
-(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: 
-T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O 
-(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
-e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 
-(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda 
-(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: 
-C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h 
-(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O 
-(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
-e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop 
-h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 
-(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S 
-d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda 
-(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) 
-O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop 
-h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r 
-(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 
-(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 
-t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: 
-C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) 
-c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T 
-(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: 
-C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: 
-C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: 
-C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus 
-(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C 
-c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) 
-x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 
-(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in 
-(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: 
-C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus 
-(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) 
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: 
-C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) 
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: 
-C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S 
-i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) 
-(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S 
-i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) 
-H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) 
-(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
-
-theorem drop_trans_ge:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
-nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 
-e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
-C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) 
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
-nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O 
-c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) 
-O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d 
-(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 
-H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall 
-(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall 
-(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 
-e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: 
-C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c 
-e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
-nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: 
-(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort 
-n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda 
-(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d 
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) 
-e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) 
-in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort 
-n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop 
-(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda 
-(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) 
-(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind 
-nat (S i0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n) 
-(CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3)))) 
-(drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall 
-(c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall 
-(e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O 
-c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d: 
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t) 
-c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop 
-(S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind 
-(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop 
-(S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
-e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda 
-(H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def 
-(eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t) 
-(drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n: 
-nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k 
-c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda 
-(_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 
-e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
-e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2: 
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind 
-nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2)) 
-(drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n))) 
-(\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n)) 
-(\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k 
-c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus 
-i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h)) 
-(\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k 
-t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to 
-(drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda 
-(H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop 
-(S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda 
-(e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
-T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r 
-k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: 
-(eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 d0 (CHead c2 k 
-t) c) \to (\forall (e3: C).((drop (S i0) O c e3) \to ((le d0 (S i0)) \to 
-(drop (S (plus i0 h0)) O (CHead c2 k t) e3))))))) IHd (CHead x0 k x1) H2) in 
-(let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0 
-k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
-nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k x1)) \to (\forall (e3: 
-C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S i0)) \to (drop (S (plus 
-i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r 
-T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k 
-t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r 
-k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 
-e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda 
-(H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 
-(le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r 
-(Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r 
-(Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h 
-(r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) 
-d))))))) c1)))) i).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma
deleted file mode 100644 (file)
index 41b0848..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/defs.ma
+++ /dev/null
@@ -1,32 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop/defs.ma".
-
-include "basic_1/lift1/defs.ma".
-
-inductive drop1: PList \to (C \to (C \to Prop)) \def
-| drop1_nil: \forall (c: C).(drop1 PNil c c)
-| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds 
-c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
-
-rec definition ptrans (hds: PList) on hds: nat \to PList \def \lambda (i: 
-nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow 
-(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) 
-with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow 
-q])))]).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma
deleted file mode 100644 (file)
index 3195918..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/fwd.ma
+++ /dev/null
@@ -1,81 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop1/defs.ma".
-
-implied rec lemma drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: 
-(\forall (c: C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: 
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: 
-C).(\forall (hds: PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons 
-h d hds) c1 c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P 
-p c c0 \def match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1 
-c2 h d0 d1 c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f 
-f0) hds c2 c3 d2))].
-
-lemma drop1_gen_pnil:
- \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq 
-PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 
-c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda 
-(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c 
-c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) 
-(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: 
-PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to 
-(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def 
-(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee with [PNil 
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H4) in 
-(False_ind (eq C c3 c5) H5)))))))))))) y c1 c2 H0))) H))).
-
-lemma drop1_gen_pcons:
- \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: 
-nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda 
-(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
-\def
- \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq 
-PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: 
-PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds 
-c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind 
-(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d 
-hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 
-hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d 
-hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee with 
-[PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds) 
-H1) in (False_ind (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: 
-C).(drop1 hds c2 c))) H2)))) (\lambda (c2: C).(\lambda (c4: C).(\lambda (h0: 
-nat).(\lambda (d0: nat).(\lambda (H1: (drop h0 d0 c2 c4)).(\lambda (c5: 
-C).(\lambda (hds0: PList).(\lambda (H2: (drop1 hds0 c4 c5)).(\lambda (H3: 
-(((eq PList hds0 (PCons h d hds)) \to (ex2 C (\lambda (c6: C).(drop h d c4 
-c6)) (\lambda (c6: C).(drop1 hds c6 c5)))))).(\lambda (H4: (eq PList (PCons 
-h0 d0 hds0) (PCons h d hds))).(let H5 \def (f_equal PList nat (\lambda (e: 
-PList).(match e with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) 
-(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat 
-(\lambda (e: PList).(match e with [PNil \Rightarrow d0 | (PCons _ n _) 
-\Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H7 \def 
-(f_equal PList PList (\lambda (e: PList).(match e with [PNil \Rightarrow hds0 
-| (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds) H4) in 
-(\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let H10 \def 
-(eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) \to (ex2 
-C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 c5))))) 
-H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: PList).(drop1 p 
-c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0 
-n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda (n: nat).(drop n 
-d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h d c2 c6)) 
-(\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) H5)))))))))))) y 
-c1 c3 H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma
deleted file mode 100644 (file)
index 83889c3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/getl.ma
+++ /dev/null
@@ -1,107 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop1/fwd.ma".
-
-include "basic_1/getl/drop.ma".
-
-lemma drop1_getl_trans:
- \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
-\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl 
-i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) 
-e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 
-(ptrans hds i) v)))))))))))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
-C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: 
-C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to 
-(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl 
-(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) 
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda 
-(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl 
-i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in 
-(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) 
-(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda 
-(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) 
-v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
-C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: 
-T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda 
-(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) 
-c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda 
-(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl 
-i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d 
-H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) 
-(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match 
-(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) 
-(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow 
-(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
-(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h 
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
-hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: 
-(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: 
-bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons 
-h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
-hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow 
-(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
-(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) 
-(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 
-i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow 
-(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow 
-(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true 
-\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
-(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d 
-(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) 
-v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def 
-(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: 
-C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x 
-(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: 
-C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) 
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h 
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: 
-C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans 
-hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def 
-(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b 
-x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda 
-(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans 
-hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S 
-(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S 
-(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 
-i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans 
-hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 
-(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 
-i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 
-x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 
-i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead 
-e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) 
-v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 
-i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) 
-false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in 
-(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: 
-C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) 
-(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl 
-(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) 
-(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: 
-(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let 
-H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) 
-(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 
-(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 
-(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans 
-hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma
deleted file mode 100644 (file)
index 41b61d8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/drop1/props.ma
+++ /dev/null
@@ -1,88 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop1/fwd.ma".
-
-include "basic_1/drop/props.ma".
-
-include "basic_1/getl/defs.ma".
-
-lemma drop1_skip_bind:
- \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: 
-C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) 
-(lift1 hds u)) (CHead e (Bind b) u)))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: 
-PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) 
-(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def 
-(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead 
-c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c 
-H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
-(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead 
-c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def 
-(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda 
-(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S 
-n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) 
-(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x 
-e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) 
-(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e 
-(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
-
-lemma drop1_cons_tail:
- \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop 
-h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to 
-(drop1 (PConsTail hds h d) c1 c3))))))))
-\def
- \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda 
-(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 
-c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def 
-(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d 
-PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: 
-((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 
-c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x 
-\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C 
-(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 
-(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n 
-n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 
-(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))).
-
-theorem drop1_trans:
- \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) 
-\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 
-(papp is1 is2) c1 c2)))))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
-C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: 
-C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: 
-C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: 
-PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def 
-(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: 
-C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: 
-C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 
-c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: 
-C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda 
-(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 
-p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 
-c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 
-c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x 
-c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) 
-H2))))))))))))) is1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/etc/performance.txt b/matita/matita/contribs/lambdadelta/basic_1/etc/performance.txt
deleted file mode 100644 (file)
index 6c46086..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/etc/performance.txt
+++ /dev/null
@@ -1,17 +0,0 @@
-full validation of lambdadelta_1
-
-command: time ../../matitac.opt basic_1
-
-- machine: "monica"
-  date   : ven  6 mar 2015, 20.31.46, CET
-
-  real    4m39.904s
-  user    3m58.580s
-  sys     0m11.473s
-
-- machine: "dev.helm"
-  date   : Sat Mar  7 16:41:46 CET 2015
-
-  real    30m36.357s
-  user    6m35.749s
-  sys     0m31.518s

diff --git a/matita/matita/contribs/lambdadelta/basic_1/etc/planes.txt b/matita/matita/contribs/lambdadelta/basic_1/etc/planes.txt
deleted file mode 100644 (file)
index d4b66b3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/etc/planes.txt
+++ /dev/null
@@ -1,13 +0,0 @@
-T s tlist tlt iso
-C r flt app
-lift cnt drop clear getl clen cimp
-lift1 drop1
-subst0 subst1 subst csubst0 csubst1 fsubst0 
-G next_plus
-sty0 sty1
-A asucc aplus leq llt aprem ex0 
-pr0 wcpr0 pr1 pr2 pr3 
-csubv arity csuba
-nf2 sn3 sc3 csubc ex2
-pc1 pc3
-ty3 csubt wf3 ex1

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma
deleted file mode 100644 (file)
index 546e039..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma
+++ /dev/null
@@ -1,32 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/A/defs.ma".
-
-include "basic_1/G/defs.ma".
-
-definition gz:
- G
-\def
- mk_G S lt_n_Sn.
-
-inductive leqz: A \to (A \to Prop) \def
-| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
-(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort 
-h2 n2))))))
-| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: 
-A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma
deleted file mode 100644 (file)
index 248d07e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma
+++ /dev/null
@@ -1,28 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ex0/defs.ma".
-
-implied rec lemma leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: 
-nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus 
-h1 n2) (plus h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall 
-(a1: A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3: 
-A).(\forall (a4: A).((leqz a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead 
-a2 a4))))))))))) (a: A) (a0: A) (l: leqz a a0) on l: P a a0 \def match l with 
-[(leqz_sort h1 h2 n1 n2 e) \Rightarrow (f h1 h2 n1 n2 e) | (leqz_head a1 a2 
-l0 a3 a4 l1) \Rightarrow (f0 a1 a2 l0 ((leqz_ind P f f0) a1 a2 l0) a3 a4 l1 
-((leqz_ind P f f0) a3 a4 l1))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma
deleted file mode 100644 (file)
index c115f9d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma
+++ /dev/null
@@ -1,188 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ex0/fwd.ma".
-
-include "basic_1/leq/fwd.ma".
-
-include "basic_1/aplus/props.ma".
-
-lemma aplus_gz_le:
- \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A 
-(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
-\def
- \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: 
-nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) 
-n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y 
-\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) 
-(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: 
-nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A 
-(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: 
-nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A 
-(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O 
-\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: 
-nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) 
-k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O 
-(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 
-n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) 
-(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: 
-nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S 
-(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) 
-(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz 
-(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: 
-nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz 
-(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S 
-k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda 
-(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz 
-(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) 
-k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: 
-A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) 
-k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S 
-n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
-
-lemma aplus_gz_ge:
- \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A 
-(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
-\def
- \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: 
-nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) 
-n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: 
-nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) 
-(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 
-h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda 
-(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz 
-(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le 
-(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: 
-nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) 
-(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 
-x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
-\Rightarrow True | (S _) \Rightarrow False])) I (S x) H0) in (False_ind (eq A 
-(asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) H2))))) (le_gen_S k0 O H))) 
-(\lambda (n0: nat).(\lambda (_: (((le (S k0) n0) \to (eq A (asucc gz (aplus 
-gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n))))).(\lambda (H0: (le (S k0) 
-(S n0))).(let H_y \def (le_S_n k0 n0 H0) in (eq_ind A (aplus gz (ASort n0 n) 
-k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0) n) k0)) a)) 
-(eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) k0) (\lambda (a: A).(eq A a 
-(aplus gz (ASort n0 n) k0))) (refl_equal A (aplus gz (ASort n0 n) k0)) (asucc 
-gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S n0) n))) 
-(ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
-
-lemma next_plus_gz:
- \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
-\def
- \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat 
-(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: 
-nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat 
-S (next_plus gz n n0) (plus n0 n) H))) h)).
-
-lemma leqz_leq:
- \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz 
-(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda 
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda 
-(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k 
-h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 
-(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def 
-(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort 
-h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 
-(le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) (le_n_S (S k) h1 H1)))))) in (let 
-H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort 
-(minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n 
-k h2 (le_S_n (S k) (S h2) (le_S (S (S k)) (S h2) (le_n_S (S k) h2 H2)))))) in 
-(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n _) 
-\Rightarrow n | (AHead _ _) \Rightarrow (minus h1 k)])) (ASort (minus h1 k) 
-n1) (ASort (minus h2 k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: 
-A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
-(ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in (\lambda (H7: (eq nat 
-(minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: nat).(leqz (ASort h1 
-n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: nat).(leqz (ASort h1 n1) 
-(ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal nat (plus h1 n1))) h2 
-(minus_minus k h1 h2 (le_S_n k h1 (le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) 
-(le_n_S (S k) h1 H1)))) (le_S_n k h2 (le_S_n (S k) (S h2) (le_S (S (S k)) (S 
-h2) (le_n_S (S k) h2 H2)))) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 
-k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a 
-(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 
-(le_S_n k h1 (le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) (le_n_S (S k) h1 
-H1)))))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: 
-A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus (minus k h2) n2)) 
-(aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat (minus h1 k) (\lambda 
-(n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) n2)))) H4 (S (minus 
-h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind A (ASort (S (minus 
-h1 (S k))) n1) (\lambda (ee: A).(match ee with [(ASort n _) \Rightarrow 
-(match n with [O \Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) 
-\Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in (False_ind 
-(leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 
-k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k 
-h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A 
-a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) 
-(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) 
-k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort 
-(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S_n (S k) (S h2) (le_S 
-(S (S k)) (S h2) (le_n_S (S k) h2 H2)))))) in (let H5 \def (sym_eq A (ASort O 
-(plus (minus k h1) n1)) (ASort (minus h2 k) n2) H4) in (let H6 \def (eq_ind 
-nat (minus h2 k) (\lambda (n: nat).(eq A (ASort n n2) (ASort O (plus (minus k 
-h1) n1)))) H5 (S (minus h2 (S k))) (minus_x_Sy h2 k H2)) in (let H7 \def 
-(eq_ind A (ASort (S (minus h2 (S k))) n2) (\lambda (ee: A).(match ee with 
-[(ASort n _) \Rightarrow (match n with [O \Rightarrow False | (S _) 
-\Rightarrow True]) | (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus 
-k h1) n1)) H6) in (False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H7))))))) 
-(\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) 
-(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus 
-k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz 
-(ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) 
-H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def 
-(f_equal A nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | 
-(AHead _ _) \Rightarrow (plus (minus k h1) n1)])) (ASort O (plus (minus k h1) 
-n1)) (ASort O (plus (minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 
-n1 n2 H1 H2 H5) in (leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: 
-A).(\lambda (a3: A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 
-a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda 
-(H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-
-lemma leq_leqz:
- \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind 
-(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda 
-(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus 
-h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A 
-(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) 
-(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A 
-(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) 
-(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus 
-(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus 
-h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) 
-(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: 
-nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus 
-h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq 
-A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz 
-n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 
-h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: 
-nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus 
-h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal 
-nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 
-n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) 
-(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) 
-(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) 
-(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus 
-(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) 
-(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 
-h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: 
-A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 
-a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda 
-(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma
deleted file mode 100644 (file)
index 4da7cc4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex1/defs.ma
+++ /dev/null
@@ -1,29 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-definition ex1_c:
- C
-\def
- CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O).
-
-definition ex1_t:
- T
-\def
- THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma
deleted file mode 100644 (file)
index 91ce174..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex1/props.ma
+++ /dev/null
@@ -1,516 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ex1/defs.ma".
-
-include "basic_1/ty3/fwd.ma".
-
-include "basic_1/pc3/fwd.ma".
-
-include "basic_1/nf2/pr3.ma".
-
-include "basic_1/nf2/props.ma".
-
-include "basic_1/arity/defs.ma".
-
-include "basic_1/leq/props.ma".
-
-fact ex1__leq_sort_SS:
- \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc 
-g (asucc g (ASort (S (S k)) n))))))
-\def
- \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g 
-(asucc g (ASort (S (S k)) n)))))).
-
-lemma ex1_arity:
- \forall (g: G).(arity g ex1_c ex1_t (ASort O O))
-\def
- \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S 
-(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) 
-(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) 
-O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g 
-(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) 
-(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) 
-O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead 
-(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) 
-(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort 
-O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort 
-O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort 
-O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) 
-O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))).
-
-lemma ex1_ty3:
- \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: 
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort 
-O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: 
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind 
-Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) 
-u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) 
-(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) 
-u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef 
-(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda 
-(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
-t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
-t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
-x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind 
-Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: 
-T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S 
-O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: 
-T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort 
-O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 
-x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
-x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef 
-(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
-T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) 
-O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
-T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) 
-x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
-(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
-x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 
-(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S 
-O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: 
-C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) 
-x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
-Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abbr) 
-x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) 
-H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda (H10: (ex3_3 C T T 
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
-T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x8) 
-x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
-(Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
-x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 
-(Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S 
-O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P (\lambda (x: 
-C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abst) 
-x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
-Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abbr) 
-x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) 
-H5))) in (False_ind P H17))))) H14)))))))) H10)) (ty3_gen_lref g (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) x6 (S (S O)) H8))))))) (ty3_gen_bind g Abst (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (TLRef (S (S O))) (TSort O) (THead (Bind Abst) x0 x1) H1)))))))) 
-H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e 
-u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e 
-u0 t)))) P (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: 
-(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (Bind Abst) (TLRef O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 
-x4)).(ex3_2_ind T T (\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 
-x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef 
-(S (S O))) t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: 
-T).(\lambda (x6: T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind 
-Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind 
-Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S 
-O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S 
-O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda 
-(x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
-(TLRef O)) (lift (S (S (S O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (CHead x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 
-x9)).(let H14 \def (getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) 
-(getl_gen_S (Bind Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind 
-C (\lambda (e: C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) 
-x8))) P (\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x 
-(CHead x7 (Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow x2 | (CHead c _ _) \Rightarrow c])) (CHead x2 
-(Bind Abst) x3) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) 
-x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) 
-H5))) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) 
-(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda 
-(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 
-(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def 
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def 
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead 
-(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) 
-(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort 
-O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead (CSort O) (Bind Abst) (TSort O)) 
-(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) x8) (TSort O) H23)) in 
-(False_ind P H24)))))))) H17))))) H14)))))))) H10)) (\lambda (H10: (ex3_3 C T 
-T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
-T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x8) 
-x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
-(Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def 
-(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind 
-Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: 
-C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P 
-(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 
-(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow x2 | (CHead c _ _) \Rightarrow c])) (CHead x2 (Bind 
-Abst) x3) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) 
-(TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) 
-in ((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) 
-(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda 
-(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 
-(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def 
-(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def 
-(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead 
-(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) 
-(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort 
-O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow x7 | (CHead c _ _) \Rightarrow c])) (CHead x7 
-(Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
-(CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in ((let H25 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow x8 | (CHead 
-_ _ t) \Rightarrow t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind 
-Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) 
-(TSort O) H23)) in (\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind 
-T x8 (\lambda (t: T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def 
-(eq_ind T x8 (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) 
-O t) x6)) H11 (TSort O) H25) in (let H29 \def (eq_ind C x7 (\lambda (c: 
-C).(ty3 g c (TSort O) x9)) H27 (CSort O) H26) in (or_ind (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O t) x4)))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C 
-T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) 
-x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P 
-(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: 
-T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: 
-T).(\lambda (x12: T).(\lambda (_: (pc3 (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x12) x4)).(\lambda (H32: 
-(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(CHead x10 (Bind Abbr) x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def 
-(eq_ind C (CHead x10 (Bind Abbr) x11) (\lambda (ee: C).(match ee with [(CSort 
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
-(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) 
-(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) 
-(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: 
-T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: 
-(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 
-\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow x10 | 
-(CHead c _ _) \Rightarrow c])) (CHead x10 (Bind Abst) x11) (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
-(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) 
-(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (CHead x10 (Bind Abst) x11) H32))) in ((let H35 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow x11 | (CHead _ _ t) 
-\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) 
-(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
-x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) 
-(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g 
-x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: 
-T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 
-(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) 
-(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
-x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind 
-b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S 
-O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) 
-O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
-t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
-Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O 
-(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
-O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) 
-(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
-O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O 
-(S O) O (le_O_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 
-(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
-(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda 
-(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead 
-(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
-O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort 
-O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead 
-(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) 
-(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
-(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) 
-(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead 
-(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort 
-O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) 
-x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
-(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) 
-(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind 
-Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) 
-(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T 
-(TLRef (S (S O))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef n) \Rightarrow (match n with [O \Rightarrow False | (S n0) 
-\Rightarrow (match n0 with [O \Rightarrow False | (S _) \Rightarrow True])]) 
-| (THead _ _ _) \Rightarrow False])) I (TLRef (S O)) H46) in (False_ind P 
-H47)))))) H43))))) (pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 
-x5 x1 H7))))))) H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) 
-H17))))) H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) 
-(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S 
-O)) H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) 
-(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
-(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead 
-(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
-Abst) (TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort 
-O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) 
-(TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma
deleted file mode 100644 (file)
index 6524d31..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex2/defs.ma
+++ /dev/null
@@ -1,28 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-definition ex2_c:
- C
-\def
- CSort O.
-
-definition ex2_t:
- T
-\def
- THead (Flat Appl) (TSort O) (TSort O).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
deleted file mode 100644 (file)
index 42c2de6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ex2/props.ma
+++ /dev/null
@@ -1,152 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ex2/defs.ma".
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/pr2/fwd.ma".
-
-include "basic_1/arity/fwd.ma".
-
-lemma ex2_nf2:
- nf2 ex2_c ex2_t
-\def
- \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) 
-(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 
-H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
-O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
-(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
-O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat 
-Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) 
-(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 
-(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) 
-x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 
-(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) 
-(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: 
-T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort 
-(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) 
-(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal 
-T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
-(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T 
-(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead 
-(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 
-x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def 
-(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) 
-(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) 
-x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 
-\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind Abst) x0 x1) H2) in (False_ind (eq T (THead (Flat 
-Appl) (TSort O) (TSort O)) (THead (Bind Abbr) (TSort O) x3)) H7)) t2 
-H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) 
-(Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) 
-(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) 
-(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq 
-T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) 
-x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) 
-(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead 
-(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in 
-(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O 
-H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) 
-t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind x0) x1 x2) H3) in (False_ind (eq T (THead (Flat Appl) 
-(TSort O) (TSort O)) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-(TSort O)) x3))) H9)) t2 H8))))))))))))))) H1)) H0))).
-
-lemma ex2_arity:
- \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: 
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat 
-Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def 
-(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda 
-(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) 
-(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) 
-(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let 
-H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O 
-(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a 
-a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P 
-(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: 
-(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def 
-(eq_ind A (ASort O O) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
-(False_ind P H7))))))) H3)))))) H0))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma
deleted file mode 100644 (file)
index 71e1028..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/flt/defs.ma
+++ /dev/null
@@ -1,29 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/C/defs.ma".
-
-definition fweight:
- C \to (T \to nat)
-\def
- \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
-
-definition flt:
- C \to (T \to (C \to (T \to Prop)))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt 
-(fweight c1 t1) (fweight c2 t2))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma
deleted file mode 100644 (file)
index a25e739..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/flt/fwd.ma
+++ /dev/null
@@ -1,49 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/flt/defs.ma".
-
-fact flt_wf__q_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C 
-\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq 
-nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall 
-(t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall 
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda 
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: 
-C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: 
-C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
-
-lemma flt_wf_ind:
- \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: 
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) 
-\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
-\def
- let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall 
-(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda 
-(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: 
-T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) 
-\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda 
-(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: 
-nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda 
-(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: 
-nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq 
-nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 
-(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 
-(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c 
-t))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma
deleted file mode 100644 (file)
index 395bb94..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/flt/props.ma
+++ /dev/null
@@ -1,105 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/flt/defs.ma".
-
-include "basic_1/C/props.ma".
-
-lemma flt_thead_sx:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c 
-(THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l 
-(tweight u) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
-u) (plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t))))))).
-
-lemma flt_thead_dx:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c 
-(THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l 
-(tweight t) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
-t) (plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t))))))).
-
-lemma flt_shift:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c 
-k u) t c (THead k u t)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat 
-(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt 
-(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus 
-(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus 
-(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight 
-c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight 
-t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S 
-(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) 
-(tweight t))))))).
-
-lemma flt_arith0:
- \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t 
-(CHead c k t) (TLRef i)))))
-\def
- \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: 
-nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
-
-lemma flt_arith1:
- \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle 
-(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: 
-nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
-\def
- \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda 
-(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: 
-K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) 
-(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H 
-(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: 
-nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) 
-(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) 
-(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S 
-O))))))))))).
-
-lemma flt_arith2:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 
-t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt 
-c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda 
-(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda 
-(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) 
-(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight 
-t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S 
-O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
-
-lemma cle_flt_trans:
- \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (c3: C).(\forall 
-(u2: T).(\forall (u3: T).((flt c2 u2 c3 u3) \to (flt c1 u2 c3 u3)))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
-c2))).(\lambda (c3: C).(\lambda (u2: T).(\lambda (u3: T).(\lambda (H0: (lt 
-(plus (cweight c2) (tweight u2)) (plus (cweight c3) (tweight 
-u3)))).(le_lt_trans (plus (cweight c1) (tweight u2)) (plus (cweight c2) 
-(tweight u2)) (plus (cweight c3) (tweight u3)) (le_plus_plus (cweight c1) 
-(cweight c2) (tweight u2) (tweight u2) H (le_n (tweight u2))) H0))))))).
-
-theorem flt_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 
-t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt 
-c1 t1 c3 t3))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: 
-T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 
-t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma
deleted file mode 100644 (file)
index 8d5cc21..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/defs.ma
+++ /dev/null
@@ -1,26 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubst0/defs.ma".
-
-inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
-| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 
-t2))
-| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 
-t1))
-| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: 
-C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma
deleted file mode 100644 (file)
index 59ca03b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/fsubst0/fwd.ma
+++ /dev/null
@@ -1,57 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/fsubst0/defs.ma".
-
-implied lemma fsubst0_ind:
- \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (t1: T).(\forall 
-(P: ((C \to (T \to Prop)))).(((\forall (t2: T).((subst0 i v t1 t2) \to (P c1 
-t2)))) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2 t1)))) \to 
-(((\forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: C).((csubst0 i v c1 
-c2) \to (P c2 t2)))))) \to (\forall (c: C).(\forall (t: T).((fsubst0 i v c1 
-t1 c t) \to (P c t)))))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (t1: T).(\lambda 
-(P: ((C \to (T \to Prop)))).(\lambda (f: ((\forall (t2: T).((subst0 i v t1 
-t2) \to (P c1 t2))))).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1 c2) 
-\to (P c2 t1))))).(\lambda (f1: ((\forall (t2: T).((subst0 i v t1 t2) \to 
-(\forall (c2: C).((csubst0 i v c1 c2) \to (P c2 t2))))))).(\lambda (c: 
-C).(\lambda (t: T).(\lambda (f2: (fsubst0 i v c1 t1 c t)).(match f2 with 
-[(fsubst0_snd x x0) \Rightarrow (f x x0) | (fsubst0_fst x x0) \Rightarrow (f0 
-x x0) | (fsubst0_both x x0 x1 x2) \Rightarrow (f1 x x0 x1 x2)]))))))))))).
-
-lemma fsubst0_gen_base:
- \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall 
-(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 
-c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 
-i v t1 t2) (csubst0 i v c1 c2)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind 
-i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v 
-t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 
-i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 
-(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 
-c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 
-i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i 
-v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 
-t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj 
-(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: 
-T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 
-i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 
-t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj 
-(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma
deleted file mode 100644 (file)
index 0915f5f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/clear.ma
+++ /dev/null
@@ -1,139 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/props.ma".
-
-include "basic_1/clear/drop.ma".
-
-lemma clear_getl_trans:
- \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to 
-(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: 
-C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 
-c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 
-c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 
-(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda 
-(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to 
-(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: 
-C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall 
-(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda 
-(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda 
-(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 
-c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) 
-\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: 
-K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) 
-c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda 
-(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to 
-(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c 
-(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def 
-(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in 
-(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) 
-(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda 
-(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n 
-H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) 
-t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 
-c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i).
-
-lemma getl_clear_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to 
-(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 
-c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all 
-c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
-C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 
-x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in 
-(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 
-(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 
-\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) 
-in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind 
-x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 
-c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 
-x1 c3 x2 H7)))))))) H4))))) H1))))))).
-
-lemma getl_clear_bind:
- \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c 
-(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) 
-\to (getl (S n) c e2))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: 
-C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: 
-C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda 
-(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) 
-(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: 
-(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) 
-(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: 
-C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: 
-C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda 
-(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear 
-(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: 
-nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 
-k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda 
-(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) 
-v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) (CHead e1 (Bind b) v) 
-(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in 
-((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 
-(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) 
-(clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in (\lambda (H6: (eq B b 
-b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: 
-C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(getl (S n) 
-(CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) n c0 e2 H8 t) b0 H6))))) H4)) 
-H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 
-(Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 
-(Bind b) v) t H2) e2 n H1) f t))) k H0))))))))))) c)).
-
-lemma getl_clear_conf:
- \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to 
-(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: 
-C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 
-c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 
-c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: 
-C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in 
-(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 
-(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 
-\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead 
-x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: 
-C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 
-(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall 
-(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) 
-\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall 
-(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) 
-c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) 
-(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) 
-c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: 
-C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: 
-C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda 
-(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: 
-K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl 
-(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) 
-t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c 
-(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 
-(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) 
-(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda 
-(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n 
-H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma
deleted file mode 100644 (file)
index a8f4df8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/dec.ma
+++ /dev/null
@@ -1,97 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/props.ma".
-
-lemma getl_dec:
- \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda 
-(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: 
-C).((getl i c d) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T 
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) 
-v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) 
-(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: 
-C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) 
-v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) 
-(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: 
-Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall 
-(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
-T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to 
-(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: 
-nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: 
-B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall 
-(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind 
-(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
-T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O 
-(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl 
-(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead 
-c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 
-(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: 
-C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e 
-(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def 
-(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: 
-B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: 
-C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: 
-C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e 
-(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to 
-(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda 
-(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T 
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) 
-v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl 
-O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O 
-(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: 
-C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind 
-x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
-T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: 
-C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro 
-C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat 
-f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) 
-O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to 
-(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: 
-B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) 
-(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) 
-(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: 
-Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t 
-(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: 
-nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
-(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: 
-C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H 
-(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda 
-(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall 
-(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T 
-(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) 
-(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to 
-(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda 
-(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind 
-C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead 
-e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
-(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: 
-C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: 
-C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 
-(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: 
-B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) 
-(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) 
-(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) 
-(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 
-(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) 
-c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: 
-C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind 
-b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: 
-Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) 
-d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) 
-i)))))) c).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma
deleted file mode 100644 (file)
index 9a21c5e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/drop/defs.ma".
-
-include "basic_1/clear/defs.ma".
-
-inductive getl (h: nat) (c1: C) (c2: C): Prop \def
-| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h 
-c1 c2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma
deleted file mode 100644 (file)
index 6b8d9a2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/drop.ma
+++ /dev/null
@@ -1,483 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/props.ma".
-
-include "basic_1/clear/drop.ma".
-
-lemma getl_drop:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
-nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
-C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to 
-(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) 
-u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) 
-e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: 
-T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 
-e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) 
-(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: 
-(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear 
-(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) 
-(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind 
-b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind b) u) (CHead 
-c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 
-\def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow b | 
-(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) \Rightarrow b1 | (Flat 
-_) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) 
-(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C 
-T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) 
-\Rightarrow t0])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind 
-b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: 
-(eq C e c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind 
-b0) t) c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) 
-c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) 
-H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e 
-(Bind b) u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead 
-e (Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O 
-(drop_refl e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) 
-(\lambda (n: nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) 
-\to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k 
-t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) 
-(\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e 
-(Bind b) u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
-
-lemma getl_drop_conf_lt:
- \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
-h d c0 e0)))))))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: 
-C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) 
-c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i 
-(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i 
-(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda 
-(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda 
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
-h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) 
-(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def 
-(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C 
-(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 
-(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: 
-C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead 
-c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to 
-((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
-C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead 
-e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) 
-(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda 
-(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind 
-b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda 
-(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
-\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda 
-(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda 
-(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: 
-K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) 
-(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: 
-B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda 
-(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def 
-(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c1 | 
-(CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) 
-t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H10 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow b | 
-(CHead _ k1 _) \Rightarrow (match k1 with [(Bind b1) \Rightarrow b1 | (Flat 
-_) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) 
-(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def 
-(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
-(CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) 
-t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in (\lambda (H12: (eq 
-B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 \def (eq_ind_r T t0 (\lambda 
-(t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind b0) t1))) H7 u H11) in (let 
-H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i O (CHead c0 k t) (CHead x0 
-(Bind b1) u))) H14 b H12) in (let H16 \def (eq_ind_r C x0 (\lambda (c2: 
-C).((drop i O (CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in (let H17 \def 
-(eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead c2 (Bind b) 
-u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) 
-d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: 
-T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) d) 
-x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: 
-(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: 
-T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) 
-H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
-T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) 
-(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind 
-b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 
-e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O 
-(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat 
-f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n 
-d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) 
-\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: 
-T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
-h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) 
-e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda 
-(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
-\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | (CHead c2 _ _) 
-\Rightarrow c2])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead 
-c0 k t) (CHead x0 (Flat f) t0) H10)) in ((let H12 \def (f_equal C K (\lambda 
-(e0: C).(match e0 with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow 
-k1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) 
-(CHead x0 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) 
-(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 
-(Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda (H15: (eq C 
-c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: C).(clear c2 (CHead c1 
-(Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) c0 H15) in 
-(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O (CHead c0 k t) c2) 
-\to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda 
-(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))))))) IHx0 c0 H15) in (let H18 \def (eq_ind K k (\lambda (k1: K).((drop O 
-O (CHead c0 k1 t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) H14) in (let H19 \def 
-(eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) (CHead c0 k1 t) e)) H9 
-(Flat f) H14) in (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e 
-(CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r 
-(Flat f) (plus O d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat 
-f) (plus O d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda 
-(H21: (eq T t (lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r 
-(Flat f) (plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) 
-t (lift h (r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e 
-(\lambda (c2: C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead 
-c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
-h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 
-(Flat f) x2) H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: 
-C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda 
-(_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t 
-(\lambda (t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead 
-c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
-h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))))))) H24 (lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O 
-c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda 
-(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O 
-(CHead x1 (Flat f) x2) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T 
-u (lift h d x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda 
-(H29: (drop h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O 
-(CHead c0 (Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) 
-t1)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 
-(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 
-(lift h d x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 
-(CHead c1 (Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) 
-(\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h 
-d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) 
-(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) 
-x2) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))) x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) 
-O H28 f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h 
-(plus O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda 
-(IHi: (((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k 
-t) (CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear 
-x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq 
-T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 
-(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus 
-(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) 
-(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) 
-\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda 
-(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) 
-e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 
-k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) 
-v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) 
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
-(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r 
-k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 
-x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S 
-i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S 
-(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 
-(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 
-(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let 
-H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to 
-((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
-C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda 
-(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T 
-t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead 
-c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
-h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) 
-(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
-e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind 
-T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k 
-x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 
-O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S 
-(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) 
-(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S 
-i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h 
-(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k 
-i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) 
-(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 
-(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d 
-H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
-v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda 
-(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: 
-C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 
-(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def 
-(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S 
-(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C 
-(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) 
-H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind 
-b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) 
-in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: 
-T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
-T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
-T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: 
-T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift 
-h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) 
-H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k 
-H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
-
-lemma getl_drop_conf_ge:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d 
-h) i) \to (getl (minus i h) e a)))))))))
-\def
- \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c 
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
-d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) 
-in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 
-a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c 
-x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i 
-x c H3 e h d H0 H1) H4)))) H2)))))))))).
-
-lemma getl_conf_ge_drop:
- \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 
-c2) \to (drop i O c2 e))))))))
-\def
- \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda 
-(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) 
-(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e 
-u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S 
-i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in 
-H3)))))))).
-
-lemma getl_drop_conf_rev:
- \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
-(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i 
-c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) 
-(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
-\def
- \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 
-e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 
-H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
-
-lemma drop_getl_trans_lt:
- \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: 
-C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda 
-(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda 
-(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
-\def
- \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
-c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i 
-c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) 
-v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: 
-C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead 
-e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d 
-(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: 
-(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 
-e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: 
-C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: 
-C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
-c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat 
-(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) 
-(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b 
-e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h 
-(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C 
-(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) 
-(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda 
-(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda 
-(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i 
-c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h 
-(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus 
-d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d 
-(le_S_n (S i) (S d) (le_S (S (S i)) (S d) (le_n_S (S i) d H)))) c1 c2 h H0 x 
-H3))))) H2)))))))))))).
-
-lemma drop_getl_trans_le:
- \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 
-e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) 
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: 
-C).(\lambda (e1: C).(clear e1 e2))))))))))))
-\def
- \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
-c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def 
-(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
-(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: 
-C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) 
-e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: 
-C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def 
-(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i 
-O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda 
-(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: 
-C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
-e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h 
-(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i 
-O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) 
-(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) 
-H2)))))))))).
-
-lemma drop_getl_trans_ge:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
-nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) 
-\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: 
-C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def 
-(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
-(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: 
-C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro 
-(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
-
-lemma getl_drop_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to 
-(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i 
-h)) O c1 e2)))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
-nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 
-e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: 
-C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: 
-C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 
-H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda 
-(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: 
-C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 
-e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall 
-(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: 
-C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead 
-c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: 
-C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) 
-t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop 
-(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead 
-c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S 
-i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) 
-H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 
-(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead 
-c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 
-t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: 
-(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: 
-nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) 
-e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
-C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S 
-i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop 
-(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 
-i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: 
-F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: 
-nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: 
-nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) 
-e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: 
-C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) 
-(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) 
-(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) 
-t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to 
-(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i 
-n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 
-(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) 
-O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S 
-(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma
deleted file mode 100644 (file)
index 6387630..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/flt.ma
+++ /dev/null
@@ -1,60 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/fwd.ma".
-
-include "basic_1/flt/props.ma".
-
-lemma getl_flt:
- \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
-C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to 
-(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) 
-u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef 
-i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: 
-T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef 
-i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) 
-(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: 
-(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear 
-(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef 
-O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e 
-(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 with 
-[(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind b) 
-u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) 
-in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 
-(Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 
-\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) 
-(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b 
-b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 
-(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t 
-(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b (\lambda (b1: B).(flt c0 t 
-(CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) 
-u H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) 
-t) (CHead e (Bind b) u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e 
-(Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) 
-k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: 
-nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u 
-(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead 
-e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind 
-b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma
deleted file mode 100644 (file)
index 7390775..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/fwd.ma
+++ /dev/null
@@ -1,154 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/defs.ma".
-
-include "basic_1/drop/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-implied lemma getl_ind:
- \forall (h: nat).(\forall (c1: C).(\forall (c2: C).(\forall (P: 
-Prop).(((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to P)))) \to 
-((getl h c1 c2) \to P)))))
-\def
- \lambda (h: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (P: 
-Prop).(\lambda (f: ((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to 
-P))))).(\lambda (g: (getl h c1 c2)).(match g with [(getl_intro x x0 x1) 
-\Rightarrow (f x x0 x1)])))))).
-
-lemma getl_gen_all:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 
-C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 
-c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
-C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda 
-(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda 
-(e0: C).(clear e0 c2)) e H0 H1)))) H)))).
-
-lemma getl_gen_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h 
-(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) 
-in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear 
-e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: 
-(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P 
-(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: 
-(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 
-(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 
-H1))))) H0)))))).
-
-lemma getl_gen_O:
- \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
-\def
- \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def 
-(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) 
-(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: 
-(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 
-(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
-
-lemma getl_gen_S:
- \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
-nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: 
-nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all 
-(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c 
-k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: 
-C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 
-x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
-
-lemma getl_gen_2:
- \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 
-B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind 
-b) v)))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 
-c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: 
-C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: 
-B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) 
-(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x 
-c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: 
-B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B 
-C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind 
-b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: 
-(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: 
-C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind 
-x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda 
-(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: 
-B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c 
-(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) 
-H3))))) H0))))).
-
-lemma getl_gen_flat:
- \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
-\def
- \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n 
-e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e 
-(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d 
-H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to 
-(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) 
-d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
-
-lemma getl_gen_bind:
- \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d 
-(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda 
-(j: nat).(getl j e d)))))))))
-\def
- \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or 
-(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: 
-nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl 
-O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: 
-C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda 
-(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl 
-(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat 
-(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e 
-(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind 
-b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d 
-(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: 
-nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n 
-O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S 
-j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e 
-(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) 
-v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl 
-j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: 
-nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n 
-H0)))))) i))))).
-
-theorem getl_mono:
- \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to 
-(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
-\def
- \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c 
-x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all 
-c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: 
-C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c 
-x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in 
-(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq 
-C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear 
-x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 
-(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: 
-C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def 
-(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 
-H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma
deleted file mode 100644 (file)
index 64e3319..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/getl.ma
+++ /dev/null
@@ -1,51 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/drop.ma".
-
-include "basic_1/getl/clear.ma".
-
-lemma getl_conf_le:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
-(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e 
-a))))))))
-\def
- \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c 
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda 
-(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda 
-(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e 
-a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x 
-e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) 
-e H4)))) H2))))))))).
-
-theorem getl_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl 
-h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 
-e2)))))))
-\def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: 
-nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 
-e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: 
-C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) 
-(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x 
-e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 
-e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: 
-C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O 
-h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to 
-(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def 
-(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x 
-H_y H3))))) i H2)))) H1)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma b/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma
deleted file mode 100644 (file)
index 56592c5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/getl/props.ma
+++ /dev/null
@@ -1,72 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/getl/fwd.ma".
-
-include "basic_1/clear/props.ma".
-
-include "basic_1/drop/props.ma".
-
-lemma getl_refl:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) 
-(CHead c (Bind b) u))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind 
-b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) 
-u)) (clear_bind b c u)))).
-
-lemma getl_head:
- \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k 
-h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
-\def
- \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
-(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k 
-h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: 
-C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: 
-(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k 
-u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
-
-lemma getl_flat:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall 
-(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c 
-e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in 
-(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) 
-(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c 
-x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to 
-(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def 
-(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in 
-(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c 
-(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: 
-(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop 
-(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat 
-f) h0 c x H3 u) H2)))) h H1)))) H0))))))).
-
-lemma getl_ctail:
- \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: 
-T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda 
-(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C 
-(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) 
-u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: 
-C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) 
-u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v 
-x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma
deleted file mode 100644 (file)
index 54a5e01..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/iso/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-inductive iso: T \to (T \to Prop) \def
-| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
-| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2)))
-| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: 
-T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma
deleted file mode 100644 (file)
index fc3550a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/iso/fwd.ma
+++ /dev/null
@@ -1,184 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/iso/defs.ma".
-
-include "basic_1/tlist/defs.ma".
-
-implied lemma iso_ind:
- \forall (P: ((T \to (T \to Prop)))).(((\forall (n1: nat).(\forall (n2: 
-nat).(P (TSort n1) (TSort n2))))) \to (((\forall (i1: nat).(\forall (i2: 
-nat).(P (TLRef i1) (TLRef i2))))) \to (((\forall (v1: T).(\forall (v2: 
-T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).(P (THead k v1 t1) 
-(THead k v2 t2)))))))) \to (\forall (t: T).(\forall (t0: T).((iso t t0) \to 
-(P t t0)))))))
-\def
- \lambda (P: ((T \to (T \to Prop)))).(\lambda (f: ((\forall (n1: 
-nat).(\forall (n2: nat).(P (TSort n1) (TSort n2)))))).(\lambda (f0: ((\forall 
-(i1: nat).(\forall (i2: nat).(P (TLRef i1) (TLRef i2)))))).(\lambda (f1: 
-((\forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).(P (THead k v1 t1) (THead k v2 t2))))))))).(\lambda (t: T).(\lambda 
-(t0: T).(\lambda (i: (iso t t0)).(match i with [(iso_sort x x0) \Rightarrow 
-(f x x0) | (iso_lref x x0) \Rightarrow (f0 x x0) | (iso_head x x0 x1 x2 x3) 
-\Rightarrow (f1 x x0 x1 x2 x3)]))))))).
-
-lemma iso_gen_sort:
- \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda 
-(n2: nat).(eq T u2 (TSort n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) 
-u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex 
-nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: 
-(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) 
-\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: 
-nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort n) \Rightarrow n | 
-(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
-n1) H1) in (ex_intro nat (\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 
-(refl_equal T (TSort n2))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda 
-(H1: (eq T (TLRef i1) (TSort n1))).(let H2 \def (eq_ind T (TLRef i1) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-True | (THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex 
-nat (\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
-K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T 
-(THead k v1 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False 
-| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort 
-n1) H1) in (False_ind (ex nat (\lambda (n2: nat).(eq T (THead k v2 t2) (TSort 
-n2)))) H2)))))))) y u2 H0))) H))).
-
-lemma iso_gen_lref:
- \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda 
-(n2: nat).(eq T u2 (TLRef n2))))))
-\def
- \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) 
-u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex 
-nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: 
-(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) 
-\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: 
-nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 
-\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n3: nat).(eq T 
-(TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: nat).(\lambda (i2: 
-nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow i1 | (TLRef n) 
-\Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in 
-(ex_intro nat (\lambda (n2: nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal 
-T (TLRef i2))))))) (\lambda (v1: T).(\lambda (v2: T).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H1: (eq T (THead k v1 t1) 
-(TLRef n1))).(let H2 \def (eq_ind T (THead k v1 t1) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ 
-_ _) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: 
-nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
-
-lemma iso_gen_head:
- \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso 
-(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
-(THead k v2 t2)))))))))
-\def
- \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda 
-(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: 
-T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: 
-T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y 
-u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to 
-(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) 
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k 
-v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda 
-(v2: T).(\lambda (t2: T).(eq T (TSort n2) (THead k v2 t2))))) H2))))) 
-(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (THead k 
-v1 t1))).(let H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda 
-(v2: T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) 
-(\lambda (v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(k0: K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def 
-(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | (TLRef 
-_) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 v0 t0) (THead 
-k v1 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) 
-\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k 
-v1 t1) H1) in (\lambda (_: (eq T v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r 
-K k (\lambda (k1: K).(ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead 
-k1 v2 t2) (THead k v3 t3)))))) (ex_2_intro T T (\lambda (v3: T).(\lambda (t3: 
-T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 
-t2))) k0 H6)))) H3)) H2)))))))) y u2 H0))) H))))).
-
-lemma iso_flats_lref_bind_false:
- \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall 
-(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind 
-b) v t)) \to (\forall (P: Prop).P)))))))
-\def
- \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda 
-(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads 
-(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) 
-(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let 
-H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in 
-(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P 
-(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let 
-H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef x) H1) in (False_ind P H2)))) H0))))) (\lambda 
-(t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f) t1 (TLRef 
-i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso 
-(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) (THead (Bind b) v 
-t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f) t0 (THeads (Flat 
-f) t1 (TLRef i)) (THead (Bind b) v t) H0) in (let H1 \def H_x in (ex_2_ind T 
-T (\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Bind b) v t) (THead (Flat 
-f) v2 t2)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T (THead 
-(Bind b) v t) (THead (Flat f) x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) 
-v t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat f) x0 x1) 
-H2) in (False_ind P H3))))) H1)))))))) vs)))))).
-
-lemma iso_flats_flat_bind_false:
- \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall 
-(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads 
-(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: 
-Prop).P)))))))))
-\def
- \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda 
-(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind 
-(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) 
-(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead 
-(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def 
-(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in 
-(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) 
-(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: 
-(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T 
-(THead (Bind b) v t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) 
-(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead 
-(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v 
-t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads 
-(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 
-\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead 
-(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 
-\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) 
-vs)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma b/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma
deleted file mode 100644 (file)
index baef0f8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/iso/props.ma
+++ /dev/null
@@ -1,52 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/fwd.ma".
-
-include "basic_1/iso/fwd.ma".
-
-lemma iso_refl:
- \forall (t: T).(iso t t)
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: 
-nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: 
-(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
-
-theorem iso_trans:
- \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 
-t3) \to (iso t1 t3)))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) 
-(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso 
-(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in 
-(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) 
-(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) 
-(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) 
-(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso 
-(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in 
-(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) 
-(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) 
-(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) 
-(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def 
-(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda 
-(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 
-x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) 
-(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
deleted file mode 100644 (file)
index 42005d6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
+++ /dev/null
@@ -1,447 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/leq/props.ma".
-
-lemma asucc_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
-(asucc g a1) (asucc g a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g 
-a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: 
-nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) 
-(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O 
-(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow 
-(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq 
-A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: 
-nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g 
-(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S 
-h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) 
-(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind 
-A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O 
-(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq 
-A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) 
-(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) 
-(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) 
-H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g 
-(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: 
-nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) 
-\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next 
-g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g 
-(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) 
-n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g 
-(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: 
-A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) 
-n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) 
-k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort 
-O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) 
-(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) 
-(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g 
-n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort 
-O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A 
-(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda 
-(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to 
-((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g 
-(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow 
-(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) 
-\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O 
-\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) 
-(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) 
-k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) 
-\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
-\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 
-(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A 
-(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) 
-(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g 
-(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O 
-n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort 
-(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) 
-(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda 
-(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
-h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) 
-\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
-\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g 
-n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 
-with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h 
-n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
-(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
-(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next 
-g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 
-n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a 
-(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) 
-(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A 
-(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g 
-(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S 
-h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) 
-(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) 
-(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda 
-(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g 
-(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: 
-(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g 
-a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-
-lemma asucc_inj:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc 
-g a2)) \to (leq g a1 a2))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g 
-(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda 
-(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) 
-(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort 
-n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) 
-(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 
-n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g 
-(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g 
-(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 
-g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind 
-nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: 
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort 
-h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) 
-(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
-[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g 
-n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match 
-e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with 
-[(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) 
-H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda 
-(n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) 
-x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A 
-(aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) 
-H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda 
-(a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) 
-(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) 
-a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in 
-(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: 
-nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) 
-\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g 
-(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O 
-(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
-O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda 
-(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g 
-(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: 
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
-x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
-\Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) 
-H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ 
-n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 
-x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 
-(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort 
-n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: 
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 
-n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) 
-(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) 
-(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 
-(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in 
-(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) 
-(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g 
-(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq 
-g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda 
-(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to 
-((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort 
-n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) 
-(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O 
-n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) 
-\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 
-n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
-(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
-n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g 
-(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus 
-g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
-\Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1 
-x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort 
-_ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _) 
-\Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda 
-(H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A 
-(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9 
-\def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2) 
-(aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A 
-(aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g 
-n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0 
-n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2) 
-(\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g 
-(ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O 
-n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_: 
-(((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g 
-(asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) 
-(ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda 
-(H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) 
-n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4) 
-n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def 
-(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind 
-nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
-(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: 
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 
-n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: 
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g 
-(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 
-n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
-[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2) 
-(ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e 
-with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 
-n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r 
-nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 
-x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A 
-(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 
-\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g 
-(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) 
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) 
-a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) 
-in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 
-H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n 
-n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda 
-(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) 
-a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a 
-a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g 
-a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) 
-(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 
-n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) 
-(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O 
-n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq 
-g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g 
-(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc 
-g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g 
-n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g 
-(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda 
-(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g 
-(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 
-x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match 
-ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
-(ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8))))))) 
-H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) 
-(asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 
-n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort 
-n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a 
-a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to 
-(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) 
-n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g 
-(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def 
-(leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in 
-(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
-nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
-(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) 
-(ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0: 
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort 
-n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g 
-a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda 
-(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
-\Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1) 
-n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: 
-A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq 
-g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g 
-a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: 
-A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a 
-a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g 
-(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 
-n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O 
-n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) 
-H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
-A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) 
-(\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3 
-a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1: 
-A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda 
-(H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A 
-(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
-(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: 
-nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) 
-\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g 
-(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a 
-(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda 
-(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq 
-g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) 
-(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g 
-a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def 
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
-(False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1)))) 
-(\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) 
-\to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc 
-g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: 
-(leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def 
-(leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def 
-H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5))) 
-(\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5: 
-A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g 
-(AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: 
-(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A 
-(AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda 
-(e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow 
-a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A 
-(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead 
-_ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda 
-(H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g 
-(asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0 
-(\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0 
-a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
-
-lemma leq_asucc:
- \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
-a0)))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: 
-A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro 
-A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) 
-(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda 
-(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A 
-(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A 
-(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g 
-(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc 
-g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) 
-(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) 
-a)).
-
-lemma leq_ahead_asucc_false:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 
-(asucc g a1)) \to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
-A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda 
-(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead 
-(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: 
-nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O 
-(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g 
-(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 
-g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind 
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: 
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
-(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
-A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda 
-(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A 
-(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in 
-(False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead 
-(ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S 
-n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) 
-a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda 
-(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) 
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
-(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort 
-n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee: 
-A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) 
-(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g 
-a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall 
-(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: 
-Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) 
-(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g 
-(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind 
-A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: 
-A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
-(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
-A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 
-x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def 
-(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
-(AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in 
-((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) 
-\Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g 
-a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A 
-x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def 
-(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in 
-(leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
-
-lemma leq_asucc_false:
- \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
-Prop).P)))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) 
-a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda 
-(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
-\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
-(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g 
-n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: 
-(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O 
-(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
-O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda 
-(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda 
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g 
-(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A 
-(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: 
-A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O])) 
-(ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e: 
-A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) 
-(ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def 
-(eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) 
-(aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 
-(\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O 
-n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g 
-n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g 
-(ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def 
-(aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 
-(\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) 
-H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow 
-(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to 
-P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def 
-(leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in 
-(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
-nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
-(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort 
-h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: 
-nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) 
-x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def 
-(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
-(AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in 
-((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2) 
-\Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1 
-x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1 
-(\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0) 
-x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A 
-(aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let 
-H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0 
-(aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2)) 
-(aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2 
-(ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le 
-n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda 
-(a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: 
-Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to 
-(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead 
-a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) 
-(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
-(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) 
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 
-x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) 
-(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1) 
-(AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) 
-(AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 
-(\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def 
-(eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) 
-H6))))))) H2))))))))) a)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma
deleted file mode 100644 (file)
index 45a4288..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma
+++ /dev/null
@@ -1,26 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/aplus/defs.ma".
-
-inductive leq (g: G): A \to (A \to Prop) \def
-| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
-(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
-| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: 
-A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 
-a4))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma
deleted file mode 100644 (file)
index 4fc6915..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma
+++ /dev/null
@@ -1,254 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/leq/defs.ma".
-
-implied rec lemma leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1: 
-nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k: 
-nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (P 
-(ASort h1 n1) (ASort h2 n2))))))))) (f0: (\forall (a1: A).(\forall (a2: 
-A).((leq g a1 a2) \to ((P a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leq 
-g a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead a2 a4))))))))))) (a: A) 
-(a0: A) (l: leq g a a0) on l: P a a0 \def match l with [(leq_sort h1 h2 n1 n2 
-k e) \Rightarrow (f h1 h2 n1 n2 k e) | (leq_head a1 a2 l0 a3 a4 l1) 
-\Rightarrow (f0 a1 a2 l0 ((leq_ind g P f f0) a1 a2 l0) a3 a4 l1 ((leq_ind g P 
-f f0) a3 a4 l1))].
-
-lemma leq_gen_sort1:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
-g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda 
-(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort 
-h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A 
-a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g 
-(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat 
-nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a 
-k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: 
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
-k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal 
-A nat (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H2) in ((let H4 \def (f_equal A 
-nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 
-h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 
-n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
-nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda 
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 
-n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort 
-n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) 
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) 
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
-(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 
-H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
-(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) 
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to 
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
-nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 
-n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def 
-(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H5) in 
-(False_ind (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(k: nat).(eq A (aplus g (AHead a1 a4) k) (aplus g (ASort h2 n2) k))))) 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a3 a5) 
-(ASort h2 n2)))))) H6))))))))))) y a2 H0))) H))))).
-
-lemma leq_gen_head1:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g 
-(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
-a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
-(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
-g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
-a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
-(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda 
-(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq 
-A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) 
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match 
-ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead a1 a2) H2) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (_: 
-A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda 
-(a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) (AHead a3 a4))))) H3))))))))) 
-(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: 
-(((eq A a0 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: 
-A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda 
-(a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))))))).(\lambda (a4: 
-A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a4 
-(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a1 
-a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda (a6: 
-A).(\lambda (a7: A).(eq A a5 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead 
-a0 a4) (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e 
-with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 
-a4) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
-with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 
-a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 a1)).(let H9 \def (eq_ind A 
-a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: 
-A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 
-a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A a5 (AHead a7 a8))))))) H4 a2 
-H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: A).(leq g a6 a5)) H3 a2 H7) 
-in (let H11 \def (eq_ind A a0 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: 
-A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
-a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 \def (eq_ind A a0 (\lambda (a6: 
-A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A A (\lambda (a6: A).(\lambda (_: 
-A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda 
-(a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) a3 a5 H12 H10 
-(refl_equal A (AHead a3 a5))))))))) H6))))))))))) y a H0))) H))))).
-
-lemma leq_gen_sort2:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
-g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) 
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
-(ASort h2 n2))))))))))
-\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
-A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda 
-(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) 
-(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq 
-A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind 
-g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat 
-nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus 
-g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: 
-nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
-nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
-k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal 
-A nat (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n2) (ASort h1 n1) H2) in ((let H4 \def (f_equal A 
-nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) 
-\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 
-h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 
-n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
-nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
-nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda 
-(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 
-n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort 
-h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
-(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) 
-(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) 
-(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 
-n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
-(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 
-H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
-(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
-h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
-A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to 
-(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
-nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 
-n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def 
-(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee with [(ASort _ _) 
-\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H5) in 
-(False_ind (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (AHead a3 a5) k))))) 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a1 a4) 
-(ASort h2 n2)))))) H6))))))))))) a2 y H0))) H))))).
-
-lemma leq_gen_head2:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a 
-(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 
-a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
-(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
-g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
-a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
-(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 
-a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda 
-(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq 
-A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) 
-(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match 
-ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead a1 a2) H2) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (_: 
-A).(leq g a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda 
-(a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) (AHead a3 a4))))) H3))))))))) 
-(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: 
-(((eq A a3 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: 
-A).(leq g a4 a1))) (\lambda (_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda 
-(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))))))).(\lambda (a4: 
-A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a5 
-(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a6 
-a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda (a6: 
-A).(\lambda (a7: A).(eq A a4 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead 
-a3 a5) (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e 
-with [(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 
-a5) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
-with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) (AHead a3 
-a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 a1)).(let H9 \def (eq_ind A 
-a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: 
-A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 
-a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A a4 (AHead a7 a8))))))) H4 a2 
-H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: A).(leq g a4 a6)) H3 a2 H7) 
-in (let H11 \def (eq_ind A a3 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
-(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: 
-A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
-a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 \def (eq_ind A a3 (\lambda (a6: 
-A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A A (\lambda (a6: A).(\lambda (_: 
-A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda 
-(a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) (AHead a6 a7)))) a0 a4 H12 H10 
-(refl_equal A (AHead a0 a4))))))))) H6))))))))))) a y H0))) H))))).
-
-lemma ahead_inj_snd:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
-(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda 
-(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def 
-(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A 
-(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda 
-(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) 
-(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda 
-(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead 
-a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e 
-with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a4) 
-(AHead x0 x1) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with 
-[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) 
-(AHead x0 x1) H3) in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 
-(\lambda (a: A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 
-(\lambda (a: A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma
deleted file mode 100644 (file)
index 6eec657..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma
+++ /dev/null
@@ -1,188 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/leq/fwd.ma".
-
-include "basic_1/aplus/props.ma".
-
-lemma leq_refl:
- \forall (g: G).(\forall (a: A).(leq g a a))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) 
-(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A 
-(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 
-a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 
-H0))))) a)).
-
-lemma leq_eq:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 
-a2))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 
-a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
-
-lemma leq_sym:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
-a2 a1))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: 
-nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: 
-nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: 
-(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: 
-A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 
-H1 a6 a5 H3))))))))) a1 a2 H)))).
-
-theorem leq_trans:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
-(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 
-a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
-nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort 
-h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g 
-(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 
-\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 
-n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 
-(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: 
-nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus 
-g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def 
-(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 
-x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 
-n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g 
-(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r 
-nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) 
-(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) 
-x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus 
-g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 
-H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) 
-(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus 
-(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g 
-(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 
-x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g 
-(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: 
-A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: 
-A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: 
-A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) 
-\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) 
-a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in 
-(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: 
-A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
-a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: 
-A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: 
-(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 
-(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead 
-a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) 
-H5))))))))))))) a1 a2 H)))).
-
-lemma leq_ahead_false_1:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 
-\to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
-A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n 
-n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
-(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
-(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) 
-a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: 
-A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
-(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O 
-n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: 
-A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: 
-nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to 
-P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1) 
-n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort (S n1) n0) 
-H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
-A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 
-a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3 
-a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) 
-n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort (S n1) n0) 
-(AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda (ee: 
-A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) 
-(\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to 
-(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: 
-A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: 
-A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: 
-Prop).(let H_x \def (leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let 
-H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead 
-a a0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq 
-g a2 x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def 
-(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
-(AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7 
-\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | 
-(AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda 
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) 
-H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a 
-a0) a3)) H3 a H8) in (H a0 H10 P))))) H6))))))) H2)))))))))) a1)).
-
-lemma leq_ahead_false_2:
- \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) 
-\to (\forall (P: Prop).P))))
-\def
- \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: 
-A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort 
-n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
-(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
-a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O 
-n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
-A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
-(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
-(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
-x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) 
-(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match 
-ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
-(AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: 
-nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to 
-P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) n0)) (ASort (S n1) 
-n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) (ASort (S n1) n0) 
-H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
-A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (ASort (S n1) n0) 
-a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3 
-a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
-x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort (S 
-n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda 
-(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n 
-H)))))) (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) 
-a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall 
-(a1: A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: 
-A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: 
-Prop).(let H_x \def (leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let 
-H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
-a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda 
-(a3: A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead 
-a a0) x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def 
-(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
-(AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7 
-\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | 
-(AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda 
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead 
-a a0) a3)) H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g 
-a1 a3)) H3 a H8) in (H0 a H9 P))))) H6))))))) H2)))))))))) a2)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma
deleted file mode 100644 (file)
index 6c59592..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/defs.ma
+++ /dev/null
@@ -1,35 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/tlist/defs.ma".
-
-include "basic_1/s/defs.ma".
-
-rec definition lref_map (f: (nat \to nat)) (d: nat) (t: T) on t: T \def match 
-t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match 
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u 
-t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))].
-
-definition lift:
- nat \to (nat \to (T \to T))
-\def
- \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: 
-nat).(plus x h)) i t))).
-
-rec definition lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def match ts 
-with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) 
-(lifts h d ts0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma
deleted file mode 100644 (file)
index beb400a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/fwd.ma
+++ /dev/null
@@ -1,634 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/props.ma".
-
-lemma lift_gen_sort:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T 
-(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind 
-(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) 
-(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort 
-n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: 
-(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort 
-n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let 
-H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (lift h d (TLRef n0)) H) in (False_ind (lt n0 d) H1)))) in (let H2 
-\def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) 
-(\lambda (_: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda 
-(t0: T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let 
-H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (lift h d (TLRef n0)) H) in (False_ind (le d n0) H1)))) in (let H2 
-\def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (TLRef (plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) 
-H2))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) 
-(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq 
-T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T 
-(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d 
-(THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d 
-t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T 
-(TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k 
-(lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k t0 t1) 
-(TSort n)) H3))))))))) t)))).
-
-lemma lift_gen_lref:
- \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
-(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le 
-(plus d h) i) (eq T t (TLRef (minus i h)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: 
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) 
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
-h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda 
-(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def 
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort 
-n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n) H0) in (False_ind (or (land 
-(lt i d) (eq T (TSort n) (TLRef i))) (land (le (plus d h) i) (eq T (TSort n) 
-(TLRef (minus i h))))) H1)))))))) (\lambda (n: nat).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H: (eq T (TLRef i) (lift h 
-d (TLRef n)))).(lt_le_e n d (or (land (lt i d) (eq T (TLRef n) (TLRef i))) 
-(land (le (plus d h) i) (eq T (TLRef n) (TLRef (minus i h))))) (\lambda (H0: 
-(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T 
-(TLRef i) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (f_equal 
-T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0) 
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H1) in 
-(eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef 
-n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) 
-(or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus d h) n) 
-(eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) (TLRef 
-n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d n)).(let H1 
-\def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H 
-(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0) 
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef (plus n h)) 
-H1) in (eq_ind_r nat (plus n h) (\lambda (n0: nat).(or (land (lt n0 d) (eq T 
-(TLRef n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus 
-n0 h)))))) (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt (plus n h) d) (eq 
-T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h) (plus n h)) (eq T 
-(TLRef n) (TLRef n0))))) (or_intror (land (lt (plus n h) d) (eq T (TLRef n) 
-(TLRef (plus n h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef 
-n))) (conj (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n)) 
-(le_plus_plus d n h h H0 (le_n h)) (refl_equal T (TLRef n)))) (minus (plus n 
-h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: ((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
-(TLRef i) (lift h d t0)) \to (or (land (lt i d) (eq T t0 (TLRef i))) (land 
-(le (plus d h) i) (eq T t0 (TLRef (minus i h))))))))))).(\lambda (t1: 
-T).(\lambda (_: ((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
-(TLRef i) (lift h d t1)) \to (or (land (lt i d) (eq T t1 (TLRef i))) (land 
-(le (plus d h) i) (eq T t1 (TLRef (minus i h))))))))))).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H1: (eq T (TLRef i) (lift 
-h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) 
-(\lambda (t2: T).(eq T (TLRef i) t2)) H1 (THead k (lift h d t0) (lift h (s k 
-d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TLRef i) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s 
-k d) t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef 
-i))) (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) 
-H3)))))))))))) t).
-
-lemma lift_gen_lref_lt:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall 
-(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n 
-d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x 
-\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) 
-(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq 
-T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind 
-(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n 
-d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: 
-(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d 
-h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus 
-d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef 
-(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq 
-T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S 
-n) d h H))) t H4))) H2)) H1)))))))).
-
-lemma lift_gen_lref_false:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
-(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall 
-(P: Prop).P)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
-n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T 
-(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d 
-h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) 
-(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land 
-(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P 
-(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H 
-H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n 
-h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda 
-(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false 
-(plus d h) n P H4 H0))) H3)) H2)))))))))).
-
-lemma lift_gen_lref_ge:
- \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
-(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
-n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 
-t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in 
-(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus 
-d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) 
-(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind 
-(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda 
-(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r 
-T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq 
-T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d 
-(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) 
-(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
-h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
-h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda 
-(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n 
-h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus 
-(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
-
-lemma lift_gen_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T 
-(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z)))))))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind 
-(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: 
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) 
-(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda 
-(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 
-\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
-T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) 
-z))))) H1))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d 
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: (lt n 
-d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead 
-k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (eq_ind T 
-(THead k u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
-(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))) 
-(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
-(t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d 
-H0)) in (let H2 \def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T 
-(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: 
-T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
-(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: 
-T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
-(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead 
-k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 
-t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: 
-T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
-\Rightarrow k1])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) 
-(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | 
-(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead k u t) 
-(THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (\lambda (H6: (eq T u 
-(lift h d t0))).(\lambda (H7: (eq K k k0)).(let H8 \def (eq_ind_r K k0 
-(\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k H7) in (eq_ind K k 
-(\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k1 
-t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))) (let H9 
-\def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq 
-T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: 
-T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h0 
-d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h0 (s k d0) 
-z))))))))) H0 (lift h (s k d) t1) H8) in (let H10 \def (eq_ind T t (\lambda 
-(t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 
-d0 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t2 (lift h0 (s k d0) z))))))))) H (lift h (s k d) 
-t1) H8) in (eq_ind_r T (lift h (s k d) t1) (\lambda (t2: T).(ex3_2 T T 
-(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t2 (lift h (s k d) z)))))) (let H11 \def (eq_ind T u 
-(\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k t2 
-(lift h (s k d) t1)) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda 
-(z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 
-(lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
-(lift h0 (s k d0) z))))))))) H10 (lift h d t0) H6) in (let H12 \def (eq_ind T 
-u (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k t2 
-(lift h (s k d) t1)) (lift h0 d0 t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda 
-(z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 
-(lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
-(lift h0 (s k d0) z))))))))) H9 (lift h d t0) H6) in (eq_ind_r T (lift h d 
-t0) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
-k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k 
-d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 
-t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
-(lift h (s k d) z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T 
-(lift h d t0)) (refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) 
-H4)) H3))))))))))) x)))).
-
-lemma lift_gen_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T 
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda 
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (S d) z)))))))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d 
-x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x 
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda 
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
-h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
-(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
-(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) 
-x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
-(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) 
-z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro 
-T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind 
-b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) 
-z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d 
-x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
-
-lemma lift_gen_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T 
-T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda 
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h d z)))))))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d 
-x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x 
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda 
-(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
-h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
-(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
-(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T 
-T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda 
-(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: 
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) 
-(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T 
-(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq 
-T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
-T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d 
-x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T 
-(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
-T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
-(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) 
-(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x 
-H1)))))) H0))))))))).
-
-lemma lift_inj:
- \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T 
-(lift h d x) (lift h d t)) \to (eq T x t)))))
-\def
- \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t 
-t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def 
-(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H 
-(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t 
-H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq 
-T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d 
-(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt 
-n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d 
-d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift 
-h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) 
-(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 
-t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
-T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) 
-(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) 
-\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: 
-nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t 
-t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
-(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
-t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
-(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
-(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
-(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) 
-(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift 
-h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r 
-T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) 
-(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind 
-b) t t0) (THead (Bind b) x0 x1) (f_equal3 K T T T THead (Bind b) (Bind b) t 
-x0 t0 x1 (refl_equal K (Bind b)) (H x0 h d H4) (H0 x1 h (S d) H5)))) t1 
-H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d 
-H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
-(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
-(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
-t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
-(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
-(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
-(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T 
-(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq 
-T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d 
-x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead 
-(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T 
-(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) 
-(THead (Flat f) x0 x1) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 x1 
-(refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)))) t1 H3)))))) 
-(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
-
-lemma lift_gen_lift:
- \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: 
-nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 
-t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 
-t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: 
-nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to 
-((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: 
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 
-t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda 
-(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 
-d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) 
-x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t 
-(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T 
-(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) 
-(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda 
-(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) 
-(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T 
-(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 
-d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T 
-(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 
-(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda 
-(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda 
-(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 
-h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) 
-(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n 
-d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t 
-(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in 
-(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift 
-h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T 
-(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
-(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: 
-T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) 
-(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef 
-n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 
-(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n 
-(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) 
-(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) 
-(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) 
-(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x 
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) 
-(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 
-T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) 
-(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) 
-(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef 
-n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) 
-t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n 
-h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) 
-(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x 
-(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) 
-(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq 
-T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) 
-(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus 
-n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) 
-h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) 
-(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda 
-(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 
-d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus 
-n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus 
-(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans 
-h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T 
-(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T 
-t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) 
-(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 
-d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n 
-h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) 
-(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) 
-h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T 
-(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) 
-(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 
-n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 
-h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) 
-(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) 
-(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: 
-T).(eq T (TLRef (plus (minus n h2) h2)) t)) (sym_eq T (TLRef (plus (minus 
-(plus (minus n h2) h2) h2) h2)) (TLRef (plus (minus n h2) h2)) (f_equal nat T 
-TLRef (plus (minus (plus (minus n h2) h2) h2) h2) (plus (minus n h2) h2) 
-(f_equal2 nat nat nat plus (minus (plus (minus n h2) h2) h2) (minus n h2) h2 
-h2 (minus_plus_r (minus n h2) h2) (refl_equal nat h2)))) (lift h2 d2 (TLRef 
-(minus (plus (minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) 
-h2) h2) h2 d2 (le_minus d2 (plus (minus n h2) h2) h2 (le_plus_plus d2 (minus 
-n h2) h2 h2 (le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n 
-(le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 
-(plus d2 h1) (minus (plus n h1) h2) (arith0 h2 d2 n H4 h1) x 
-H5)))))))))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall 
-(x: T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: 
-nat).((le d1 d2) \to ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 
-T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift 
-h2 d2 t2))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: 
-T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: 
-nat).((le d1 d2) \to ((eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) x)) \to 
-(ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 
-(lift h2 d2 t2))))))))))))).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2: 
-nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (H1: (le d1 d2)).(\lambda 
-(H2: (eq T (lift h1 d1 (THead k t t0)) (lift h2 (plus d2 h1) x))).(K_ind 
-(\lambda (k0: K).((eq T (lift h1 d1 (THead k0 t t0)) (lift h2 (plus d2 h1) 
-x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: 
-T).(eq T (THead k0 t t0) (lift h2 d2 t2)))))) (\lambda (b: B).(\lambda (H3: 
-(eq T (lift h1 d1 (THead (Bind b) t t0)) (lift h2 (plus d2 h1) x))).(let H4 
-\def (eq_ind T (lift h1 d1 (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 
-(lift h2 (plus d2 h1) x))) H3 (THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) 
-t0)) (lift_bind b t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: 
-T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) z)))) (ex2 T (\lambda 
-(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) t 
-t0) (lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
-x (THead (Bind b) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus 
-d2 h1) x0))).(\lambda (H7: (eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 
-h1)) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead 
-(Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift 
-h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: 
-T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
-(THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T 
-x0 (lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T 
-(lift h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind 
-b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t t0) 
-(lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t3))) 
-(\lambda (t3: T).(eq T (THead (Bind b) t2 t0) (lift h2 d2 t3))))) (let H10 
-\def (refl_equal nat (plus (S d2) h1)) in (let H11 \def (eq_ind nat (S (plus 
-d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) t0) (lift h2 n x1))) H7 (plus 
-(S d2) h1) H10) in (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 (S d1) t2))) 
-(\lambda (t2: T).(eq T t0 (lift h2 (S d2) t2))) (ex2 T (\lambda (t2: T).(eq T 
-(THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
-(THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3: 
-T).(\lambda (H12: (eq T x1 (lift h1 (S d1) x3))).(\lambda (H13: (eq T t0 
-(lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S d1) x3) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) t2) (lift h1 d1 t3))) 
-(\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2 
-t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: 
-T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 
-t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t2) (lift h2 d2 
-t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) 
-(lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) 
-(lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 t2))) (THead (Bind b) x2 x3) 
-(eq_ind_r T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (\lambda 
-(t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) t2)) 
-(refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3))) (lift h1 
-d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 d1)) (eq_ind_r T (THead 
-(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (\lambda (t2: T).(eq T (THead 
-(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) t2)) (refl_equal T (THead (Bind 
-b) (lift h2 d2 x2) (lift h2 (S d2) x3))) (lift h2 d2 (THead (Bind b) x2 x3)) 
-(lift_bind b x2 x3 h2 d2))) t0 H13) x1 H12)))) (H0 x1 h1 h2 (S d1) (S d2) 
-(le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x 
-H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S d1) t0) x h2 (plus d2 
-h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead (Flat f) t 
-t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead 
-(Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 
-(THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t t0 h1 d1)) in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 
-h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0) (lift h2 
-(plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda 
-(t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda 
-(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T 
-(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0 
-x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) 
-(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T 
-(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 
-d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1 
-t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) 
-(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T 
-t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3: 
-T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 
-x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 
-d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0) 
-(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2))) 
-(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T 
-(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
-(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3: 
-T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2 
-d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: 
-T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: 
-T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T 
-(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat 
-f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T 
-(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda 
-(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 
-t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) 
-(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1 
-d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 
-x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift 
-h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1)) 
-(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2: 
-T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T 
-(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f) 
-x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 
-H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f 
-(lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2))))))))))))) 
-t1).
-
-lemma lifts_inj:
- \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: 
-nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
-\def
- \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: 
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h 
-d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: 
-TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts 
-h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: 
-nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda 
-(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: 
-nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) 
-(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match 
-ee with [TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons 
-(lift h d t) (lifts h d t0)) H0) in (False_ind (eq TList TNil (TCons t t0)) 
-H1)))))))) ts)) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall 
-(ts: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t0) 
-(lifts h d ts)) \to (eq TList t0 ts))))))).(\lambda (ts: TList).(TList_ind 
-(\lambda (t1: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h 
-d (TCons t t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts 
-h d t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d 
-t0)) (\lambda (ee: TList).(match ee with [TNil \Rightarrow False | (TCons _ 
-_) \Rightarrow True])) I TNil H0) in (False_ind (eq TList (TCons t t0) TNil) 
-H1))))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: ((\forall (h: 
-nat).(\forall (d: nat).((eq TList (TCons (lift h d t) (lifts h d t0)) (lifts 
-h d t2)) \to (eq TList (TCons t t0) t2)))))).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H1: (eq TList (TCons (lift h d t) (lifts h d t0)) (TCons (lift 
-h d t1) (lifts h d t2)))).(let H2 \def (f_equal TList T (\lambda (e: 
-TList).(match e with [TNil \Rightarrow (lref_map (\lambda (x: nat).(plus x 
-h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) 
-(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList 
-TList (\lambda (e: TList).(match e with [TNil \Rightarrow (lifts h d t0) | 
-(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons 
-(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h 
-d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2))) 
-(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1 
-(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
deleted file mode 100644 (file)
index 610e020..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
+++ /dev/null
@@ -1,323 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/defs.ma".
-
-include "basic_1/s/props.ma".
-
-include "basic_1/T/fwd.ma".
-
-lemma lift_sort:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
-n)) (TSort n))))
-\def
- \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort 
-n)))).
-
-lemma lift_lref_lt:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T 
-(lift h d (TLRef n)) (TLRef n)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n 
-d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true 
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T 
-(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
-
-lemma lift_lref_ge:
- \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
-(lift h d (TLRef n)) (TLRef (plus n h))))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d 
-n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true 
-\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) 
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false 
-(le_bge d n H)))))).
-
-lemma lift_head:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
-(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) 
-t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
-
-lemma lift_bind:
- \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
-(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) 
-(lift h (S d) t)))))))
-\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
-
-lemma lift_flat:
- \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
-(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) 
-(lift h d t)))))))
-\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
-
-lemma thead_x_lift_y_y:
- \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall 
-(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
-\def
- \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: 
-T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) 
-\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) 
-(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
-(TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H) in (False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TLRef n))) 
-(TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
-(TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H) in (False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: 
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
-h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: 
-((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
-h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 
-t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (lift h d (THead 
-k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead 
-_ t2 _) \Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 
-t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d 
-t0) (lref_map (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) 
-\Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d t0) (lref_map 
-(\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) \Rightarrow 
-t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in 
-(\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k 
-(\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall (d0: nat).((eq 
-T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: Prop).P0)))))) H0 k0 
-H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: 
-T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head 
-k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) H2)))))))))))) 
-t)).
-
-lemma lift_r:
- \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) 
-t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda 
-(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) 
-(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef 
-n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) 
-(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T 
-t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) 
-(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) 
-t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) 
-t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) 
-t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) 
-(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) 
-(lift O (s k d) t1)) (THead k t0 t1) (f_equal3 K T T T THead k k (lift O d 
-t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))))) (lift O 
-d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
-
-lemma lift_lref_gt:
- \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef 
-(pred n))) (TLRef n))))
-\def
- \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef 
-(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus 
-(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n 
-(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S 
-(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) 
-(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d 
-(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) 
-(S_pred n d H))))))).
-
-lemma lift_tle:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(tle t (lift h d t))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
-nat).(le (tweight t0) (tweight (lift h d t0)))))) (\lambda (_: nat).(\lambda 
-(_: nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (_: nat).(\lambda (_: 
-nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(le (tweight t0) 
-(tweight (lift h d t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: 
-nat).(\forall (d: nat).(le (tweight t1) (tweight (lift h d t1))))))).(\lambda 
-(h: nat).(\lambda (d: nat).(let H_y \def (H h d) in (let H_y0 \def (H0 h (s k 
-d)) in (le_n_S (plus (tweight t0) (tweight t1)) (plus (tweight (lref_map 
-(\lambda (x: nat).(plus x h)) d t0)) (tweight (lref_map (\lambda (x: 
-nat).(plus x h)) (s k d) t1))) (le_plus_plus (tweight t0) (tweight (lref_map 
-(\lambda (x: nat).(plus x h)) d t0)) (tweight t1) (tweight (lref_map (\lambda 
-(x: nat).(plus x h)) (s k d) t1)) H_y H_y0))))))))))) t).
-
-lemma lifts_tapp:
- \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
-TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
-(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
-t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
-t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
-(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
-v)) H)))) vs)))).
-
-lemma lift_free:
- \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
-(lift h d t)) (lift (plus k h) d t))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
-nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
-(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: 
-nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: 
-nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T 
-(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort 
-n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
-(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) 
-(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k 
-h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) 
-(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: 
-nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d 
-h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef 
-n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T 
-(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef 
-n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
-(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
-(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus 
-k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 
-H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d 
-n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift 
-(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda 
-(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n 
-(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal 
-nat T TLRef (plus (plus n h) k) (plus n (plus k h)) 
-(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) 
-(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
-(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 
-(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda 
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
-nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
-(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda 
-(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e 
-(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda 
-(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d 
-h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k 
-d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 
-t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift 
-h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 
-t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) 
-t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k 
-e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h 
-d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift 
-(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) 
-(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le 
-k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift 
-(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e 
-(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
-h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h 
-d))))))))))))) t).
-
-lemma lift_free_sym:
- \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
-(lift h d t)) (lift (plus h k) d t))))))))
-\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
-nat).(\lambda (e: nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d 
-e)).(eq_ind_r nat (plus k h) (\lambda (n: nat).(eq T (lift k e (lift h d t)) 
-(lift n d t))) (lift_free t h k d e H H0) (plus h k) (plus_sym h k)))))))).
-
-lemma lift_d:
- \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) 
-(lift k e (lift h d t))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k 
-d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda 
-(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: 
-(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) 
-(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
-T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: 
-T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
-T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k 
-e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) 
-(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) 
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
-nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h 
-(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda 
-(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) 
-(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef 
-n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d 
-(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k 
-e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
-(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift 
-h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) 
-(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k 
-d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e 
-n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) 
-t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: 
-nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) 
-(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d 
-(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda 
-(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) 
-(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef 
-(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T 
-(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d 
-(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) 
-(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: 
-(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 
-(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda 
-(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef 
-(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) 
-(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) 
-(plus_permute_2_in_3 n k h)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
-(plus n h) k e (le_plus_trans e n h H0))) (lift h d (TLRef n)) (lift_lref_ge 
-n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge (plus n k) h 
-(plus d k) (le_plus_plus d n k k H1 (le_n k)))))) (plus k d) (plus_sym k d)) 
-(lift k e (TLRef n)) (lift_lref_ge n k e H0)))))))))) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
-nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 
-d) (lift k0 e t0)) (lift k0 e (lift h d t0)))))))))).(\lambda (t1: 
-T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
-nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e 
-t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda (k0: 
-nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(eq_ind_r T 
-(THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h 
-(plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) (eq_ind_r T (THead k 
-(lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
-e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d (THead k t0 t1))))) 
-(eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) (\lambda (t2: T).(eq 
-T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) 
-(lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead k (lift k0 e (lift 
-h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda (t2: T).(eq T (THead 
-k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
-e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda (n: nat).(eq T (THead 
-k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift k0 (s k e) t1))) (THead 
-k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))))) 
-(f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e 
-(lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) t1)) (lift k0 (s k 
-e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) (H0 h k0 (s k d) (s 
-k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 d)) (lift k0 e 
-(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
-h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h d)) 
-(lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 (s k e) t1))) (lift_head 
-k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 d))) (lift k0 e (THead k t0 
-t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma
deleted file mode 100644 (file)
index 5adcfd6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/tlt.ma
+++ /dev/null
@@ -1,284 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/props.ma".
-
-include "basic_1/tlt/props.ma".
-
-lemma lift_weight_map:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to 
-nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat 
-(weight_map f (lift h d t)) (weight_map f t))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
-(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) 
-(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
-nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
-O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: 
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
-nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
-O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f 
-(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat 
-(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) 
-(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq 
-nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda 
-(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) 
-(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda 
-(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: 
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
-(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f 
-t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: 
-nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
-(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f 
-t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
-nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
-O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 
-t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead 
-(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat 
-(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda 
-(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift 
-h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) 
-t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map 
-(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f 
-(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with 
-[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S 
-(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) 
-(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) 
-(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: 
-nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S 
-(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) 
-(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: 
-nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) 
-(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus 
-(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) 
-(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) 
-(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) 
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
-n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda 
-(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda 
-(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) 
-(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r 
-nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map 
-f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) 
-t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map 
-(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 
-nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map 
-(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat 
-(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h 
-(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat 
-(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd 
-f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le 
-d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x 
-H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) 
-(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus 
-(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus 
-(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f 
-t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f 
-(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd 
-f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) 
-(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: 
-nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S 
-n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: 
-nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S 
-x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d 
-m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h 
-d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s 
-(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f 
-(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d 
-t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) 
-(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) 
-(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) 
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
-k)))))))))) t).
-
-lemma lift_weight:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d 
-t)) (weight t))))
-\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d 
-(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat 
-O)))))).
-
-lemma lift_weight_add:
- \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: 
-nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
-(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to 
-(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat 
-(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
-\def
- \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: 
-nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat 
-(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) 
-\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d 
-t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: 
-nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) 
-w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f 
-m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) 
-(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m 
-d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: 
-((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d 
-(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d 
-(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) 
-(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef 
-n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d 
-H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: 
-T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f 
-(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda 
-(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f 
-(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) 
-(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift 
-h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda 
-(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat 
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to 
-(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d 
-m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) 
-(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: 
-((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall 
-(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f 
-m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g 
-(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift 
-(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat 
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: 
-nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) 
-w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f 
-m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 
-t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: 
-B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) 
-(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead 
-(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) 
-(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) 
-(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind 
-(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus 
-(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d 
-t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h 
-d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S 
-(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) 
-t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) 
-d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) 
-(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) 
-(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus 
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) 
-t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map 
-(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus 
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift 
-(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map 
-f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S 
-(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S 
-(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 
-H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S 
-(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S 
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) 
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d 
-t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m 
-O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift 
-(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat 
-nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq 
-nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g 
-H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S 
-m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat 
-m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g 
-(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda 
-(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r 
-nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d 
-t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) 
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
-n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: 
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
-x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) 
-n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus 
-(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus 
-(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) 
-t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g 
-(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map 
-(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) 
-(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) 
-(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) 
-(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat 
-O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m 
-H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda 
-(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) 
-(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: 
-nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S 
-x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) 
-H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
-m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
-n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S 
-x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g 
-n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat 
-S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) 
-t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S 
-h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) 
-(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) 
-(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S 
-d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S 
-d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) 
-(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda 
-(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) 
-(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: 
-nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda 
-(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O 
-m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda 
-(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) 
-(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: 
-nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S 
-n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: 
-nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
-x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) 
-(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head 
-(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind 
-b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) 
-(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) 
-(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead 
-(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: 
-T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) 
-d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d 
-t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) 
-(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f 
-(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) 
-(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) 
-(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) 
-(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
-k))))))))))))) t)).
-
-lemma lift_weight_add_O:
- \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to 
-nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) 
-O t))))))
-\def
- \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to 
-nat))).(lift_weight_add (minus (wadd f w O) O) t h O f (wadd f w) (\lambda 
-(m: nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) 
-(minus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal 
-nat (f m)))))))).
-
-lemma lift_tlt_dx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
-(d: nat).(tlt t (THead k u (lift h d t)))))))
-\def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight 
-(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) 
-(lift_weight t h d)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma
deleted file mode 100644 (file)
index 4ab2589..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/defs.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/defs.ma".
-
-rec definition trans (hds: PList) on hds: nat \to nat \def \lambda (i: 
-nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let 
-j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false 
-\Rightarrow (plus j h)]))]).
-
-rec definition lift1 (hds: PList) on hds: T \to T \def \lambda (t: T).(match 
-hds with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 
-hds0 t))]).
-
-rec definition lifts1 (hds: PList) (ts: TList) on ts: TList \def match ts 
-with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) 
-(lifts1 hds ts0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma
deleted file mode 100644 (file)
index f5839c7..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/drop1.ma
+++ /dev/null
@@ -1,127 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/props.ma".
-
-include "basic_1/drop1/defs.ma".
-
-lemma lift1_lift1:
- \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
-(lift1 is2 t)) (lift1 (papp is1 is2) t))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
-PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) 
-t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
-((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 
-(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat 
-T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal 
-nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
-
-lemma lift1_xhg:
- \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
-(lift (S O) O (lift1 hds t))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T 
-(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: 
-T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) 
-(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T 
-(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S 
-O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: 
-nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d 
-(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda 
-(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift 
-(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 
-p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S 
-d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
-
-lemma lifts1_xhg:
- \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts 
-(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
-\def
- \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq 
-TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) 
-(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq 
-TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds 
-t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList 
-(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 
-hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O 
-(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 
-hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds 
-t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O 
-(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) 
-(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
-
-lemma lift1_free:
- \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
-(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: 
-nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O 
-(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T 
-(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
-PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 
-(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) 
-t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 
-i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift 
-(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | 
-false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 
-i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans 
-hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt 
-(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) 
-O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans 
-hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with 
-[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
-false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind 
-(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d 
-(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with 
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
-h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) 
-(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S 
-(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h 
-n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
-(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
-(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
-(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S 
-(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) 
-(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans 
-hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S 
-(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d 
-(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: 
-(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans 
-hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S 
-(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus 
-h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) 
-t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) 
-(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
-(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans 
-hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 
-(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) 
-(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S 
-(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 
-i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) 
-(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda 
-(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d 
-(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) 
-(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans 
-hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) 
-hds).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
deleted file mode 100644 (file)
index 286dfbe..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
+++ /dev/null
@@ -1,140 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift1/defs.ma".
-
-include "basic_1/lift/props.ma".
-
-lemma lift1_sort:
- \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
-\def
- \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T 
-(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: 
-nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p 
-(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 
-n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
-
-lemma lift1_lref:
- \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
-(trans hds i))))
-\def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T 
-(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T 
-(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
-(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p 
-i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq 
-T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow 
-(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T 
-(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false 
-\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
-
-lemma lift1_bind:
- \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
-(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss 
-hds) t))))))
-\def
- \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) 
-(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal 
-T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
-PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
-(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda 
-(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) 
-t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p 
-u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n 
-n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 
-(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) 
-(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 
-(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) 
-(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u 
-t)) (H u t)))))))) hds)).
-
-lemma lift1_flat:
- \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
-(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds 
-t))))))
-\def
- \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) 
-(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T 
-(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
-PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
-(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: 
-T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) 
-(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) 
-(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p 
-u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift 
-n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) 
-(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) 
-(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 
-p (THead (Flat f) u t)) (H u t)))))))) hds)).
-
-lemma lift1_cons_tail:
- \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq 
-T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
-\def
- \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: 
-PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) 
-(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 
-(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d 
-t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d 
-t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p 
-h d) t) H))))) hds)))).
-
-lemma lifts1_flat:
- \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: 
-TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds 
-ts) (lift1 hds t))))))
-\def
- \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: 
-TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 
-t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 
-hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds 
-(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds 
-t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) 
-t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads 
-(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) 
-(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 
-hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) 
-(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat 
-f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) 
-(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 
-(THeads (Flat f) t1 t)))))) ts)))).
-
-lemma lifts1_nil:
- \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
-\def
- \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) 
-t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: 
-(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq 
-TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 
-PNil t0) H)))) ts).
-
-lemma lifts1_cons:
- \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: 
-TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: 
-TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) 
-(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d 
-(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: 
-TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 
-hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d 
-(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) 
-H)))) ts)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma
deleted file mode 100644 (file)
index 7d5bc55..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/A/defs.ma".
-
-rec definition lweight (a: A) on a: nat \def match a with [(ASort _ _) 
-\Rightarrow O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight 
-a2)))].
-
-definition llt:
- A \to (A \to Prop)
-\def
- \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma
deleted file mode 100644 (file)
index 8d45d37..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma
+++ /dev/null
@@ -1,46 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/llt/defs.ma".
-
-fact llt_wf__q_ind:
- \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to 
-Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 
-a))))) P n))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: 
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) 
-n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight 
-a)))))).
-
-lemma llt_wf_ind:
- \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 
-a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
-\def
- let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: 
-A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to 
-Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) 
-(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind 
-(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: 
-A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
-\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat 
-(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
-(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P 
-a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt 
-(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight 
-a1))))))))))))) a)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma
deleted file mode 100644 (file)
index 0c7cef6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma
+++ /dev/null
@@ -1,65 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/llt/defs.ma".
-
-include "basic_1/leq/fwd.ma".
-
-lemma lweight_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat 
-(lweight a1) (lweight a2)))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight 
-a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: 
-nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: 
-A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight 
-a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda 
-(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight 
-a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus 
-(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 
-H)))).
-
-lemma llt_repl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
-(a3: A).((llt a1 a3) \to (llt a2 a3))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
-a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 
-\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 
-(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
-
-theorem llt_trans:
- \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 
-a3) \to (llt a1 a3)))))
-\def
- \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight 
-a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans 
-(lweight a1) (lweight a2) (lweight a3) H H0))))).
-
-lemma llt_head_sx:
- \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) 
-(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
-
-lemma llt_head_dx:
- \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
-\def
- \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) 
-(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma
deleted file mode 100644 (file)
index 04e15b3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/next_plus/defs.ma
+++ /dev/null
@@ -1,21 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/G/defs.ma".
-
-rec definition next_plus (g: G) (n: nat) (i: nat) on i: nat \def match i with 
-[O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma
deleted file mode 100644 (file)
index 2f42fc6..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/next_plus/props.ma
+++ /dev/null
@@ -1,59 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/next_plus/defs.ma".
-
-lemma next_plus_assoc:
- \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq 
-nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2))))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: 
-nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) 
-(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus 
-g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat 
-(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda 
-(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus 
-g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda 
-(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) 
-(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) 
-(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n 
-n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) 
-(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) 
-n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g 
-(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) 
-(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))).
-
-lemma next_plus_next:
- \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g 
-(next g n) h) (next g (next_plus g n h)))))
-\def
- \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus 
-g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n 
-h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n 
-(S O)) h) (next_plus_assoc g n (S O) h)))).
-
-lemma next_plus_lt:
- \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next 
-g n) h))))
-\def
- \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: 
-nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) 
-(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next 
-g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) 
-n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next 
-g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next 
-g n0) n)) (next_plus_next g (next g n0) n))))) h)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
deleted file mode 100644 (file)
index c5b3b0a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/arity.ma
+++ /dev/null
@@ -1,486 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/fwd.ma".
-
-include "basic_1/arity/subst0.ma".
-
-lemma arity_nf2_inv_all:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
-a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat 
-(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
-A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda 
-(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort 
-n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort 
-n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n 
-(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) 
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: 
-T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: 
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: 
-nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef 
-i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads 
-(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 
-ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
-i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u 
-(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef 
-i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda 
-(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T 
-(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda 
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
-i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T 
-(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda 
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
-i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) 
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda 
-(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 
-(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 
-(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind 
-b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) 
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads 
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 
-(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead 
-c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u 
-t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 
-(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind 
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
-nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws 
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda 
-(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq 
-B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 
-a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 
-H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) 
-u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 
-w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
-(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u 
-t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g 
-(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u 
-t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False with []) in 
-H9)))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 
-(Bind Void) u) t0 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let 
-H9 \def (arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O 
-(getl_refl Void c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O 
-v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind 
-Void) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
-(ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u 
-t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) (\lambda (x: T).(\lambda (H10: (eq T t0 (lift (S O) O x))).(let H11 
-\def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Bind Void) u t1))) H8 
-(lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t1) 
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
-(\lambda (n: nat).(eq T (THead (Bind Void) u t1) (TSort n)))) (ex3_2 TList 
-nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u t1) 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))) (nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Bind Void) u (lift (S O) O x)) (TSort n)))) (ex3_2 TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u (lift (S O) O x)) 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))) t0 H10)))) H9))))) b H0 H3 H5))))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda 
-(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
-(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda 
-(a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: 
-(((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda (w: 
-T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w) 
-u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind Abst) u) 
-ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind Abst) u) 
-(TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u t0))).(let H5 
-\def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 (CHead c0 (Bind 
-Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
-(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort 
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead 
-(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 
-(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
-(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind 
-Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind 
-Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) 
-\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
-Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
-T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
-nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda 
-(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def 
-(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) 
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
-(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList 
-nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def 
-(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
-w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: 
-(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
-c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq 
-T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
-t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq 
-T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: 
-(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: 
-(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: 
-T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in 
-(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 
-(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) 
-(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
-(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
-(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
-Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind 
-Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
-w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind 
-Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat 
-Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) 
-H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind 
-nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: 
-nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda 
-(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def 
-(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) 
-H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def 
-(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) 
-in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq 
-g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead 
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda 
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda 
-(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
-x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 
-x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee with 
-[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 
-x1) H16) in (False_ind (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
-(THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
-c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
-(TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) H17))))))) H13))) t0 H10))))) 
-H9)) (\lambda (H9: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
-u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: 
-TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 
-(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef 
-x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) 
-u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T 
-t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 
-(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda 
-(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
-Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u 
-t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
-(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) 
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
-(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u 
-(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads 
-(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda 
-(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u 
-(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) 
-H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda 
-(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
-(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
-(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda 
-(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) 
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
-T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 
-u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
-Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
-c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
-(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u 
-t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
-(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 
-t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T 
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind 
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: 
-nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda 
-(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) 
-u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind 
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
-T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind 
-Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind 
-Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w 
-u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead 
-(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) 
-u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) 
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind 
-Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
-c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) 
-t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort 
-n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
-(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
-(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
-n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T 
-(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: 
-T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 
-c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) 
-(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T 
-(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) 
-(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: 
-T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort 
-x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda 
-(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x 
-(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: 
-T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) 
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads 
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 
-ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda 
-(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 
-(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef 
-x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 
-(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w 
-u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
-(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads 
-(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: 
-T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
-c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) 
-x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
-(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws 
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda 
-(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef 
-x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
-i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 
-H6)))))) H5)) H4))))))))))) c t a H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
deleted file mode 100644 (file)
index d721184..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/dec.ma
+++ /dev/null
@@ -1,193 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/pr2/clen.ma".
-
-include "basic_1/pr0/dec.ma".
-
-include "basic_1/C/props.ma".
-
-lemma nf2_dec:
- \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
-T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
-\def
- \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall 
-(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda 
-(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in 
-(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) 
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to 
-(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T 
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 
-(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 
-H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) 
-(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to 
-(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: 
-T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T 
-t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) 
-(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x 
-H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or 
-(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 
-t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H 
-t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T 
-t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) 
-t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: 
-((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: 
-K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 
-T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: 
-B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 
-t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def 
-(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: 
-T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) 
-(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) 
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda 
-(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq 
-T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) 
-(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) 
-(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def 
-H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t 
-c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind 
-Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
-t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t 
-c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 
-(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
-t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) 
-x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 
-(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in 
-(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) 
-(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt 
-(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym 
-(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t 
-(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) 
-(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 
-(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 
-(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda 
-(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T 
-t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall 
-(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T 
-(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) 
-(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
-(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda 
-(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 
-\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let 
-H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda 
-(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) 
-(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift 
-(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) 
-t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
-Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) 
-(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda 
-(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) 
-(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: 
-(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 
-t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) 
-(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: 
-T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x 
-x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) 
-H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) 
-(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) 
-(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen 
-c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) 
-t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) 
-H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) 
-\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda 
-(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def 
-(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
-(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
-K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
-(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: 
-K).(match ee with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow 
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
-\Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) 
-H4)) H3)))))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 
-t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda 
-(t2: T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def 
-(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
-(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) 
-(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
-(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
-(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
-(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
-K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
-(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
-(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: 
-K).(match ee with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow 
-False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) 
-\Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) 
-H4)) H3)))))) b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail 
-(Flat f) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) 
-t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 
-t2)).(let H_x0 \def (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def 
-H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind 
-Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 
-t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: 
-(ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 
-t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: 
-T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
-T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
-(eq K (Flat f) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 
-(clen c0) t x0 t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match 
-ee with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind 
-Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) k)) (\lambda 
-(H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) (or (\forall 
-(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
-T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t 
-c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) \to (\forall (P: 
-Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall (t2: T).((pr2 
-(CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2))) 
-(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 x H3 k 
-t)))))) H1)) H0)))))))) c).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma
deleted file mode 100644 (file)
index 5b4849d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/defs.ma
+++ /dev/null
@@ -1,27 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/defs.ma".
-
-definition nf2:
- C \to (T \to Prop)
-\def
- \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 
-t2)))).
-
-rec definition nfs2 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
deleted file mode 100644 (file)
index 06487b7..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/fwd.ma
+++ /dev/null
@@ -1,173 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/pr2/clen.ma".
-
-include "basic_1/subst0/dec.ma".
-
-include "basic_1/T/props.ma".
-
-lemma nf2_gen_lref:
- \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 
-c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: 
-Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 
-(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef 
-i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
-
-lemma nf2_gen_abst:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u 
-t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
-T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) 
-t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: 
-T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: 
-T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
-_ t0 _) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) t2 t) (H 
-(THead (Bind Abst) t2 t) (pr2_head_1 c u t2 H0 (Bind Abst) t))) in (let H2 
-\def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u t0)) H0 u H1) in (eq_ind T u 
-(\lambda (t0: T).(eq T u t0)) (refl_equal T u) t2 H1))))) (\lambda (t2: 
-T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t t2)).(let H1 \def (f_equal T 
-T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) u t) 
-(THead (Bind Abst) u t2) (H (THead (Bind Abst) u t2) (let H_y \def 
-(pr2_gen_cbind Abst c u t t2 H0) in H_y))) in (let H2 \def (eq_ind_r T t2 
-(\lambda (t0: T).(pr2 (CHead c (Bind Abst) u) t t0)) H0 t H1) in (eq_ind T t 
-(\lambda (t0: T).(eq T t t0)) (refl_equal T t) t2 H1))))))))).
-
-lemma nf2_gen_cast:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u 
-t)) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead 
-(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t 
-(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
-
-lemma nf2_gen_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c 
-(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) 
-\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) 
-(H (THead (Bind Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind 
-Abst) v t)) (THead (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t 
-(pr0_refl t))))) in (False_ind P H0))))))).
-
-lemma nf2_gen_flat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c 
-(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) 
-u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall 
-(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c 
-u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Flat f) u t) (THead (Flat f) t2 t) (H (THead (Flat f) t2 t) 
-(pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) (\lambda (t2: T).(\lambda (H0: 
-(pr2 c t t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) 
-(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
-
-fact nf2_gen__nf2_gen_aux:
- \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T 
-(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: 
-T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
-(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort 
-n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
-d (TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H) in (False_ind P H0))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TLRef n))) (TLRef 
-n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
-d (TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
-H) in (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: 
-((\forall (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) 
-t) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
-(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to 
-(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: 
-(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
-t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | 
-(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u (lift (S O) d (THead k t 
-t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) 
-\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
-t0) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t) 
-(lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (TLRef _) 
-\Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t) 
-(lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) 
-\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
-t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) 
-H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t 
-t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift 
-(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 
-P)))))) H3)) H2))))))))))) x)).
-
-lemma nf2_gen_abbr:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u 
-t)) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
-T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) 
-t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x 
-in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t 
-(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift 
-(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O 
-x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O 
-x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (H (THead 
-(Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind Abbr) u t) (THead 
-(Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) t t (pr0_refl t) 
-(lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda (t0: T).(subst0 
-O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in (subst0_refl u (lift (S 
-O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) O x))).(let H3 \def 
-(eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c (THead (Bind Abbr) u 
-t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H (lift (S O) O x) H2) in 
-(nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c (THead (Bind Abbr) u (lift 
-(S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) u))) P))) H1))) 
-H0))))))).
-
-lemma nf2_gen_void:
- \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u 
-(lift (S O) O t))) \to (\forall (P: Prop).P))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
-T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind 
-Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux 
-Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t 
-(pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
deleted file mode 100644 (file)
index 39b8275..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/iso.ma
+++ /dev/null
@@ -1,125 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/pr3.ma".
-
-include "basic_1/iso/props.ma".
-
-lemma nf2_iso_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
-TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso 
-(THeads (Flat Appl) vs (TLRef i)) u))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads 
-(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) 
-(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def 
-(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda 
-(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda 
-(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) 
-u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat 
-Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
-t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
-i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u 
-(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat 
-Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 
-x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 
-(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 
-(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda 
-(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 
-t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda 
-(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 
-x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
-u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in 
-(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
-T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift 
-(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 
-x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 
-(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 
-H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-u))))))))))))))) H3)) H2))))))) vs)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
deleted file mode 100644 (file)
index 9e680d7..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/lift1.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/props.ma".
-
-include "basic_1/drop1/fwd.ma".
-
-lemma nf2_lift1:
- \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
-hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
-\def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
-t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c 
-e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in 
-(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: 
-C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
-t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) 
-c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) 
-in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda 
-(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: 
-C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x 
-(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
deleted file mode 100644 (file)
index 5c6a686..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
+++ /dev/null
@@ -1,50 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/pr3/pr3.ma".
-
-lemma nf2_pr3_unfold:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
-t1) \to (eq T t1 t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
-t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t 
-(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) 
-\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def 
-(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def 
-(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T 
-t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
-
-theorem nf2_pr3_confluence:
- \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) 
-\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: 
-T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t 
-t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) 
-(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: 
-(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 
-x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 
-H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) 
-in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 
-(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: 
-T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 
-t1 H1))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
deleted file mode 100644 (file)
index ccaa9eb..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/props.ma
+++ /dev/null
@@ -1,309 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/nf2/defs.ma".
-
-include "basic_1/pr2/fwd.ma".
-
-lemma nf2_sort:
- \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
-\def
- \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort 
-n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal 
-T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
-
-lemma nf2_csort_lref:
- \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
-\def
- \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort 
-n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq 
-T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) 
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
-i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r 
-T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
-H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
-(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
-n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
-(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 
-(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T 
-(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) 
-(lift (S i) O x1))) t2 H3))))) H1)) H0))))).
-
-theorem nf2_abst:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: 
-T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind 
-Abst) u t))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
-\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda 
-(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t 
-t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) 
-t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead 
-(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 
-(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: 
-((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t 
-x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead 
-(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t 
-x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) 
-H2)))))))))).
-
-theorem nf2_abst_shift:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c 
-(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
-\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 
-(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda 
-(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 
-H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 
-c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T 
-(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) 
-u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 
-H3)))))) H2)))))))).
-
-lemma nfs2_tapp:
- \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) 
-\to (land (nfs2 c ts) (nf2 c t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: 
-TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: 
-(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True 
-(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I 
-H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c 
-(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c 
-t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c 
-(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: 
-(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let 
-H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c 
-t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj 
-(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) 
-H6))) H4))))) H1)))))) ts))).
-
-lemma nf2_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
-TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads 
-(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 
-(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in 
-(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c 
-t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def 
-(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
-t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t 
-x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: 
-T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 
-(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) 
-(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
-i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: 
-T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 
-(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 
-H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
-t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) 
-x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 
-c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
-u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind 
-(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T 
-(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 
-x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda 
-(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 
-x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) TNil (TLRef i))) (THead (Bind Abbr) 
-x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c 
-(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef 
-i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead (Flat Appl) t (THeads (Flat 
-Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 x3)))))).(\lambda (_: (nf2 c 
-(THeads (Flat Appl) (TCons t1 t3) (TLRef i)))).(\lambda (H13: (eq T (THeads 
-(Flat Appl) (TCons t1 t3) (TLRef i)) (THead (Bind Abst) x0 x1))).(let H14 
-\def (eq_ind T (THead (Flat Appl) t1 (THeads (Flat Appl) t3 (TLRef i))) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x0 
-x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) (TCons 
-t1 t3) (TLRef i))) (THead (Bind Abbr) x2 x3)) H14))))))) t0 H_y H8) t2 
-H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) 
-t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: 
-B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: 
-(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) 
-t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: 
-TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat 
-Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t 
-(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) 
-(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil 
-(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead 
-(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ 
-_ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) TNil (TLRef i))) (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3))) H16)))) (\lambda (t1: T).(\lambda 
-(t3: TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq 
-T (THeads (Flat Appl) t3 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead 
-(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3))))))).(\lambda (_: (nf2 c (THeads (Flat 
-Appl) (TCons t1 t3) (TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) 
-(TCons t1 t3) (TLRef i)) (THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T 
-(THead (Flat Appl) t1 (THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T 
-(THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) 
-t2 H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
-
-theorem nf2_appl_lref:
- \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c 
-(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: 
-nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 
-(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
-
-lemma nf2_lref_abst:
- \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c 
-(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c 
-(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 
-(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d 
-(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O 
-u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T 
-(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
-H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
-(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
-(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
-(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
-(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) 
-O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) 
-(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c 
-c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H 
-(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) 
-u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k 
-_) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e 
-(Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) H3)) in (False_ind (eq T (TLRef 
-i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) H1)))))))).
-
-lemma nf2_lift:
- \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: 
-nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
-\def
- \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) 
-\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: 
-nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c 
-(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind 
-T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) 
-(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i 
-x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq 
-T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x 
-(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq 
-T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) 
-H2)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma
deleted file mode 100644 (file)
index 7fb3cd2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc1/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr1/defs.ma".
-
-definition pc1:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda 
-(t: T).(pr1 t2 t)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
deleted file mode 100644 (file)
index 86e794a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
+++ /dev/null
@@ -1,116 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc1/defs.ma".
-
-include "basic_1/pr1/pr1.ma".
-
-lemma pc1_pr0_r:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T 
-(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) 
-(pr1_refl t2)))).
-
-lemma pc1_pr0_x:
- \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T 
-(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) 
-(pr1_pr0 t2 t1 H)))).
-
-lemma pc1_refl:
- \forall (t: T).(pc1 t t)
-\def
- \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: 
-T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
-
-lemma pc1_pr0_u:
- \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
-t3) \to (pc1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: 
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
-T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
-T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda 
-(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) 
-H3)))) H1)))))).
-
-lemma pc1_s:
- \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in 
-(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 
-t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 
-x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 
-H1)))) H0)))).
-
-lemma pc1_head_1:
- \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall 
-(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: 
-T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) 
-(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda 
-(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T 
-(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 
-t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) 
-H0)))))).
-
-lemma pc1_head_2:
- \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall 
-(k: K).(pc1 (THead k u t1) (THead k u t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: 
-T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) 
-(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: 
-T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda 
-(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead 
-k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
-
-theorem pc1_t:
- \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 
-t3) \to (pc1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: 
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
-T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
-T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in 
-(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 
-t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 
-x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 
-t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x 
-x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 
-(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x 
-H2))))) H4))))) H1)))))).
-
-lemma pc1_pr0_u2:
- \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 
-t2) \to (pc1 t1 t2)))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: 
-T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
-
-theorem pc1_head:
- \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall 
-(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 
-t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead 
-k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) 
-(pc1_head_2 t1 t2 H0 u2 k)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma
deleted file mode 100644 (file)
index c8cca31..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/dec.ma
+++ /dev/null
@@ -1,146 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/arity_props.ma".
-
-include "basic_1/nf2/fwd.ma".
-
-theorem pc3_dec:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c 
-u1 u2) ((pc3 c u1 u2) \to False)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
-u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 
-t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T 
-(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) 
-((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 
-x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 
-\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c 
-u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda 
-(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) 
-in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: 
-Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x 
-x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let 
-H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl 
-(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) 
-(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 
-u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 
-in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) 
-False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 
-x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 
-in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) 
-False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 
-x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def 
-(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def 
-(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 
-H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))).
-
-theorem pc3_abst_dec:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
-\to False))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
-u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 
-H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T 
-(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T 
-(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
-\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c 
-x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T 
-(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T 
-(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
-\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 
-c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T 
-(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x 
-(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda 
-(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: 
-T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: 
-T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c 
-v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) 
-(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 
-t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or 
-(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 
-u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) 
-t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: 
-T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind 
-Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead 
-(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 
-(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: 
-T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def 
-(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
-(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
-(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
-\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead 
-(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind 
-Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda 
-(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) 
-x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 
-x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda 
-(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: 
-Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 
-(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead 
-(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) 
-(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 
-(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c 
-u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: 
-T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False 
-(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead 
-(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: 
-T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda 
-(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 
-\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def 
-(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda 
-(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: 
-(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c 
-(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t 
-x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 
-x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead 
-c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda 
-(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 
-x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
-u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: 
-T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 
-(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: 
-T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def 
-(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) 
-x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind 
-Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in 
-(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) 
-(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K 
-(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) 
-H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) 
-H12)))))) H9)))))) H6)))))) H3)))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma
deleted file mode 100644 (file)
index a0c9723..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/defs.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/defs.ma".
-
-definition pc3:
- C \to (T \to (T \to Prop))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 
-c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
-
-inductive pc3_left (c: C): T \to (T \to Prop) \def
-| pc3_left_r: \forall (t: T).(pc3_left c t t)
-| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
-| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma
deleted file mode 100644 (file)
index 61df150..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/fsubst0.ma
+++ /dev/null
@@ -1,697 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/left.ma".
-
-include "basic_1/fsubst0/fwd.ma".
-
-include "basic_1/csubst0/getl.ma".
-
-lemma pc3_pr2_fsubst0:
- \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall 
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
-c2 t2 t)))))))))))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 
-t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 
-t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
-t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
-(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
-T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda 
-(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
-\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
-t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
-u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
-T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c 
-t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 
-w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
-t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: 
-T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x 
-t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 
-(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl 
-u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: 
-C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 
-(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
-t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: 
-C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 
-T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 
-t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 
-H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
-T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda 
-(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 
-t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x 
-H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c 
-c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) 
-(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) 
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
-T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
-t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
-T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda 
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
-u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 
-t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
-u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 
-H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 
-t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
-c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
-u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
-(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
-(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
-(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
-H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
-(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
-((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
-(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 
-(S i)) u0 t5 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: 
-C).(getl i c0 (CHead c3 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in 
-(ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S 
-(plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda 
-(H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) 
-i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
-(\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in 
-(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 
-x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead 
-e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 
-t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda 
-(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
-i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
-x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
-(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
-b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B 
-Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda 
-(t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) H10 u H14) in (let H18 \def 
-(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H11 d 
-H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
-(Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 
-t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
-(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i)) 
-u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
-_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) 
-\Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
-(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
-i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
-(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
-t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
-u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
-u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
-(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
-c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
-T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: 
-(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
-Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 
-\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
-(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
-(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 
-c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 
-(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 
-t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
-(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
-(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
-(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow 
-c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
-((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: 
-(eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 
-(S i)) u0 t6 x3)) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: 
-C).(getl i c0 (CHead c3 (Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in 
-(ex2_ind T (\lambda (t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S 
-(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda 
-(H21: (subst0 i x3 t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) 
-i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
-(\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in 
-(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c 
-c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 
-(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 
-x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
-e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back 
-t3 t0 u i H2 x3 u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) 
-(\lambda (H9: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) 
-u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 
-(CHead x2 (Bind x0) x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 
-x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) 
-\Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
-(eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) H11 u 
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
-i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
-(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 
-(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) 
-(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
-e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
-x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: 
-(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
-((let H16 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H10) in (\lambda (H17: (eq B Abbr x0)).(\lambda (H18: 
-(eq C d x1)).(let H19 \def (eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 
-(S i)) u0 t6 x4)) H12 u H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: 
-C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in 
-(ex2_ind T (\lambda (t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S 
-(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda 
-(H22: (subst0 i x4 t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) 
-i)) u0 t0 x)).(let H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
-(\lambda (n: nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in 
-(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c 
-c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 
-(pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 
-x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
-e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back 
-t3 t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) 
-H8))) (\lambda (H7: (le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 
-(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 
-t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
-(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
-H2))))))))))) c2 t4 H3)))))))))))))))) c1 t1 t H)))).
-
-lemma pc3_pr2_fsubst0_back:
- \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall 
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
-c2 t t2)))))))))))
-\def
- \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t 
-t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 
-t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 
-t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
-(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
-T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda 
-(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
-\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 
-t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
-u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 
-(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: 
-C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c 
-(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) 
-(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda 
-(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e 
-(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 
-t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e 
-(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) 
-(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
-T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
-t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
-T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda 
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
-u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 
-t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
-u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 
-H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 
-(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 
-(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
-c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
-u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
-(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
-(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
-(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
-H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
-u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
-(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
-(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
-((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
-(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 
-(S i)) u0 t5 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: 
-C).(getl i c0 (CHead c3 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in 
-(ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S 
-(plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda 
-(H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) 
-i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
-(\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in 
-(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 
-x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead 
-e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 
-t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda 
-(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
-i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
-x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
-(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
-b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B 
-Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda 
-(t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) H10 u H14) in (let H18 \def 
-(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H11 d 
-H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
-(Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 
-t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
-(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i)) 
-u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
-_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) 
-\Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
-(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
-H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
-i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
-(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
-i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
-t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
-H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
-nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
-t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
-c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
-u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
-u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
-(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
-c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
-T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: 
-(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
-Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 
-\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
-(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
-B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
-(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u 
-c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 
-(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C 
-T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
-u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
-(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
-(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
-(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow 
-c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
-((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: 
-(eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 
-(S i)) u0 t6 x3)) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: 
-C).(getl i c0 (CHead c3 (Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in 
-(ex2_ind T (\lambda (t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S 
-(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda 
-(H21: (subst0 i x3 t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) 
-i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
-(\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in 
-(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 
-t0 x (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
-e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 
-t0 u i H2 x3 u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda 
-(H9: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
-i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
-x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
-x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def 
-(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
-(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 with 
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
-b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in (\lambda (H16: (eq B 
-Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda 
-(t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) H11 u H15) in (let H19 \def 
-(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H12 d 
-H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
-(Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 (pr2_delta c0 x2 u i H20 
-t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge 
-i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) 
-t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t5) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H12: 
-(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: (csubst0 (minus i0 (S i)) 
-u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
-_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) 
-\Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
-(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def 
-(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u 
-H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
-i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda 
-(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
-i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 
-t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
-H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
-nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x 
-t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x 
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 
-(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
-(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 
-t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) 
-(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i 
-(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
-H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n 
-i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 
-H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
-
-lemma pc3_fsubst0:
- \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall 
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
-t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
-c2 t2 t)))))))))))
-\def
- \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 
-t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 
-t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
-t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: 
-C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u 
-c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e 
-(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 
-i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) 
-u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 
-H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: 
-C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) 
-(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda 
-(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e 
-(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i 
-(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 
-H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: 
-(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: 
-((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: 
-T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
-Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: 
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 
-t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: 
-C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: 
-T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 
-(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 
-t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda 
-(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
-(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 
-(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 
-c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda 
-(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: 
-(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 
-t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 
-(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda 
-(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: 
-T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall 
-(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 
-t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: 
-T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda 
-(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) 
-\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 
-t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) 
-u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 
-t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda 
-(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
-(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 
-t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 
-(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: 
-(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 
-c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) 
-u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 
-t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst 
-i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma
deleted file mode 100644 (file)
index 99239c2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/fwd.ma
+++ /dev/null
@@ -1,304 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/props.ma".
-
-lemma pc3_gen_sort:
- \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort 
-n)) \to (eq nat m n))))
-\def
- \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c 
-(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c 
-(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: 
-T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let 
-H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n 
-H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat 
-(\lambda (e: T).(match e with [(TSort n0) \Rightarrow n0 | (TLRef _) 
-\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H3) in 
-H4))))) H0))))).
-
-lemma pc3_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall 
-(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to 
-(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
-t1 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 
-t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) 
-u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c 
-u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) 
-(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda 
-(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 
-x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: 
-T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 
-x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) 
-(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
-t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead 
-(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def 
-(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead 
-(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ t _) 
-\Rightarrow t])) (THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in 
-((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x3 | (TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) 
-(THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: 
-(eq T x2 x0)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 
-\def (eq_ind T x2 (\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 
-u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) 
-(pc3_pr3_t c u1 x0 H16 u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t 
-(CHead c (Bind b) u) t1 x1 (H15 b u) t2 (H6 b u))))))))) H12)))))))) 
-H7))))))) H3))))) H0))))))).
-
-lemma pc3_gen_abst_shift:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c 
-(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind 
-Abst) u) t1 t2)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def 
-(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) 
-(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 
-(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: 
-((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 
-Abst u))) H0))))))).
-
-lemma pc3_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall 
-(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d 
-c e) \to (pc3 e t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: 
-C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: 
-T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e 
-t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: 
-(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in 
-(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e 
-t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d 
-x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e 
-H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: 
-T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift 
-h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: 
-T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 
-(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e 
-t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
-
-lemma pc3_gen_not_abst:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: 
-T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) 
-u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S 
-O) O (THead (Bind Abst) u2 t2))))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
-(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: 
-T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead 
-c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda 
-(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) 
-u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
-T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind 
-Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind 
-Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) 
-x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
-(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead 
-c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
-(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 
-x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def 
-(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
-c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 
-(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 
-x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind 
-Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T 
-(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow False | 
-Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind 
-Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) H14)))))))) H9))))))) 
-H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))).(let 
-H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c 
-(Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 
-x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def (eq_ind T x 
-(\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O t))) H5 (THead 
-(Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) t1 (lift (S O) 
-O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind Abst) u2 t2)) 
-(pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind Abbr) O c c 
-(drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 x1) 
-(pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) H4))))) 
-H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: C).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pc3 
-c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def (match 
-(H (refl_equal B Abst)) in False with []) in H1)))))))) (\lambda (_: (not (eq 
-B Void Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) 
-(THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
-T).(pr3 c (THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind 
-Abst) u2 t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind 
-Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) 
-x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
-(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) 
-t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead 
-(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
-c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 
-x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x 
-H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
-Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind 
-Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x 
-(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: 
-((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let 
-H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 
-(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) 
-x2 x3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow (match b0 with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
-Void) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) 
-O (THead (Bind Abst) u2 t2))) H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 
-(CHead c (Bind Void) u1) t1 (lift (S O) O x))).(let H6 \def (pr3_gen_abst c 
-u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O 
-(THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: 
-(eq T x (THead (Bind Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda 
-(H9: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 
-x1))))).(let H10 \def (eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) 
-u1) t1 (lift (S O) O t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t 
-(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 
-(lift (S O) O (THead (Bind Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c 
-(S O) O (drop_drop (Bind Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 
-t2) (THead (Bind Abst) x0 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 
-Abst x0)))))))))) H6))) H4))))) H1))))))))) b).
-
-lemma pc3_gen_lift_abst:
- \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall 
-(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) 
-\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda 
-(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
-T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
-t1)))))))))))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind 
-Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H 
-in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 
-c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: 
-T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
-T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) 
-(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c 
-(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) 
-in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 
-e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind 
-Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) 
-(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T 
-x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 
-t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
-(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e 
-t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 
-(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 
-x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x 
-(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y 
-z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: 
-T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: 
-T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
-t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead 
-(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: 
-(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 
-(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c 
-u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: 
-T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T 
-(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) 
-(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: 
-T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 
-x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
-
-lemma pc3_gen_sort_abst:
- \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c 
-(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda 
-(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 
-\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: 
-T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c 
-(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def 
-(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u 
-x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
-u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) 
-(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind 
-T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H7) in (False_ind P H8)))))))) H3))))) H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
deleted file mode 100644 (file)
index 579d42b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
+++ /dev/null
@@ -1,117 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/props.ma".
-
-implied rec lemma pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall 
-(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to 
-(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: 
-(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
-T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: 
-pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow 
-(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1 
-((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1) 
-\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
-
-fact pc3_ind_left__pc3_left_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda 
-(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 
-c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: 
-(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_trans:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
-(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: 
-T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda 
-(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
-t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 
-t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: 
-(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda 
-(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 
-t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_sym:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
-(pc3_left c t2 t1))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) 
-(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
-t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 
-H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 
-t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 
-(pc3_left_r c t3))))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_pc3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to 
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
-T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
-x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x 
-(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x 
-(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
-
-fact pc3_ind_left__pc3_pc3_left:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
-(pc3 c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda 
-(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c 
-t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 
-c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 
-(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
-
-lemma pc3_ind_left:
- \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 
-t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
-T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: 
-T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to 
-((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c 
-t t0) \to (P t t0))))))))
-\def
- \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall 
-(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 
-t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 
-t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
-\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 
-t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t 
-t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: 
-T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 
-(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left 
-c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 
-(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 
-(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma
deleted file mode 100644 (file)
index 3ce4180..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/nf2.ma
+++ /dev/null
@@ -1,46 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/defs.ma".
-
-include "basic_1/nf2/pr3.ma".
-
-lemma pc3_nf2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
-t1) \to ((nf2 c t2) \to (eq T t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in 
-(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T 
-t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 
-x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x 
-(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x 
-(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c 
-t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 
-H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 
-H_y0))))))))) H2))))))).
-
-lemma pc3_nf2_unfold:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
-t2) \to (pr3 c t1 t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: 
-T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: 
-T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def 
-(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: 
-T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma
deleted file mode 100644 (file)
index a629475..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/pc1.ma
+++ /dev/null
@@ -1,33 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/defs.ma".
-
-include "basic_1/pc1/defs.ma".
-
-include "basic_1/pr3/pr1.ma".
-
-lemma pc3_pc1:
- \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 
-t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: 
-C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: 
-T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 
-x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) 
-(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) 
-H0))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma
deleted file mode 100644 (file)
index 8d86935..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/props.ma
+++ /dev/null
@@ -1,410 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/defs.ma".
-
-include "basic_1/pr3/pr3.ma".
-
-lemma clear_pc3_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to 
-(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind 
-T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 
-t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 
-t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 
-H0))))) H1))))))).
-
-lemma pc3_pr2_r:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c 
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
-t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
-
-lemma pc3_pr2_x:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c 
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 
-t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
-t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
-
-lemma pc3_pr3_r:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c 
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
-t2 H (pr3_refl c t2))))).
-
-lemma pc3_pr3_x:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c 
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 
-t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
-t1 (pr3_refl c t1) H)))).
-
-lemma pc3_pr3_t:
- \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall 
-(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 
-t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: 
-T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
-
-lemma pc3_refl:
- \forall (c: C).(\forall (t: T).(pc3 c t t))
-\def
- \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) 
-(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
-
-lemma pc3_s:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c 
-t2 t1))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 
-t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
-T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
-x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) 
-(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
-
-lemma pc3_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
-(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda 
-(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u 
-t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
-x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead 
-(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead 
-(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) 
-H0))))))).
-
-lemma pc3_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
-u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda 
-(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) 
-(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: 
-(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) 
-(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x 
-H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl 
-(CHead c k x) t)))))) H0))))))).
-
-lemma pc3_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T 
-(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) 
-t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: 
-(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 
-T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u 
-t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) 
-(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
-
-lemma pc3_pr2_u:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) 
-x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
-
-theorem pc3_t:
- \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall 
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
-x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
-T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 
-x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) 
-(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: 
-(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c 
-H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) 
-H4))))) H1))))))).
-
-lemma pc3_pr2_u2:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
-(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
-t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x 
-c t1 t0 H) t2 H0)))))).
-
-lemma pc3_pr3_conf:
- \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall 
-(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t 
-t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c 
-t2 t H0) t1 H)))))).
-
-theorem pc3_head_12:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
-(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
-(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c 
-u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
-
-theorem pc3_head_21:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
-(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
-(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c 
-u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
-
-lemma pc3_pr0_pr2_t:
- \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: 
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
-(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
-t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda 
-(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: 
-(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 
-(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k 
-u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
-(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k 
-u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in 
-(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) 
-u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k 
-u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) 
-t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
-u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead 
-c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda 
-(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
-u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
-((let H12 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
-((let H13 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
-(\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H16 \def (eq_ind 
-T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) in (eq_ind B Abbr 
-(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
-T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) (pc3 (CHead c (Bind 
-Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O u1 t4 x)).(\lambda 
-(H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x (pr3_pr2 (CHead c 
-(Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl 
-Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) u1) t x (pr2_free 
-(CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 t4 t O H16 u1 H)) b 
-H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: (clear (CHead c (Flat 
-f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t3 
-t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t (pr2_delta (CHead d (Bind Abbr) u) 
-d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) (CHead c (Flat f) u1) (clear_flat 
-c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H10) 
-f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H8)))) (\lambda 
-(i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) 
-\to ((subst0 i0 u t4 t) \to (pc3 (CHead c k u1) t3 t))))).(\lambda (H8: (getl 
-(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) 
-u t4 t)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind 
-Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c k0 u1) t3 t)))) \to 
-((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) 
-(\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind 
-Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Bind b) u1) t3 
-t))))).(\lambda (H10: (getl (r (Bind b) i0) c (CHead d (Bind Abbr) 
-u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d 
-u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H10 u1) t3 t4 H3 t 
-H9))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead 
-d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Flat f) u1) t3 
-t))))).(\lambda (H10: (getl (r (Flat f) i0) c (CHead d (Bind Abbr) 
-u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u 
-(r (Flat f) i0) H10 t3 t4 H3 t H9) f u1))))) k IHi (getl_gen_S k c (CHead d 
-(Bind Abbr) u) u2 i0 H8)))))) i H7 H4)))))))))))))) y t1 t2 H1))) H0)))))))).
-
-lemma pc3_pr2_pr2_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 
-u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
-T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 
-(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
-K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 
-t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
-(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
-(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: 
-C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: 
-C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
-T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 
-t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
-t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 
-t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
-(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
-t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
-c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
-(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: 
-nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
-t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
-(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 
-(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
-(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
-c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 
-(CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 \def (f_equal C B (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
-(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead 
-d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
-(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead 
-d0 (Bind Abbr) u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
-C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
-H11 t1 H15) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 
-t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 
-t6 t7)) (pc3 (CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: 
-(subst0 O t2 t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: 
-T).(subst0 O t t5 t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 
-(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t 
-t5 x0)).(\lambda (H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal 
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S 
-i) H23) in (pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 
-(Bind Abbr) t) c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x 
-(CHead c0 (Bind Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) 
-(getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 
-H24)))))))) (subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd 
-t1 t5 t6 O H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda 
-(H12: (clear (CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) 
-u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 
-(Bind Abbr) u0) t4 t6 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl 
-Abbr d0 u0) t4 t5 H6 t6 H11)) (CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 
-(Bind Abbr) u0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f 
-t)))) k (getl_gen_O (CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) 
-(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind 
-Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pc3 (CHead c0 k t) t4 
-t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t1) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl 
-(r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c0 k0 t) t4 t6))) 
-(\lambda (b: B).(\lambda (H12: (getl (r (Bind b) i1) c0 (CHead d0 (Bind Abbr) 
-u0))).(pc3_pr2_r (CHead c0 (Bind b) t) t4 t6 (pr2_delta (CHead c0 (Bind b) t) 
-d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind Abbr) u0) H12 t) t4 t5 
-H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (r (Flat f) i1) c0 (CHead 
-d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 
-t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) H12 t4 t5 H6 t6 H11) f t)))) k 
-(getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 i1 H10)))))) i0 H9 
-H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 H)))).
-
-lemma pc3_pr2_pr3_t:
- \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to 
-(pc3 (CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
-(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 
-(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
-u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
-(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) 
-\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 
-u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
-u1 H3)))))))))) t1 t2 H)))))).
-
-lemma pc3_pr3_pc3_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 
-u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
-(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 
-t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
-K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
-(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
-t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 
-(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: 
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 
-t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c 
-k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) 
-t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: 
-(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 
-x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 
-H0)))))) H4))))))))))))) u2 u1 H)))).
-
-lemma pc3_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift 
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 
-t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: 
-T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda 
-(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) 
-(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H 
-t2 x H3))))) H1))))))))).
-
-lemma pc3_eta:
- \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t 
-(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead 
-(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: 
-(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v 
-w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O 
-(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef 
-O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) 
-(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead 
-(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) 
-(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S 
-O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) 
-H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w 
-(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) 
-(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O 
-(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl 
-c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma
deleted file mode 100644 (file)
index 1df3512..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/subst1.ma
+++ /dev/null
@@ -1,45 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/props.ma".
-
-include "basic_1/pr3/subst1.ma".
-
-lemma pc3_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall 
-(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d 
-c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c 
-a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: 
-T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda 
-(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda 
-(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: 
-T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T 
-(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 
-x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d 
-x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x 
-(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda 
-(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 
-a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 
-(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 
-H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) 
-(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma
deleted file mode 100644 (file)
index a72a7e9..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/wcpr0.ma
+++ /dev/null
@@ -1,87 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/props.ma".
-
-include "basic_1/wcpr0/getl.ma".
-
-fact pc3_wcpr0__pc3_wcpr0_t_aux:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall 
-(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 
-(CHead c2 k u) t1 t2))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: 
-K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
-(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: 
-T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 
-t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: 
-(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead 
-c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 
-t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead 
-c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda 
-(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 
-k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda 
-(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d 
-(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 
-t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c 
-(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead 
-d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
-T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
-(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: 
-T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 
-t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t 
-x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 
-t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x 
-H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k 
-u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) 
-H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
-
-lemma pc3_wcpr0_t:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
-T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 
-t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: 
-C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: 
-T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def 
-(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T 
-(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k 
-u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 
-(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x 
-(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 
-(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x 
-H6)))))) H4))))))))))))) c1 c2 H))).
-
-lemma pc3_wcpr0:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
-T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind 
-T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 
-t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 
-x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 
-(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
deleted file mode 100644 (file)
index 889e820..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/dec.ma
+++ /dev/null
@@ -1,520 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/props.ma".
-
-include "basic_1/subst0/dec.ma".
-
-include "basic_1/T/dec.ma".
-
-include "basic_1/T/props.ma".
-
-lemma nf0_dec:
- \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t1 t2))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to 
-(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl 
-(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) 
-t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T 
-(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl 
-(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) 
-t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T 
-(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: 
-T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 
-t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or 
-(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) 
-(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: 
-B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) 
-t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind 
-Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: 
-T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in 
-(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) 
-O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead 
-(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t 
-t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 
-(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S 
-O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) 
-t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind 
-Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let 
-H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
-| (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind 
-Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)) H4) in (let H6 \def 
-(eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S O) O x))) H3 (lift (S 
-O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 P))))) (pr0_delta t t 
-(pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) (\lambda (H3: (eq T 
-t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) \to (\forall (P: 
-Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) t3)))) (ex_intro2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) O x)) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t (lift (S 
-O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t (lift (S O) O x)) 
-x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S O) O H4 P))) 
-(pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) H1)))) (let 
-H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to 
-(eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
-(Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 t t2) 
-\to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) 
-\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead 
-(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda 
-(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
-(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) 
-(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 
-t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def 
-(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 
-H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: 
-T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: 
-T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead 
-(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) 
-(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 
-t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: 
-T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) 
-(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead 
-(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t 
-t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) 
-t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: 
-Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 
-(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind 
-Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t 
-x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
-\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) t x) H7) in 
-(let H9 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H6 t0 H8) in (let 
-H10 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
-Prop).P0))) H5 t0 H8) in (H10 (refl_equal T t0) P)))))) (pr0_comp t t 
-(pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
-(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: 
-T).(\lambda (H3: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr0 
-t x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq 
-T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind 
-Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind 
-Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Bind Abst) t t0) t2)) (THead (Bind Abst) x t0) (\lambda (H5: (eq T (THead 
-(Bind Abst) t t0) (THead (Bind Abst) x t0))).(\lambda (P: Prop).(let H6 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
-_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) 
-(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: 
-T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T 
-t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) 
-(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x 
-\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or 
-(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall 
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) 
-(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 
-(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift 
-(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T 
-(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind 
-Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let 
-H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to 
-(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
-(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) 
-\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) 
-\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead 
-(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda 
-(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) 
-(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t 
-t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 
-t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t 
-x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def 
-(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 
-t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead 
-(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda 
-(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: 
-T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead 
-(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) 
-(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: 
-(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let 
-H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 
-(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq 
-T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) 
-(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
-(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead 
-(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 
-H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
-t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall 
-(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) 
-(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: 
-Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 
-(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind 
-Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) 
-t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
-\Rightarrow t2])) (THead (Bind Void) t t0) (THead (Bind Void) t x0) H10) in 
-(let H12 \def (eq_ind_r T x0 (\lambda (t2: T).(pr0 t0 t2)) H9 t0 H11) in (let 
-H13 \def (eq_ind_r T x0 (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
-Prop).P0))) H8 t0 H11) in (H13 (refl_equal T t0) P)))))) (pr0_comp t t 
-(pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6))) (\lambda (H5: (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
-(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda 
-(x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: Prop).P)))).(\lambda 
-(H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) 
-t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
-(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) x0 t0) (\lambda (H8: 
-(eq T (THead (Bind Void) t t0) (THead (Bind Void) x0 t0))).(\lambda (P: 
-Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) 
-(THead (Bind Void) t t0) (THead (Bind Void) x0 t0) H8) in (let H10 \def 
-(eq_ind_r T x0 (\lambda (t2: T).(pr0 t t2)) H7 t H9) in (let H11 \def 
-(eq_ind_r T x0 (\lambda (t2: T).((eq T t t2) \to (\forall (P0: Prop).P0))) H6 
-t H9) in (H11 (refl_equal T t) P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) 
-(Bind Void))))))) H5)) H4))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(let 
-H4 \def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to 
-(eq T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: 
-Prop).P))) (\lambda (t3: T).(pr0 t2 t3))))) H0 (lift (S O) O x) H3) in 
-(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or (\forall (t3: T).((pr0 
-(THead (Bind Void) t t2) t3) \to (eq T (THead (Bind Void) t t2) t3))) (ex2 T 
-(\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3) \to (\forall (P: 
-Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2) t3))))) (or_intror 
-(\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O x)) t2) \to (eq T 
-(THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T (\lambda (t2: T).((eq T 
-(THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2))) (ex_intro2 
-T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S 
-O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) 
-x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Void) x t (S O) O H5 P))) 
-(pr0_zeta Void not_void_abst x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) 
-(\lambda (f: F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead 
-(Flat f0) t t0) t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda 
-(t2: T).((eq T (THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) 
-(\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2))))) (let H_x \def 
-(binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: 
-B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
-(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
-u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat 
-Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: 
-T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T 
-(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 
-u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq 
-T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
-t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
-T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 
-\def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq 
-T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) 
-(\lambda (t3: T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r 
-T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead 
-(Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T 
-(\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: 
-Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind 
-(\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to 
-(eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
-(Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
-(THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 
-x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
-b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
-Appl) t (THead (Bind b) x1 x2)) t2)))))) (\lambda (_: (or (\forall (t2: 
-T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1 x2) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2) 
-t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind 
-Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) 
-x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
-Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
-(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 
-x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
-(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead 
-(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
-Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
-(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
-I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
-(False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 
-(pr0_refl x1) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 
-(THead (Bind Abst) x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 
-T (\lambda (t2: T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror 
-(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
-\to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
-\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead 
-(Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
-Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda 
-(t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead 
-(Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) 
-x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T 
-(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ 
-_) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) 
-(pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or 
-(\forall (t2: T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind 
-Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) 
-t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 
-x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead 
-(Bind Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 
-x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
-Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: 
-T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) 
-x1 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
-(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead 
-(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
-Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
-(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
-I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
-(False_ind P H7)))) (pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 
-(pr0_refl x1) x2 x2 (pr0_refl x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: 
-((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
-u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: 
-T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: 
-T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
-(\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def 
-H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T 
-(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to 
-(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
-(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
-(THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) 
-\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t 
-t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
-T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 
-(THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) 
-(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 
-t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t 
-x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def 
-(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 
-t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: 
-T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3: 
-T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead 
-(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) 
-(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead 
-(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 
-x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead 
-(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall 
-(t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in 
-(let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind 
-Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind 
-Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl 
-(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 
-x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: 
-B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T 
-t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 
-(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda 
-(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 
-(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat 
-Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4: 
-T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let 
-H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) 
-x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 
-(THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind 
-x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t 
-(THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
-x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) 
-(\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
-t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall 
-(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
-t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
-(\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: 
-Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 
-(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T 
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat 
-Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t 
-x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
-\Rightarrow t2])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in 
-(let H11 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let 
-H12 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
-Prop).P0))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t 
-(pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T 
-(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
-(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
-(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
-Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: 
-T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 
-t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq 
-T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
-Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead 
-(Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
-_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) 
-(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: 
-T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq 
-T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) 
-P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) 
-H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq 
-T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
-Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
-Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
-(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) 
-t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 
-(pr0_refl t0) t))) f)) k)))))) t1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma
deleted file mode 100644 (file)
index 5f6bd58..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/defs.ma
+++ /dev/null
@@ -1,40 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/defs.ma".
-
-inductive pr0: T \to (T \to Prop) \def
-| pr0_refl: \forall (t: T).(pr0 t t)
-| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) 
-(THead k u2 t2))))))))
-| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to 
-(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
-| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: 
-T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 
-u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)))))))))))))
-| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to 
-(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
-| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall 
-(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O 
-t1)) t2))))))
-| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: 
-T).(pr0 (THead (Flat Cast) u t1) t2)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
deleted file mode 100644 (file)
index ad2f0e8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/fwd.ma
+++ /dev/null
@@ -1,1561 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/defs.ma".
-
-include "basic_1/subst0/fwd.ma".
-
-implied rec lemma pr0_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
-t))) (f0: (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to 
-(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall 
-(k: K).(P (THead k u1 t1) (THead k u2 t2)))))))))))) (f1: (\forall (u: 
-T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall 
-(t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))))))) (f2: 
-(\forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
-T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 
-u2) \to ((P u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P 
-t1 t2) \to (P (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))))))))))))))) (f3: (\forall 
-(u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (w: T).((subst0 
-O u2 t2 w) \to (P (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 
-w))))))))))))) (f4: (\forall (b: B).((not (eq B b Abst)) \to (\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead 
-(Bind b) u (lift (S O) O t1)) t2))))))))) (f5: (\forall (t1: T).(\forall (t2: 
-T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead (Flat Cast) u 
-t1) t2))))))) (t: T) (t0: T) (p: pr0 t t0) on p: P t t0 \def match p with 
-[(pr0_refl t1) \Rightarrow (f t1) | (pr0_comp u1 u2 p0 t1 t2 p1 k) 
-\Rightarrow (f0 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 
-((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) k) | (pr0_beta u v1 v2 p0 t1 t2 
-p1) \Rightarrow (f1 u v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) v1 v2 p0) t1 
-t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1)) | (pr0_upsilon b n v1 v2 p0 
-u1 u2 p1 t1 t2 p2) \Rightarrow (f2 b n v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
-f5) v1 v2 p0) u1 u2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p1) t1 t2 p2 
-((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p2)) | (pr0_delta u1 u2 p0 t1 t2 p1 w 
-s0) \Rightarrow (f3 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 
-p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) w s0) | (pr0_zeta b n t1 t2 p0 
-u) \Rightarrow (f4 b n t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p0) u) 
-| (pr0_tau t1 t2 p0 u) \Rightarrow (f5 t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
-f5) t1 t2 p0) u)].
-
-lemma pr0_gen_sort:
- \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq 
-T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
-(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
-T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
-t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in 
-(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) 
-t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
-t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 
-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
-(TSort n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t1)) (TSort n))).(let H6 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
-v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
-t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
-(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H8) in 
-(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
-(TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
-(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda 
-(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
-Abbr) u1 t1) (TSort n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in 
-(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
-H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
-(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
-(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 
-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
-O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (u: 
-T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TSort n))).(let H4 \def 
-(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
-H4)))))))) y x H0))) H))).
-
-lemma pr0_gen_lref:
- \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
-\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq 
-T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
-(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
-T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
-t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in 
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) 
-t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
-t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 
-H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
-(TLRef n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t1)) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
-v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
-t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
-(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in 
-(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
-(TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
-(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda 
-(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
-Abbr) u1 t1) (TLRef n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in 
-(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
-H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
-(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
-(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 
-b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
-O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (u: 
-T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TLRef n))).(let H4 \def 
-(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
-H4)))))))) y x H0))) H))).
-
-lemma pr0_gen_abst:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
-t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: 
-T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
-t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda 
-(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def 
-(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r 
-T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind 
-Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 
-(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t 
-H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
-(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
-(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) in ((let H7 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
-(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
-(THead (Bind Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) 
-in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Abst))).(eq_ind_r 
-K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind 
-Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 
-u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
-T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) 
-H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
-v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
-T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead 
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: 
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) 
-H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
-w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
-t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
-[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H6) in (False_ind 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
-(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H7))))))))))))) (\lambda (b: 
-B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abst) u1 
-t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H4: (eq T 
-(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(let H5 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
-_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Abst) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
-_ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
-Abst) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
-(TLRef _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
-(THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Abst) u1 t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b 
-Abst)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 
-Abst H9) in (let H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead 
-(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in 
-(eq_ind T (lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t 
-t3))))) (let H12 \def (match (H10 (refl_equal B Abst)) in False with []) in 
-H12) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T 
-(THead (Flat Cast) u t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T 
-(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) H4)))))))) y x H0))) H)))).
-
-lemma pr0_gen_appl:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
-t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) 
-v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2))))))))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: 
-T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
-t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T 
-t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: 
-T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T 
-T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead 
-(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T 
-t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
-T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl 
-t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 
-u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
-Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to 
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
-T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat 
-Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
-\Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
-(THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: 
-(eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T 
-T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Appl) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
-(THead k0 u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))) (let H11 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let H12 \def (eq_ind 
-T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 
-(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 
-(\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda 
-(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead 
-(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 
-(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) 
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: 
-T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 
-t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
-(THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
-(Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ 
-_ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
-(Flat Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T 
-v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 
-(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 
-(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 
-\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
-b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in 
-(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) 
-(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead 
-(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind 
-Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) 
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 
-y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 
-(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 
-t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not 
-(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 
-v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
-(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to 
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) 
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
-(THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H8) in ((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
-(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ 
-t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
-(Flat Appl) u1 t1) H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind 
-T v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 
-(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 
-(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead 
-(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def 
-(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
-T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in 
-(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 
-t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead 
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in 
-(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat 
-Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat 
-Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat 
-Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
-Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind 
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) 
-(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) 
-(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) 
-H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
-t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: 
-(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
-(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 
-t1) H6) in (False_ind (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Bind Abbr) u2 w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-u2 w) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind 
-Abbr) u2 w) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))) H7))))))))))))) 
-(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) 
-u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq 
-T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 
-\def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
-(Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: 
-T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 
-t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
-
-lemma pr0_gen_cast:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) 
-(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda 
-(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
-t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat 
-Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat 
-Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: 
-T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat 
-Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead 
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) 
-u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T 
-(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda 
-(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda 
-(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
-(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) in ((let H7 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
-(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
-(THead (Flat Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) 
-in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat Cast))).(eq_ind_r 
-K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 (\lambda (t: 
-T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
-t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 
-u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 
-t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
-(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
-Cast) u1 t1) H5) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (THead (Bind Abbr) v2 t2))) H6)))))))))))) (\lambda (b: 
-B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead 
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 v2))))).(\lambda (u0: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
-(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
-Cast) u1 t1) H8) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead 
-(Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
-(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
-w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 
-t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H6) in (False_ind (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
-(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) u2 
-w))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
-Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u 
-(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(let H5 \def (eq_ind T (THead 
-(Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Cast) u1 t1) H4) in (False_ind (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda 
-(H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(let H4 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) 
-(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 
-t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def (eq_ind T t0 (\lambda (t: 
-T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: 
-T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) 
-H8))))) H4)))))))) y x H0))) H)))).
-
-lemma pr0_gen_lift:
- \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
-(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(pr0 t1 t2)))))))
-\def
- \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t 
-x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat 
-d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T 
-x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: 
-T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq 
-T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: 
-T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 
-x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: 
-T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq 
-T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) 
-(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: 
-((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T 
-(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 
-t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
-t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda 
-(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))))) 
-(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1 
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
-b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
-(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T 
-t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t: 
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h 
-(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) 
-x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) 
-x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t: 
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4: 
-T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda 
-(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T 
-(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) 
-t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) 
-x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
-x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind 
-b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5 
-x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h 
-x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3 
-H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6)))) 
-(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1 
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
-f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda 
-(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
-(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T 
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-T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: 
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-(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) 
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-(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h 
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-(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h 
-x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S 
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-(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1 
-(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda 
-(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
-v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h 
-x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2: 
-T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
-u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 
-x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: 
-T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
-t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T 
-(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) 
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-x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead 
-(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
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-(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0 
-z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: 
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda 
-(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1) 
-x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: 
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T 
-(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) 
-(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 
-(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift 
-h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq 
-T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: 
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S 
-x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead 
-(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 
-x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: 
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift 
-(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: 
-T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O 
-v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T 
-v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) 
-(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) 
-(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T 
-(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t 
-(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
-Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat 
-Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
-(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda 
-(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat 
-Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
-(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat 
-Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead 
-(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S 
-x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat 
-Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 
-x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1) 
-x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1 
-x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 
-H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) 
-(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1 
-(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: 
-T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
-T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w: 
-T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1 
-x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
-Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda 
-(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
-(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: 
-(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda 
-(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 
-(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: 
-T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
-(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 
-(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 
-(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T 
-(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
-T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: 
-(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 
-x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let 
-H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 
-(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in 
-(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) 
-(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq 
-T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 
-T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 
-t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: 
-T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: 
-(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: 
-T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T 
-(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O 
-x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) 
-t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 
-x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) 
-(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) 
-w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 
-H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 
-H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
-T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: 
-T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u 
-(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda 
-(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq 
-T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) 
-(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 
-x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T 
-(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift 
-h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S 
-O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (le_antisym (S x1) (plus (S O) 
-x1) (le_n (plus (S O) x1)) (le_n (S x1))) (plus x1 (S O)) (plus_sym x1 (S 
-O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: nat).(eq T (lift (S O) O 
-t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in (ex2_ind T (\lambda (t4: T).(eq 
-T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq T t2 (lift h x1 t4))) (ex2 T 
-(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind 
-b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H10: (eq T x3 (lift (S O) O 
-x4))).(\lambda (H11: (eq T t2 (lift h x1 x4))).(eq_ind_r T (lift (S O) O x4) 
-(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
-(t4: T).(pr0 (THead (Bind b) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T 
-t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq 
-T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
-x4)) t4))) (\lambda (x5: T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda 
-(H12: (pr0 x4 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
-(t4: T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 
-(lift (S O) O x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) 
-t4)) x5 (refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 
-H_x)))) (H3 x4 x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n 
-x1) H9)))) x0 H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 
-H4)))))))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
-t3)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
-x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
-T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H3: (eq T (THead (Flat Cast) u t2) (lift h x1 x0))).(ex3_2_ind 
-T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Cast) y0 z)))) 
-(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T t3 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) x2 x3))).(\lambda (_: (eq T 
-u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead 
-(Flat Cast) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h 
-x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 
-(lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T 
-t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4))) 
-(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H7: (pr0 
-x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
-T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) 
-t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x4) (lift h x1 t4))) 
-(\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)) x4 (refl_equal T (lift h 
-x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) 
-(lift_gen_flat Cast u t2 x0 h x1 H3)))))))))) y x H0))))) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
deleted file mode 100644 (file)
index d3a8f78..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/pr0.ma
+++ /dev/null
@@ -1,2303 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/subst0.ma".
-
-include "basic_1/lift/tlt.ma".
-
-include "basic_1/tlt/fwd.ma".
-
-fact pr0_confluence__pr0_cong_upsilon_refl:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
-T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to 
-(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) 
-\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) 
-t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t5)) t)))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda 
-(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda 
-(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda 
-(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead 
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 
-t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) 
-(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S 
-O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind 
-b))))))))))))))).
-
-fact pr0_confluence__pr0_cong_upsilon_cong:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
-T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall 
-(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: 
-T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) 
-(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t5)) t)))))))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda 
-(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 
-x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 
-x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda 
-(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) 
-(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
-t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) 
-(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat 
-Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp 
-(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat 
-Appl)) (Bind b))))))))))))))))))).
-
-fact pr0_confluence__pr0_cong_upsilon_delta:
- (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
-T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: 
-T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 
-x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to 
-((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))))
-\def
- \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: 
-T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: 
-T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: 
-(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 
-x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda 
-(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T 
-(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 
-(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead 
-(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H 
-u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O 
-v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) 
-(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind 
-Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: 
-T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda 
-(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 
-(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: 
-(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: 
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) 
-(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon 
-Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift 
-(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) 
-O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) 
-(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 
-(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 
-H5))))))))))))))))))).
-
-fact pr0_confluence__pr0_cong_upsilon_zeta:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
-T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 
-u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: 
-T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat 
-Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) (lift (S O) O x))) t)))))))))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda 
-(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda 
-(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: 
-T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: 
-(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: 
-T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
-T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead 
-(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O 
-(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 
-t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat 
-Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) 
-(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) 
-O)))))))))))))))).
-
-fact pr0_confluence__pr0_cong_delta:
- \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
-(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall 
-(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) u3 w) t))))))))))))))
-\def
- \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 
-t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda 
-(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 
-x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: 
-T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) 
-(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 
-x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) 
-(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w 
-w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) 
-(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 
-x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta 
-u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) 
-(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
-
-fact pr0_confluence__pr0_upsilon_upsilon:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
-T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: 
-T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to 
-(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 
-x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) 
-(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t2)) t)))))))))))))))))))
-\def
- \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
-(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 
-x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 
-x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T 
-(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) 
-t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) 
-x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat 
-Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) 
-(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 
-H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O 
-x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S 
-O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
-
-fact pr0_confluence__pr0_delta_delta:
- \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
-(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to 
-(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) 
-\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))))))))))))))))
-\def
- \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 
-t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: 
-(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: 
-(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 
-x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: 
-T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 
-x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: 
-T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w 
-x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp 
-u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) 
-(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O 
-x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 
-O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: 
-(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
-u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) 
-(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x 
-H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: 
-T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda 
-(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 
-w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: 
-T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 
-w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 
-w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 
-H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda 
-(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda 
-(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: 
-T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) 
-(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 
-x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in 
-(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x 
-H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda 
-(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x 
-x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: 
-T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
-t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: 
-T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 
-x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta 
-u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) 
-(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) 
-(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 
-w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
-(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 
-(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 
-x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) 
-(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
-
-fact pr0_confluence__pr0_delta_tau:
- \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
-(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T 
-(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 
-t)))))))))
-\def
- \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 
-t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda 
-(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda 
-(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
-(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S 
-O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: 
-T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w 
-(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
-(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda 
-(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) 
-(pr0_gen_lift t4 t3 (S O) O H0)))))))).
-
-theorem pr0_confluence:
- \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 
-t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
-\def
- \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to 
-(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) 
-(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall 
-(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
-v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 
-t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: 
-T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 with [(pr0_refl t3) 
-\Rightarrow (\lambda (H2: (eq T t3 t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T 
-t (\lambda (t4: T).((eq T t4 t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) 
-(\lambda (t5: T).(pr0 t2 t5))))) (\lambda (H4: (eq T t t1)).(eq_ind T t1 
-(\lambda (_: T).(ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 
-t5)))) (let H5 \def (match H1 with [(pr0_refl t4) \Rightarrow (\lambda (H5: 
-(eq T t4 t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T 
-t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 
-t6))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T 
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def 
-(eq_ind T t (\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind 
-T t (\lambda (t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t 
-(\lambda (t5: T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t 
-(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
-\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 
-(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
-\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda 
-(t5: T).(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) 
-(let H13 \def (eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in 
-(ex_intro2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 
-(pr0_refl t1) (pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T 
-t4 t H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq 
-T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T 
-(THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) 
-\to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T 
-(THead k u2 t5) (\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda 
-(H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k 
-u1 t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: 
-T).(pr0 (THead k u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t3 t6)) H2 (THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t 
-(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T 
-(\lambda (t6: T).(pr0 (THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 
-t5) t6)) (THead k u2 t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k 
-u2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 
-H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead 
-(Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T 
-(THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda 
-(t6: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda 
-(H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) 
-H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda 
-(t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) 
-(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t 
-(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t4)) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 
-t5) t6)) (THead (Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl 
-(THead (Bind Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | 
-(pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: 
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) 
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) 
-(\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda 
-(H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 
-t4)) H9) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda 
-(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall 
-(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 
-t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in 
-(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 
-(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta 
-u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 
-t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead 
-(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to 
-((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T 
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
-(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda 
-(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 
-w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead 
-(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: 
-T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) 
-H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: 
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
-T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T 
-(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 
-(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 
-t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) 
-t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: 
-(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 
-t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 
-t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 
-t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to 
-(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) 
-(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) 
-O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: 
-T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
-H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u 
-(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: 
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
-T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in 
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) 
-(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl 
-t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 
-H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda 
-(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T 
-t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
-(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda 
-(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T 
-(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 
-t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda 
-(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def 
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u 
-t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) 
-(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 
-H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) 
-(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | 
-(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 
-t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) 
-(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) 
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) 
-(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda 
-(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda 
-(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
-(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
-(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) 
-(\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 
-(\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda 
-(t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 
-t2)) H11 (THead k u1 t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: 
-T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 
-t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k 
-u1 t3) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
-t9)))))))))) H (THead k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 
-(THead k u2 t4) t6)) (\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 
-t4) (pr0_refl (THead k u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t 
-(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 
-H10 k0) \Rightarrow (\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: 
-(eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T 
-(THead k0 u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
-(H13: (eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: 
-T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
-t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 
-u3)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead k u1 t3) t7)) H4 (THead k0 u0 t5) H11) in (let H17 \def 
-(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef 
-_) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 
-u0 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
-_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-k0 u0 t5) H16) in (\lambda (H20: (eq T u1 u0)).(\lambda (H21: (eq K k 
-k0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead k0 u0 t5) H11) in (eq_ind_r K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: 
-T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) 
-(let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let 
-H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T 
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
-(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) 
-t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 
-x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda 
-(H28: (pr0 u3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
-(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 
-H27 t4 x H25 k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 
-u0 t5) u2 H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k 
-H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 
-t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
-T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
-t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
-(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))) (\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in (let H17 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H19 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in (\lambda (H20: (eq T u1 
-v1)).(\lambda (H21: (eq K k (Flat Appl))).(let H22 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t5)) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
-(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-v2 t6) t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 
-H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead 
-(Bind Abst) u t5) H19) in (let H25 \def (match H24 with [(pr0_refl t7) 
-\Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda (H26: 
-(eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq T t8 
-t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda 
-(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T (THead 
-(Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: 
-T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 
-t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat 
-Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: 
-(pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)) 
-(THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 H15) (pr0_comp v2 x H29 
-t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 
-(THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 H27)) t7 (sym_eq T t7 (THead 
-(Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 H25 t7 t8 H26 k0) \Rightarrow 
-(\lambda (H27: (eq T (THead k0 u0 t7) (THead (Bind Abst) u t5))).(\lambda 
-(H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | 
-(THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) 
-H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) 
-(THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | (TLRef _) 
-\Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) (THead 
-(Bind Abst) u t5) H27) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq T u0 u) 
-\to ((eq T t7 t5) \to ((eq T (THead k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 
-t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: 
-(eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead 
-(Bind Abst) u3 t8) t4) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda 
-(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
-(Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T t7 t5)).(eq_ind T t5 
-(\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to 
-((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda 
-(H34: (eq T (THead (Bind Abst) u3 t8) t4)).(eq_ind T (THead (Bind Abst) u3 
-t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H36: (pr0 t5 
-t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) 
-t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x: 
-T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda 
-(t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: T).(\lambda (H39: (pr0 
-u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) (pr0_beta u3 u2 x0 H39 t8 x 
-H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat 
-Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14))))) (H22 t5 (tlt_trans 
-(THead (Bind Abst) u t5) t5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) 
-(tlt_head_dx (Bind Abst) u t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) 
-u t5))) t8 H36 t6 H15)))) t4 H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 
-u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | 
-(pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda 
-(H28: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
-\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abst) u t5) H27) in (False_ind ((eq T (THead (Bind 
-Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25 v0 v3 H26 u0 
-u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat Appl) v0 
-(THead (Bind b) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H30: (eq T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4)).((let H31 
-\def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind Abst) u t5) H29) in (False_ind ((eq T 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not 
-(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) | 
-(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead 
-(Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq T (THead 
-(Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
-Abst) u t5) H28) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t4) \to ((pr0 
-u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
-Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 H27))) | (pr0_zeta b H25 t7 t8 H26 
-u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind b) u0 (lift (S O) O t7)) 
-(THead (Bind Abst) u t5))).(\lambda (H28: (eq T t8 t4)).((let H29 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
-(\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map 
-(\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) 
-(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let 
-H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 
-| (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) 
-u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
-_) \Rightarrow b | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O 
-t7)) (THead (Bind Abst) u t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T 
-u0 u) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 
-Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
-u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) 
-(\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O 
-t7) t5) \to ((eq T t8 t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to 
-(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
-T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S 
-O) O t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to 
-((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
-t6) t10))))))) (\lambda (H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: 
-T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
-t6) t10)))))) (\lambda (H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 
-t4)).(let H37 \def (match (H35 (refl_equal B Abst)) in False with []) in 
-H37))) t8 (sym_eq T t8 t4 H34))) t5 H33)) u0 (sym_eq T u0 u H32))) b (sym_eq 
-B b Abst H31))) H30)) H29)) H28 H25 H26))) | (pr0_tau t7 t8 H25 u0) 
-\Rightarrow (\lambda (H26: (eq T (THead (Flat Cast) u0 t7) (THead (Bind Abst) 
-u t5))).(\lambda (H27: (eq T t8 t4)).((let H28 \def (eq_ind T (THead (Flat 
-Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in (H25 (refl_equal T 
-(THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) H18)) H17))))) t2 
-H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) 
-\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
-(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to 
-((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda 
-(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0 
-u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) H13) in (let H21 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let 
-H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
-| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H23 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 
-v1)).(\lambda (H25: (eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
-b) u0 t5)) H13) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
-(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 
-(\lambda (t7: T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def 
-(match H28 with [(pr0_refl t7) \Rightarrow (\lambda (H29: (eq T t7 (THead 
-(Bind b) u0 t5))).(\lambda (H30: (eq T t7 t4)).(eq_ind T (THead (Bind b) u0 
-t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t9))))) (\lambda (H31: (eq T (THead (Bind b) u0 
-t5) t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).(ex2 T (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead 
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))) (ex2_ind T 
-(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t5)) t8)) (\lambda (t8: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8))) 
-(\lambda (x: T).(\lambda (H32: (pr0 u2 x)).(\lambda (H33: (pr0 v2 
-x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5 t6 H19 u2 v2 x 
-H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t5)) u2 
-H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5) H29) H30))) | 
-(pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31: (eq T (THead k0 
-u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead k0 u5 t8) 
-t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
-(THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
-\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
-(Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in (eq_ind K 
-(Bind b) (\lambda (k1: K).((eq T u4 u0) \to ((eq T t7 t5) \to ((eq T (THead 
-k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T 
-u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind 
-b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
-b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: 
-(eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead (Bind b) u5 t8) t4) 
-\to ((pr0 u0 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T (THead 
-(Bind b) u5 t8) t4)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: T).((pr0 
-u0 u5) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) 
-u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (pr0 u0 u5)).(\lambda (H40: 
-(pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 
-t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 
-t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda 
-(H42: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: 
-T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 
-x0)).(\lambda (H44: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) 
-(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
-u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda 
-(H45: (pr0 u2 x1)).(\lambda (H46: (pr0 v2 
-x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 H46 t8 t6 x 
-H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind 
-b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) u0 t5) u0 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5) 
-(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H26 
-t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b) 
-u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
-b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4 (sym_eq T 
-u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29 H30))) | 
-(pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 t5))).(\lambda 
-(H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def (eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
-\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T (THead (Bind 
-Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) H32 H29 H30))) | 
-(pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) \Rightarrow (\lambda (H33: 
-(eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead (Bind b) u0 
-t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) 
-O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind 
-b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) 
-(lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to 
-((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 H32))) | (pr0_delta 
-u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq T (THead (Bind 
-Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T (THead (Bind Abbr) 
-u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow 
-t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef 
-_) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind Abbr) u4 
-t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | 
-(THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) 
-H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) \to ((eq T t7 t5) \to 
-((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to 
-((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 
-t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T u4 u0)).(eq_ind T u0 
-(\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to 
-((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda 
-(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
-(Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))))) 
-(\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead 
-(Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 O u5 t8 
-w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
-(t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) t4)).(eq_ind T 
-(THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to 
-((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 
-t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 u5)).(\lambda (H41: 
-(pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 \def (eq_ind_r B b 
-(\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind 
-b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v 
-t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 
-t11)))))))))) H26 Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: 
-B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr H36) in (let H45 \def 
-(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H36) in 
-(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 t8 x)).(\lambda (H47: (pr0 
-t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) 
-t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H48: (pr0 u5 x0)).(\lambda 
-(H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: 
-T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H50: (pr0 
-u2 x1)).(\lambda (H51: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_delta 
-H45 u5 t8 w H42 u2 v2 x1 H50 H51 t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H27 v2 H17))))) 
-(H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5) u0 (THead (Flat Appl) v1 (THead 
-(Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) 
-v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3 H18))))) (H43 t5 (tlt_trans (THead 
-(Bind Abbr) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) 
-(tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
-Abbr) u0 t5))) t8 H41 t6 H19)))))))) t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 
-(sym_eq T u4 u0 H37))) b H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 
-H29 t7 t8 H30 u) \Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S 
-O) O t7)) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T t8 t4)).((let H33 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
-(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
-(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) 
-\Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 
-t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) 
-(THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in ((let 
-H35 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 
-| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
-b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S 
-O) O t7)) (THead (Bind b) u0 t5) H31) in (eq_ind B b (\lambda (b1: B).((eq T 
-u u0) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b1 
-Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
-u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
-(S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T u u0)).(eq_ind T u0 
-(\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq 
-B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T (lift (S O) O 
-t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not 
-(eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T t8 
-t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
-t10)))))) (\lambda (H39: (not (eq B b Abst))).(\lambda (H40: (pr0 t7 
-t4)).(let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(\forall (v: T).((tlt v 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t9))) \to (\forall (t10: T).((pr0 v 
-t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
-t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H26 (lift (S O) O t7) H37) in 
-(let H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T t3 (THead (Bind b) u0 
-t9))) H23 (lift (S O) O t7) H37) in (let H43 \def (eq_ind_r T t5 (\lambda 
-(t9: T).(pr0 t9 t6)) H19 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9: 
-T).(eq T t6 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: 
-T).(\lambda (H44: (eq T t6 (lift (S O) O x))).(\lambda (H45: (pr0 t7 
-x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
-b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)))) (ex2_ind T (\lambda 
-(t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) (\lambda 
-(x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T 
-(\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) 
-(\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 v2 
-x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 H49 
-x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 
-(lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) u0 
-(lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O 
-t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead 
-(Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) (pr0_gen_lift t7 
-t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0 
-H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_tau t7 t8 
-H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead (Bind 
-b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T (THead 
-(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
-| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32)) 
-H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T 
-t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
-(pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
-(Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u3 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
-t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T 
-(THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to 
-((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
-(\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 u3)).(\lambda (H16: 
-(pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 \def (eq_ind_r T t 
-(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind Abbr) u0 t5) H12) 
-in (let H19 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
-\Rightarrow t7])) (THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in (\lambda 
-(H22: (eq T u1 u0)).(\lambda (H23: (eq K k (Bind Abbr))).(let H24 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
-u0 t5) H12) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: 
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) 
-t7)))) (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in 
-(let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in 
-(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u3 w) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 
-x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda 
-(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
-x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x0 H29 H30 t4 x H27 
-H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H15))))) (H24 t5 
-(tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k H23))))) H20)) H19)))))) 
-t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow 
-(\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: 
-(eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: 
-T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
-Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b 
-Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in 
-(let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in ((let H18 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef 
-_) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
-(Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S 
-O) O t5)) H16) in (\lambda (H20: (eq T u1 u)).(\lambda (H21: (eq K k (Bind 
-b))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind b) u (lift (S O) O t5)) H11) in (eq_ind_r K (Bind b) (\lambda 
-(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: 
-T).(pr0 t2 t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 
-u H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift 
-(S O) O t5) H19) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O t7))) 
-(\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
-t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T 
-t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) 
-(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) 
-(\lambda (t8: T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
-(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
-(lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
-T).(\lambda (H27: (pr0 x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: 
-T).(pr0 t2 t7)) x0 (pr0_zeta b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx 
-(Bind b) u t5 (S O) O) x H26 t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O 
-H24)))) k H21))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 
-H10))) | (pr0_tau t5 t6 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat 
-Cast) u t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u 
-t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: 
-(eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda 
-(H13: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-k u1 t3) t7)) H4 (THead (Flat Cast) u t5) H10) in (let H15 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
-Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead k u1 t3) (THead (Flat Cast) u t5) H14) in ((let H17 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) 
-(THead (Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: 
-(eq K k (Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall 
-(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: 
-T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
-T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat 
-Cast) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
-(\lambda (t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 
-t7 t4)) H8 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: 
-T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 
-x)).(\lambda (H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
-(Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) 
-H24)))) (H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) 
-H16)) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal 
-T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 
-t4 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead 
-(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
-(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T 
-(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda 
-(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda 
-(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
-(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
-(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 
-t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind 
-T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda 
-(t6: T).(eq T t6 t2)) H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) 
-in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: 
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
-T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) 
-H9 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def 
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind 
-Abbr) v2 t4)) (pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 
-H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) 
-\Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T 
-(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
-k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
-(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 
-u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 
-t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ 
-_) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k 
-u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 
-t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow (THead (Bind 
-Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead 
-(Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq T v1 
-u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: 
-K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
-T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda (k0: 
-K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def (eq_ind_r 
-T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H19) in (let 
-H24 \def (match H23 with [(pr0_refl t7) \Rightarrow (\lambda (H24: (eq T t7 
-(THead (Bind Abst) u t3))).(\lambda (H25: (eq T t7 t6)).(eq_ind T (THead 
-(Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t6) t9))))) (\lambda (H26: (eq T (THead (Bind Abst) u t3) 
-t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
-(THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u t3) H19) in (let H28 
-\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda 
-(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead 
-(Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let H29 \def (eq_ind T v1 
-(\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t8: T).(pr0 
-v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: (pr0 v2 x)).(\lambda 
-(H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 
-t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u 
-t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 t4 (pr0_refl t4) 
-(Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 (tlt_head_sx (Flat 
-Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 H26)) t7 (sym_eq T 
-t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 H24 t7 t8 H25 k0) 
-\Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead (Bind Abst) u 
-t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) 
-\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead 
-(Bind Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 
-\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) 
-(THead (Bind Abst) u t3) H26) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq 
-T u0 u) \to ((eq T t7 t3) \to ((eq T (THead k1 u3 t8) t6) \to ((pr0 u0 u3) 
-\to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda 
-(H31: (eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t3) \to ((eq T 
-(THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T t7 t3)).(eq_ind T 
-t3 (\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to 
-((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda 
-(H33: (eq T (THead (Bind Abst) u3 t8) t6)).(eq_ind T (THead (Bind Abst) u3 
-t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u2 t9) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H35: (pr0 t3 
-t8)).(let H36 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) 
-u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) in (let H37 \def (eq_ind_r T t 
-(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v 
-t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
-t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u1 (THead 
-(Bind Abst) u t3)) H36) in (let H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 
-t9 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: 
-T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) 
-(\lambda (x: T).(\lambda (H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 
-x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x0: 
-T).(\lambda (H41: (pr0 t8 x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T 
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
-(THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x 
-x0) (pr0_comp v2 x H39 t4 x0 H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 
-H41))))) (H37 t3 (tlt_trans (THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 
-(THead (Bind Abst) u t3)) (tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat 
-Appl) u1 (THead (Bind Abst) u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx 
-(Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 
-(sym_eq T t7 t3 H32))) u0 (sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) 
-H30))) H29)) H28)) H27 H24 H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) 
-\Rightarrow (\lambda (H26: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
-t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 
-t8) t6)).((let H28 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
-t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t3) 
-H26) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to 
-((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) 
-| (pr0_upsilon b H24 v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda 
-(H28: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) 
-u t3))).(\lambda (H29: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v3) t8)) t6)).((let H30 \def (eq_ind T (THead (Flat Appl) v0 (THead 
-(Bind b) u0 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-Abst) u t3) H28) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) 
-(lift (S O) O v3) t8)) t6) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to 
-((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) 
-H30)) H29 H24 H25 H26 H27))) | (pr0_delta u0 u3 H24 t7 t8 H25 w H26) 
-\Rightarrow (\lambda (H27: (eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) 
-u t3))).(\lambda (H28: (eq T (THead (Bind Abbr) u3 w) t6)).((let H29 \def 
-(eq_ind T (THead (Bind Abbr) u0 t7) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
-(match k0 with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
-Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind 
-Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | 
-(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind 
-b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 
-t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) 
-\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
-t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 
-Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
-\Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
-t3) H26) in ((let H30 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow 
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
-(Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3) H26) in (eq_ind B 
-Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t7) t3) \to ((eq 
-T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
-Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda (_: 
-T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B Abst Abst)) 
-\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) (\lambda 
-(H32: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: 
-T).((eq T t8 t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T t8 t6)).(eq_ind T 
-t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u2 t6) t10)))))) (\lambda (H34: (not (eq B Abst 
-Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match (H34 (refl_equal B 
-Abst)) in False with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 H32)) u0 
-(sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 H25))) | 
-(pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat Cast) u0 
-t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let H27 \def 
-(eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
-(match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) 
-\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in (H24 
-(refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) H18)) 
-H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 H10) 
-\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
-t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind 
-Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 t6)).(let H16 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in 
-(let H17 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 
-(Bind Abst) u0 t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
-_ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
-H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
-t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: (eq T u 
-u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t (\lambda (t7: 
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
-T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
-H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H21) 
-in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in 
-(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 
-x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) 
-(\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda 
-(x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 
-t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 
-(tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) 
-(H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead 
-(Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) 
-v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) H18)) H17))))) t2 H13)) 
-t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) 
-\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b) u1 
-t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B b 
-Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (_: (pr0 v0 
-v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H20 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H13) in (let H21 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | 
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 
-t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) 
-\Rightarrow (match t7 with [(TSort _) \Rightarrow Abst | (TLRef _) 
-\Rightarrow Abst | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) 
-in ((let H23 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match 
-t7 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) 
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
-(Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
-t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 
-(THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u u1)).(\lambda (H26: (eq 
-B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst (\lambda (b0: B).(ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))) (let H28 
-\def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abst H26) in 
-(let H29 \def (match (H28 (refl_equal B Abst)) in False with []) in H29)) b 
-H26))))) H23)) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
-(pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
-(Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u2 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to 
-((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 
-u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
-Abst) u t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind 
-T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
-\Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H18) in (False_ind (ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7))) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | 
-(pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) 
-u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind 
-b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b 
-Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
-v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 
-t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) 
-in (let H17 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift 
-(S O) O t5)) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))) t6 (sym_eq T t6 t2 
-H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
-(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
-(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
-T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
-T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
-u t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
-False])])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
-H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) 
-(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 
-u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 
-t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b 
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
-(t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not 
-(eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 
-u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 with [(pr0_refl t5) 
-\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind 
-T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r 
-T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) 
-u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
-(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 
-\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: 
-T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
-(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
-T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) 
-in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 
-(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) 
-H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T 
-(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T 
-(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda 
-(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 
-t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def 
-(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Flat Appl) 
-| (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H22 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) | 
-(TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t7) \Rightarrow 
-t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in 
-(\lambda (H24: (eq T v1 u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K 
-(Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
-k0 u3 t6) t7)))) (let H26 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 
-u0 t5) t)) H15 (Flat Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: 
-T).(pr0 t7 t6)) H19 (THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 
-with [(pr0_refl t7) \Rightarrow (\lambda (H28: (eq T t7 (THead (Bind b) u1 
-t3))).(\lambda (H29: (eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda 
-(t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u3 t6) t9))))) (\lambda (H30: (eq T (THead (Bind b) u1 t3) 
-t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t8) t9)))) (let H31 \def 
-(eq_ind_r T t5 (\lambda (t8: T).(eq T (THead (Flat Appl) u0 t8) t)) H26 
-(THead (Bind b) u1 t3) H23) in (let H32 \def (eq_ind_r T t (\lambda (t8: 
-T).(\forall (v: T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall 
-(t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda 
-(t11: T).(pr0 t10 t11)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 
-t3)) H31) in (let H33 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H10 u0 
-H24) in (ex2_ind T (\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u3 
-t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 (THead 
-(Bind b) u1 t3)) t8))) (\lambda (x: T).(\lambda (H34: (pr0 v2 x)).(\lambda 
-(H35: (pr0 u3 x)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 
-t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 
-H34))))) (H32 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H33 
-u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead (Bind b) u1 t3) H28) H29))) | 
-(pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow (\lambda (H30: (eq T (THead k0 
-u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead k0 u5 t8) 
-t6)).((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
-(THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
-\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
-(Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in (eq_ind K 
-(Bind b) (\lambda (k1: K).((eq T u4 u1) \to ((eq T t7 t3) \to ((eq T (THead 
-k1 u5 t8) t6) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) (\lambda (H35: 
-(eq T u4 u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead 
-(Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda 
-(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) (\lambda 
-(H36: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead (Bind b) u5 
-t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda 
-(t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T 
-(THead (Bind b) u5 t8) t6)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: 
-T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) (\lambda (H38: (pr0 u1 
-u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def (eq_ind_r T t5 (\lambda (t9: 
-T).(eq T (THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 t3) H23) in 
-(let H41 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to 
-(\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T 
-(\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H 
-(THead (Flat Appl) u0 (THead (Bind b) u1 t3)) H40) in (let H42 \def (eq_ind T 
-v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: 
-T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x: 
-T).(\lambda (H43: (pr0 v2 x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda 
-(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) 
-(\lambda (x0: T).(\lambda (H45: (pr0 t8 x0)).(\lambda (H46: (pr0 t4 
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind 
-b) u5 t8)) t9))) (\lambda (x1: T).(\lambda (H47: (pr0 u5 x1)).(\lambda (H48: 
-(pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8))) 
-(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x H44 H43 t8 t4 x0 H45 H46 
-u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead (Bind b) u1 t3) u1 (THead 
-(Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx (Bind b) u1 t3) 
-(tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 u2 H11))))) (H41 
-t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) u0 (THead (Bind b) 
-u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind 
-b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx (Flat Appl) u0 (THead 
-(Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 (sym_eq T t7 t3 H36))) u4 
-(sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) H34))) H33)) H32)) H31 H28 
-H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) \Rightarrow (\lambda (H30: (eq T 
-(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u1 
-t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H32 \def 
-(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T 
-(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) H32)) H31 
-H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) \Rightarrow 
-(\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead 
-(Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead (Flat Appl) 
-(lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat Appl) v0 
-(THead (Bind b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 
-with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead 
-(Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat 
-Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) 
-\to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 H30 H31))) | 
-(pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: (eq T (THead 
-(Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind 
-Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
-\Rightarrow t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in 
-((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow Abbr | 
-(TLRef _) \Rightarrow Abbr | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
-b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 
-(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
-\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) 
-\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 
-u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind 
-Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to 
-(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq 
-T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 
-O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
-Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) 
-t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to 
-((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
-(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
-T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 
-u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 
-\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 
-Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
-Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T 
-(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr 
-H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat 
-Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def 
-(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: 
-T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: 
-T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat 
-Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 
-(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 
-v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead 
-(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
-T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: 
-T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda 
-(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
-T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) 
-(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead 
-(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 
-x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead 
-(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x 
-H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind 
-Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx 
-(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 
-H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat 
-Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) 
-(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) 
-(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 
-H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b 
-H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) 
-\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
-(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
-nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
-nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
-b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H33 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift 
-(S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
-\Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) 
-O t7)) (THead (Bind b) u1 t3) H30) in (eq_ind B b (\lambda (b1: B).((eq T u 
-u1) \to ((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b1 
-Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u3 t6) t9))))))))) (\lambda (H35: (eq T u u1)).(eq_ind T u1 
-(\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq 
-B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T (lift (S O) O t7) 
-t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq 
-B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
-(THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T 
-t6 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda 
-(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda 
-(H38: (not (eq B b Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def 
-(eq_ind_r T t3 (\lambda (t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift 
-(S O) O t7) H36) in (let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T 
-(THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) 
-in (let H42 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
-\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to 
-(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
-t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) 
-H41) in (let H43 \def (eq_ind_r T t3 (\lambda (t9: T).(pr0 t9 t4)) H12 (lift 
-(S O) O t7) H36) in (ex2_ind T (\lambda (t9: T).(eq T t4 (lift (S O) O t9))) 
-(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
-(Flat Appl) u3 t6) t9))) (\lambda (x: T).(\lambda (H44: (eq T t4 (lift (S O) 
-O x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: 
-T).(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t9)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
-t10)))) (let H46 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) 
-in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-t6) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) 
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
-t6) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 
-x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x)))) 
-(pr0_confluence__pr0_cong_upsilon_zeta b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 
-x1 H49 H50))))) (H42 t7 (tlt_trans (THead (Bind b) u1 (lift (S O) O t7)) t7 
-(THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) (lift_tlt_dx 
-(Bind b) u1 t7 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift 
-(S O) O t7)))) x H45 t6 H39))))) (H42 u0 (tlt_head_sx (Flat Appl) u0 (THead 
-(Bind b) u1 (lift (S O) O t7))) v2 H46 u3 H18))) t4 H44)))) (pr0_gen_lift t7 
-t4 (S O) O H43)))))))) t8 (sym_eq T t8 t6 H37))) t3 H36)) u (sym_eq T u u1 
-H35))) b0 (sym_eq B b0 b H34))) H33)) H32)) H31 H28 H29))) | (pr0_tau t7 t8 
-H28 u) \Rightarrow (\lambda (H29: (eq T (THead (Flat Cast) u t7) (THead (Bind 
-b) u1 t3))).(\lambda (H30: (eq T t8 t6)).((let H31 \def (eq_ind T (THead 
-(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
-| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-b) u1 t3) H29) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) \to (ex2 T 
-(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) H31)) H30 
-H28)))]) in (H28 (refl_equal T (THead (Bind b) u1 t3)) (refl_equal T t6))))) 
-k H25)))) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | (pr0_beta u v0 v3 
-H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v0 (THead 
-(Bind Abst) u t5)) t)).(\lambda (H16: (eq T (THead (Bind Abbr) v3 t6) 
-t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) (\lambda (_: 
-T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T 
-(THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda 
-(t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let 
-H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead 
-(Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) 
-in (let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead 
-(Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) 
-\Rightarrow (match t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b 
-| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H23 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow 
-t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
-(THead (Bind Abst) u t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow t3 | (TLRef 
-_) \Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) 
-in (\lambda (_: (eq T u1 u)).(\lambda (H26: (eq B b Abst)).(\lambda (H27: (eq 
-T v1 v0)).(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt 
-v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let 
-H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H27) in (eq_ind_r 
-B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq 
-B b0 Abst))) H9 Abst H26) in (let H31 \def (match (H30 (refl_equal B Abst)) 
-in False with []) in H31)) b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 
-H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) 
-\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 
-t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S 
-O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
-v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) 
-(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
-t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) 
-(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: 
-(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) 
-\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
-in ((let H26 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match 
-t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
-\Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
-b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
-(THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
-(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u1 | (TLRef 
-_) \Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
-in ((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
-t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 
-u0)).(\lambda (H30: (eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) 
-v0 (THead (Bind b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: 
-T).(pr0 t7 v2)) H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) 
-(lift (S O) O v3) t6)) t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not 
-(eq B b1 Abst))) H9 b0 H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: 
-T).(pr0 t7 u2)) H11 u0 H29) in (let H36 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t4)) H12 t5 H28) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) 
-(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda 
-(x: T).(\lambda (H37: (pr0 t4 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T 
-(\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) 
-O v3) t6)) t7))) (\lambda (x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: 
-(pr0 u3 x0)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 
-v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead 
-(Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 
-v2 x1)).(\lambda (H42: (pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 
-H34 v2 v3 x1 H41 H42 u2 u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 
-(tlt_head_sx (Flat Appl) v0 (THead (Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 
-u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead (Bind 
-b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead 
-(Bind b0) u0 t5))) u2 H35 u3 H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 
-t5) t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) 
-u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 
-H23))))) b H30))))))) H27)) H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 
-H16))) | (pr0_delta u0 u3 H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq 
-T (THead (Bind Abbr) u0 t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T 
-(THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda 
-(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 
-u3)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) t7)) H6 (THead (Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 
-H17 H13 H14 H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: 
-(eq T (THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 
-t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T 
-t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 
-(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 
-Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind 
-b0) u (lift (S O) O t5)) H15) in (let H21 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b0) u (lift (S O) O t5)) H20) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 (sym_eq T t6 t2 H17))) t H15 H16 
-H13 H14))) | (pr0_tau t5 t6 H13 u) \Rightarrow (\lambda (H14: (eq T (THead 
-(Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind T (THead (Flat 
-Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
-t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) u t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))) H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 
-(refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | 
-(pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead 
-(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) 
-t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to 
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) 
-(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 
-t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 
-t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda 
-(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 with [(pr0_refl t5) 
-\Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T t5 t2)).(eq_ind 
-T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
-(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H13: (eq T 
-t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t 
-(\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T 
-(THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def 
-(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) 
-H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v 
-t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to 
-(ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H 
-(THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
-(Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) 
-(THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 
-H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) 
-H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T 
-(THead k u0 t5) t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T 
-(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) 
-t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) 
-t2)).(eq_ind T (THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 
-t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 
-t3) t7)) H5 (THead k u0 t5) H13) in (let H19 \def (f_equal T K (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow 
-(Bind Abbr) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) 
-(THead k u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) H18) in ((let H21 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t3) (THead k u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda 
-(H23: (eq K (Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 
-(THead k0 u3 t6) t7)))) (let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T 
-(THead k0 u0 t5) t)) H13 (Bind Abbr) H23) in (let H25 \def (eq_ind_r T t 
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in 
-(let H26 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let 
-H27 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T 
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
-(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u3 t6) t7))) (\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: 
-(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 
-u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
-t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w 
-H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) 
-u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 
-H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta 
-u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
-T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
-T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
-t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
-u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let 
-H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in 
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 
-H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) 
-\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
-t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
-O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b 
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 
-t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
-(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead 
-(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) 
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 
-H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow 
-(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T 
-(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda 
-(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) 
-\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead 
-(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: 
-T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda 
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) 
-(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: 
-(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
-(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | 
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in ((let H22 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
-(THead (Bind Abbr) u0 t5) H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
-u0 t5) H14) in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 
-H23) in (let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) 
-in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 
-t4 x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
-u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda 
-(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
-x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19 x0 H29 H30 x 
-H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H17))))) (H24 
-t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H18))))))) H21)))))) t2 H16)) t 
-H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda 
-(H13: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 
-t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 
-t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
-Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b 
-Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
-T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O 
-t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
-(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let 
-H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
-| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
-Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let H21 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
-_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
-t3) (THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 
-u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: 
-B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b 
-(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr 
-H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def 
-(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind 
-T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T 
-(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) 
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
-T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O 
-x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: 
-T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: 
-T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
-T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 
-x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl 
-(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 
-t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 
-(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) 
-\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: 
-(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
-t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 
-w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
-(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
-(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 
-\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) 
-t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) 
-(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 
-u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) 
-t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O 
-t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) 
-\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) 
-(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b 
-Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
-(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: 
-(pr0 t3 t1)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow (\lambda 
-(H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: 
-T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
-T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
-T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
-H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u 
-(lift (S O) O t3)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
-(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 
-(THead (Bind b) u (lift (S O) O t3)) H4) in (let H14 \def (eq_ind_r T t 
-(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
-\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) 
-H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
-(THead (Bind b) u (lift (S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 
-t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | 
-(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 
-t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) 
-(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) 
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
-(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
-(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Bind b) u (lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def 
-(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Bind b) | 
-(TLRef _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead 
-(Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
-O) O t3)) (THead k u1 t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x 
-(S O))) O t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S 
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
-t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K 
-(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
-t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r 
-K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) 
-in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: 
-T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
-(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S 
-O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda 
-(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let 
-H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
-(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) 
-(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
-(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
-(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: 
-T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift 
-(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx 
-(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) 
-O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta 
-u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 
-(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
-t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: 
-T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
-(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind 
-Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: 
-(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
-(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) 
-u (lift (S O) O t3)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
-[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 
-H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 
-H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead 
-(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead 
-(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) 
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 
-Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not 
-(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) 
-u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 
-(THead (Bind b0) u1 t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t6)) t7))) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta 
-u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) 
-u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T 
-(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) 
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda 
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T 
-(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
-(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T 
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: 
-(pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 
-w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
-(lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
-_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
-t3)) (THead (Bind Abbr) u1 t5) H18) in ((let H20 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead 
-(Bind Abbr) u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O 
-t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) 
-| (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) 
-(THead (Bind Abbr) u1 t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: 
-(eq B b Abbr)).(let H24 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 
-(lift (S O) O t3) H21) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O 
-t7))) (\lambda (t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda 
-(H25: (eq T t6 (lift (S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def 
-(eq_ind_r T t5 (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift 
-(S O) O t3) H21) in (let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 
-\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) 
-H25) in (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
-Abbr H23) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 
-t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
-Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 
-t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 
-(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S 
-O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 
-H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t 
-H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda 
-(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T 
-t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq 
-T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) 
-t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal 
-T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef _) 
-\Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T 
-T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
-O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
-(\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow (lref_map 
-(\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) \Rightarrow t7])) 
-(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) 
-H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b b0)).(let H22 \def 
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) 
-u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3 (\lambda (t7: 
-T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24 \def (eq_ind 
-B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in (ex2_ind T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: 
-(pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind 
-b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 
-H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
-(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
-(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
-(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
-(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) 
-in (let H15 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
-_) \Rightarrow False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 
-T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 
-(sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal 
-T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) 
-\Rightarrow (\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: 
-(eq T t4 t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 
-t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
-T).(pr0 t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: 
-T).((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
-T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 with 
-[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5 
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t 
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6: 
-T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T (THead (Flat 
-Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
-(t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 
-t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def (eq_ind_r T t (\lambda 
-(t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to 
-(\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u t3) H3) in 
-(ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Flat 
-Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 H10)) t (sym_eq 
-T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 t5 t6 H8 k) 
-\Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda (H10: (eq T 
-(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
-k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T 
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
-T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5 
-t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
-t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow (Flat Cast) | (TLRef _) \Rightarrow 
-(Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Cast) u t3) 
-(THead k u1 t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) 
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
-Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u u1)).(\lambda (H19: 
-(eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0: K).(ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H20 
-\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9 (Flat Cast) 
-H19) in (let H21 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
-t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def (eq_ind T t3 
-(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 
-t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x: 
-T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 x)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) 
-t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx (Flat Cast) u1 t5) 
-t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 H10 H7 H8))) | 
-(pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T (THead (Bind 
-Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) 
-(\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
-(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) 
-(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def (eq_ind_r T 
-t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
-Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
-False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind 
-Abst) u0 t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 
-H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow 
-(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
-t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
-(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
-t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
-\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
-(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b 
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
-t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) 
-u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11) in (let H19 
-\def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) v1 (THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda 
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 
-H10))) | (pr0_delta u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T 
-(THead (Bind Abbr) u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) 
-t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) 
-(\lambda (H12: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 
-t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 
-t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: 
-(subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
-(Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def 
-(eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Abbr) u1 t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) 
-t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: 
-(eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 
-t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T 
-t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
-(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def 
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
-(Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
-Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 
-(lift (S O) O t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
-(\lambda (t7: T).(pr0 t2 t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 
-H8))) | (pr0_tau t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
-Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 
-t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 
-t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
-t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) 
-(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
-(THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) 
-u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def (eq_ind_r T t (\lambda 
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in 
-(let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H14) in 
-(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: 
-T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 x)).(ex_intro2 T 
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H18 H19)))) 
-(H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) H13)))) t6 
-(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T 
-t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) 
-(refl_equal T t1))))))))) t0).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
deleted file mode 100644 (file)
index d1c31fc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/props.ma
+++ /dev/null
@@ -1,534 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/fwd.ma".
-
-include "basic_1/subst0/props.ma".
-
-lemma pr0_lift:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
-(d: nat).(pr0 (lift h d t1) (lift h d t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) 
-(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
-nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
-(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: 
-T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) 
-(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k 
-d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
-t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) 
-(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h 
-d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
-d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
-t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
-(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead 
-(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u 
-t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r 
-T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s 
-(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
-(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
-d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
-(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s 
-(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) 
-u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) 
-d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d 
-(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s 
-(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s 
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) 
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda 
-(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: 
-nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) 
-(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) 
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) 
-(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead 
-(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead 
-(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) 
-t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O 
-v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead 
-(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift 
-h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) 
-(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h 
-d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d 
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) 
-(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat 
-Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) 
-t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
-d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d 
-u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) 
-t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d 
-v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) 
-d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S 
-O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) 
-(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) 
-h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
-u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) 
-(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
-t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
-(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
-(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift 
-h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) 
-u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) 
-d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind 
-Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
-d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in 
-(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) 
-(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d 
-(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (le_antisym d 
-d (le_n d) (le_n d)) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind 
-Abbr) u2 w)) (lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) 
-u1 t3)) (lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (pr0 t3 t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
-nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: 
-nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s 
-(Bind b) d) (lift (S O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) 
-(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d 
-u) (lift h n (lift (S O) O t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O 
-(lift h d t3)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d 
-t4))) (pr0_zeta b H0 (lift h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift 
-h (plus (S O) d) (lift (S O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) 
-(refl_equal nat (S d))) (lift h d (THead (Bind b) u (lift (S O) O t3))) 
-(lift_head (Bind b) u (lift (S O) O t3) h d))))))))))) (\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H1: ((\forall (h: 
-nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: 
-T).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Flat Cast) (lift h 
-d u) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(pr0 t (lift h d t4))) 
-(pr0_tau (lift h (s (Flat Cast) d) t3) (lift h d t4) (H1 h d) (lift h d u)) 
-(lift h d (THead (Flat Cast) u t3)) (lift_head (Flat Cast) u t3 h d))))))))) 
-t1 t2 H))).
-
-lemma pr0_gen_abbr:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
-(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda 
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S 
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: 
-T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T 
-T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead 
-(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 
-t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind 
-Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t 
-H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
-(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
-t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
-Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H5) in ((let H7 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
-(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
-(THead (Bind Abbr) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H5) 
-in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Abbr))).(eq_ind_r 
-K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
-T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
-t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let H11 \def (eq_ind T 
-t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 t1 H8) in (let 
-H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def 
-(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 
-u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind 
-Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
-t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind 
-Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) 
-H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O 
-v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def 
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
-O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: 
-(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
-T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
-(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
-u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
-O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
-T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
-t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
-t1) H8) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind 
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) 
-(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: 
-(((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
-t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
-t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
-T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) 
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u0 t0) 
-(THead (Bind Abbr) u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) 
-u1 t1) H6) in (\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda 
-(t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
-t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T 
-t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 
-(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 u1 H9) in (let 
-H13 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or_introl 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
-(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
-(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S 
-O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda 
-(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or 
-(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
-u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind Abbr) u2 w)) H13 (or_intror 
-(pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 
-y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
-u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: 
-(not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
-t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: 
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
-T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: 
-T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
-Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
-\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in 
-((let H6 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
-(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in ((let 
-H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
-(lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow 
-(lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
-\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
-t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let 
-H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 
-(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) 
-(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: 
-T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) 
-(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 
-H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
-(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
-(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
-H4)))))))) y x H0))) H)))).
-
-lemma pr0_gen_void:
- \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
-t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
-\def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
-(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: 
-T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
-u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) 
-O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda 
-(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda 
-(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) 
-u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
-(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 
-t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda 
-(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal 
-T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind 
-Void) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
-\Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
-(THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: 
-(eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda (k0: K).(or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Void) 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) 
-(let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Void) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))))) H4 t1 
-H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in 
-(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Void) u1 
-t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O u2))))) H2 u1 
-H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind 
-Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) u2 
-t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
-(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
-(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead 
-(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B 
-b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda 
-(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
-u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
-v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T 
-(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
-(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) 
-H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
-u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
-(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T 
-(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
-Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind 
-Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) 
-u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
-Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda 
-(H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T 
-(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1))).(let H5 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
-_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
-t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
-_ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
-Void) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
-(TLRef _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
-(THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
-(Bind Void) u1 t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b 
-Void)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 
-Void H9) in (let H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead 
-(Bind Void) u1 t)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O 
-t2))))) H3 (lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: 
-T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) 
-(lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) 
-(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: 
-(((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda (H3: (eq T 
-(THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(let H4 \def (eq_ind T 
-(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind Void) u1 t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x H0))) H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
deleted file mode 100644 (file)
index 95fb675..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst0.ma
+++ /dev/null
@@ -1,1647 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/props.ma".
-
-include "basic_1/subst0/subst0.ma".
-
-lemma pr0_subst0_back:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T 
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) 
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 
-v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
-T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
-((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
-(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
-u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
-(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 
-H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
-T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 
-(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
-(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t 
-t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t 
-(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
-x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) 
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
-T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
-T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
-(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t 
-(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
-x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
-(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
-t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 
-t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
-H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-
-lemma pr0_subst0_fwd:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
-T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T 
-(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) 
-(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v 
-u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
-T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) 
-(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
-((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
-(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
-(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
-u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: 
-T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T 
-(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
-(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 
-x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda 
-(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 
-t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
-T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 
-(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
-x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
-(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) 
-(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
-T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
-T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
-(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T 
-(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T 
-(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead 
-k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
-x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
-(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
-t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda 
-(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) 
-t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 
-t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
-
-theorem pr0_subst0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
-(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
-v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t2 w2))))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 
-w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
-nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 
-v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd 
-v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: 
-(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 
-w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
-nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
-t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
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-(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
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-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead 
-(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
-v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 
-H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
-i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda 
-(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 
-(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 
-T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind 
-Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 
-i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) 
-w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
-t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) 
-x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: 
-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T 
-x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
-x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
-(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: 
-(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T 
-(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s 
-(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
-(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or 
-(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 
-(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) 
-(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 
-v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
-H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) 
-H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 
-v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
-(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
-Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 
-t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 
-x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) 
-x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) 
-(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: 
-T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T 
-(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
-T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 
-(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
-u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: 
-(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: 
-T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in 
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: 
-T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 
-x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 
-H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
-x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) 
-x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) 
-(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind 
-Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: 
-(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
-(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
-Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead 
-(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
-v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat 
-Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda 
-(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) 
-(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
-(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
-t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
-Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s 
-(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: 
-(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
-Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) 
-(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda 
-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
-t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) 
-(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 
-v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
-H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
-t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) 
-x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: 
-(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T 
-x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) 
-x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
-w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
-(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
-(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: 
-(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
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-(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
-(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 
-x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
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-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
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-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
-T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
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-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
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-(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
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-(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) 
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-(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
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-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x 
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-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
-(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
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-(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
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-(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x 
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-H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 
-i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
-Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda 
-(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 
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-(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall 
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-(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 
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-(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 
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-(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat 
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-x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: 
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-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead 
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-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
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-(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind 
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-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
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-(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
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-(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
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-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: 
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-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) 
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-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift 
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-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T 
-(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
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-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq 
-T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 
-(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead 
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-T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 
-(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T 
-(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s 
-(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind 
-b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq 
-T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 
-x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 
-t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 
-(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 
-w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 
-u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
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-(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) 
-x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda 
-(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) 
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-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
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-(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 
-H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S 
-O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 
-H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) 
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
-t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda 
-(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x 
-(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) 
-i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead 
-(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead 
-(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
-x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) 
-(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
-(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
-x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) 
-(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind 
-b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
-x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 
-x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) 
-(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 
-(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s 
-(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind 
-b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 
-u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) 
-v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 
-x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: 
-(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x 
-(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 
-x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
-(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
-i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) 
-x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 
-H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 
-w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead 
-(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
-(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda 
-(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 
-x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
-b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst 
-v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) 
-(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
-T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 
-x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
-(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s 
-(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat 
-Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 
-H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S 
-O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat 
-Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
-i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
-(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
-x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 
-x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 
-(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) 
-(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 
-(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S 
-O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) 
-O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 
-(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) 
-(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda 
-(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) 
-u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
-(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) 
-u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 
-x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat 
-Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq 
-T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 
-u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda 
-(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T 
-(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 
-(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead 
-(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or 
-(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda 
-(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) 
-i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
-Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or 
-(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: 
-(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 
-t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
-t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda 
-(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
-T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
-(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 
-H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
-(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
-O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 
-(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
-(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
-t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda 
-(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
-x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
-v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst 
-v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
-(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
-(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both 
-v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
-Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O 
-v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat 
-Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) 
-i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T 
-x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) 
-u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) 
-(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: 
-T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s 
-(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: 
-T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in 
-(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or 
-(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 
-x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind 
-b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) 
-u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
-w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x 
-x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
-(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i 
-v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 
-(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
-u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 
-H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) 
-O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) 
-v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
-x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s 
-(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 
-x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) 
-(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
-(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead 
-(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) 
-v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
-Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind 
-b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda 
-(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind 
-T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) 
-x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) 
-(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
-O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
-(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s 
-(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
-x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) 
-(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
-i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
-Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
-(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
-v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 
-T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
-v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
-(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 
-x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
-T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
-w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
-b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd 
-(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift 
-(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind 
-b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) 
-u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: 
-T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s 
-(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
-x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
-(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
-O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda 
-(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
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-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
-(let H16 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
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-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x t3) w2)) (\lambda (w2: T).(subst0 
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-(subst0_subst0_back t4 w u2 O H4 x0 v2 i H12))))) H10)) (H1 v1 x i H9 v2 H6)) 
-w1 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind 
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-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H8: 
-(eq T w1 (THead (Bind Abbr) u1 x))).(\lambda (H9: (subst0 (s (Bind Abbr) i) 
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-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
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-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
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-(pr0 (THead (Bind Abbr) u1 x) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2))) (pr0_delta u1 u2 H0 x t4 H10 w H4))) (\lambda (H10: 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
-i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
-x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
-(\lambda (x0: T).(\lambda (H11: (pr0 x x0)).(\lambda (H12: (subst0 (s (Bind 
-Abbr) i) v2 t4 x0)).(ex2_ind T (\lambda (t: T).(subst0 O u2 x0 t)) (\lambda 
-(t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead (Bind Abbr) u1 x) 
-(THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 
-x) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) 
-(\lambda (x1: T).(\lambda (H13: (subst0 O u2 x0 x1)).(\lambda (H14: (subst0 
-(s (Bind Abbr) i) v2 w x1)).(or_intror (pr0 (THead (Bind Abbr) u1 x) (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) u1 x) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) u2 x1) (pr0_delta u1 u2 
-H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) 
-(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq 
-nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T 
-w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 
-u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 
-t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
-(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or 
-(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 
-x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind 
-Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or 
-(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
-(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) 
-(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
-i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) 
-(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 
-t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: 
-(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 
-w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
-T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 
-w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
-T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
-(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: 
-T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T 
-(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
-w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 
-O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def 
-(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
-(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
-x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x 
-H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) 
-(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 
-H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: 
-T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 
-w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: 
-(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: 
-T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind 
-Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 
-w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 
-x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x 
-x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 
-(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
-(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) 
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) 
-(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd 
-(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind 
-Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: 
-(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
-u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
-x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O 
-x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
-Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 
-x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal 
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S 
-i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) 
-(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead 
-(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) 
-w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda 
-(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: 
-(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
-Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
-i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 
-H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 
-(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) 
-(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S 
-i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i 
-H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) 
-(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: 
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: 
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
-(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift 
-(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
-(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) 
-u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or 
-(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) 
-u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
-(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
-(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u 
-x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda 
-(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda 
-(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) 
-i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
-(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift 
-(S O) O t3) x)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: 
-(lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x (S O) O (s (Bind b) 
-i) (le_O_n (s (Bind b) i)) H8 H7 (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 
-w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) (\lambda (Hle: (le (S O) (s 
-(Bind b) i))).(let H_x \def (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S O) 
-O H7 Hle) in (let H8 \def H_x in (ex2_ind T (\lambda (t5: T).(eq T x (lift (S 
-O) O t5))) (\lambda (t5: T).(subst0 (minus i O) v1 t3 t5)) (or (pr0 w1 t4) 
-(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) O x0))).(\lambda (H10: 
-(subst0 (minus i O) v1 t3 x0)).(let H11 \def (eq_ind T x (\lambda (t: T).(eq 
-T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H9) in (eq_ind_r T (THead 
-(Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda 
-(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (let H12 \def 
-(eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3 x0)) H10 i 
-(minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u (lift (S O) 
-O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H13: (pr0 x0 
-t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H13 u))) (\lambda (H13: (ex2 T 
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
-t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 
-x1)).(\lambda (H15: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
-(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
-(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H14 u) H15))))) H13)) (H2 v1 
-x0 i H12 v2 H4))) w1 H11))))) H8)))))))) H5)) (\lambda (H5: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) 
-v1 (lift (S O) O t3) x1)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 
-T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (H9: (lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x1 (S 
-O) O (s (Bind b) i) (le_O_n (s (Bind b) i)) H9 H8 (or (pr0 w1 t4) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) 
-(\lambda (Hle: (le (S O) (s (Bind b) i))).(let H_x \def (subst0_gen_lift_ge 
-v1 t3 x1 (s (Bind b) i) (S O) O H8 Hle) in (let H9 \def H_x in (ex2_ind T 
-(\lambda (t5: T).(eq T x1 (lift (S O) O t5))) (\lambda (t5: T).(subst0 (minus 
-i O) v1 t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
-(w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H10: (eq T x1 (lift 
-(S O) O x))).(\lambda (H11: (subst0 (minus i O) v1 t3 x)).(let H12 \def 
-(eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift (S O) 
-O x) H10) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda (t: 
-T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))))) (let H13 \def (eq_ind_r nat (minus i O) (\lambda 
-(n: nat).(subst0 n v1 t3 x)) H11 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 
-T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or 
-(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2)))) (\lambda (H14: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S 
-O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O 
-x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H14 x0))) 
-(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
-i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x 
-x2)).(\lambda (H16: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 
-(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift 
-(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda 
-(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H15 x0) H16))))) H14)) (H2 v1 
-x i H13 v2 H4))) w1 H12))))) H9)))))))))) H5)) (subst0_gen_head (Bind b) v1 u 
-(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
-T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
-\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
-(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) 
-w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda 
-(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
-u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
-(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda 
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: 
-(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: 
-T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat 
-Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T 
-(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda 
-(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: 
-T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 
-T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: 
-(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T 
-w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 
-t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat 
-Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T 
-(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: 
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) 
-(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) 
-i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
-(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
-(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
-v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 
-x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T 
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 
-w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead 
-(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 
-x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: 
-T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat 
-Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
-T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) 
-i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
-(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 
-x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda 
-(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 
-(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) 
-x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) 
-(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 
-(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
-(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat 
-Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) 
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 
-x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead 
-(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) 
-w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: 
-T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) 
-x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) 
-w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 
-t2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
deleted file mode 100644 (file)
index 05a9317..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
+++ /dev/null
@@ -1,93 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/subst0.ma".
-
-include "basic_1/subst1/fwd.ma".
-
-lemma pr0_delta1:
- \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
-(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead 
-(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: 
-(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind 
-Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind 
-Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H 
-t1 t2 H0 t0 H2))) w H1)))))))).
-
-lemma pr0_subst1_back:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
-T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
-(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda 
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda 
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x 
-H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-
-lemma pr0_subst1_fwd:
- \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
-i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
-\def
- \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
-T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
-(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 
-T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 
-(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
-i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda 
-(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda 
-(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: 
-T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x 
-H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
-
-theorem pr0_subst1:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
-(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
-v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 
-w2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: 
-T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 
-w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to 
-(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) 
-(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 
-t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) 
-(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda 
-(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 
-T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 
-(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
-(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 
-w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 
-w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: 
-(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: 
-T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i 
-v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma
deleted file mode 100644 (file)
index b466ad9..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/defs.ma".
-
-inductive pr1: T \to (T \to Prop) \def
-| pr1_refl: \forall (t: T).(pr1 t t)
-| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
-T).((pr1 t2 t3) \to (pr1 t1 t3))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma
deleted file mode 100644 (file)
index 8c9640c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/fwd.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr1/defs.ma".
-
-implied rec lemma pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
-t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
-T).((pr1 t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t 
-t0) on p: P t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | 
-(pr1_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr1_ind P f f0) t2 
-t3 p1))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma
deleted file mode 100644 (file)
index 60dbdbf..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/pr1.ma
+++ /dev/null
@@ -1,64 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr1/props.ma".
-
-include "basic_1/pr0/pr0.ma".
-
-lemma pr1_strip:
- \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 
-t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda 
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: 
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda 
-(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) 
-(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda 
-(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T 
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: 
-T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda 
-(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
-t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let 
-H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: 
-T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) 
-(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x 
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
-H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 
-t2 H0)))))))))) t0 t1 H))).
-
-theorem pr1_confluence:
- \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 
-t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda 
-(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: 
-T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda 
-(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) 
-(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: 
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: 
-(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda 
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda 
-(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def 
-H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) 
-(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: 
-T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x 
-H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda 
-(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
-t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x 
-x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
-H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma
deleted file mode 100644 (file)
index 97a5e45..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr1/props.ma
+++ /dev/null
@@ -1,108 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr1/fwd.ma".
-
-include "basic_1/pr0/subst1.ma".
-
-include "basic_1/subst1/props.ma".
-
-include "basic_1/T/props.ma".
-
-lemma pr1_pr0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H 
-t2 (pr1_refl t2)))).
-
-theorem pr1_t:
- \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 
-t3) \to (pr1 t1 t3)))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) 
-(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda 
-(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 
-t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 
-t5 H3)))))))))) t1 t2 H))).
-
-lemma pr1_head_1:
- \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall 
-(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: 
-T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k 
-t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda 
-(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda 
-(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing 
-(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k 
-t3 t) H2))))))) u1 u2 H))))).
-
-lemma pr1_head_2:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall 
-(k: K).(pr1 (THead k u t1) (THead k u t2))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: 
-T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u 
-t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: 
-(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing 
-(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k 
-u t4) H2))))))) t1 t2 H))))).
-
-theorem pr1_comp:
- \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: 
-T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: 
-T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall 
-(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda 
-(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 
-t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 
-t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: 
-T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead 
-k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t 
-u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k 
-t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 
-H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 
-t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k 
-t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp 
-t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v 
-w H))).
-
-lemma pr1_eta:
- \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in 
-(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) 
-(TLRef O) (lift (S O) O t))) t)))))
-\def
- \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in 
-(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift 
-(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v 
-(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H 
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) 
-(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) 
-(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) 
-(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef 
-O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) 
-u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind 
-Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) 
-(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl 
-(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u 
-(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr 
-not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O 
-(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
deleted file mode 100644 (file)
index b518e43..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
+++ /dev/null
@@ -1,151 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/props.ma".
-
-include "basic_1/clen/getl.ma".
-
-lemma pr2_gen_ctail:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
-(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: 
-T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 
-(clen c) u t t2)))))))))
-\def
- \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) 
-(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T 
-(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda 
-(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
-(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind 
-Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 
-t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) 
-(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) 
-(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
-(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 
-t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda 
-(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let 
-H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind 
-(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c 
-(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
-(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
-(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
-T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d 
-(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) 
-u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: 
-C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
-T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
-T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k 
-u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c 
-t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 
-t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 
-H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen 
-c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
-(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq 
-nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: 
-nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 
-T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
-(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: 
-(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T 
-u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda 
-(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 
-(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind 
-Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind 
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
-t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
-Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
-t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda 
-(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 
-(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 
-t2 H0))) H)))))).
-
-lemma pr2_gen_cbind:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
-(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) 
-(THead (Bind b) v t2)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c 
-(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead 
-(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y 
-t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
-(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v 
-t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead 
-(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 
-(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def 
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
-c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) 
-v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d 
-(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S 
-j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead 
-(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C 
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C 
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) 
-(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead 
-d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow 
-c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in ((let H11 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in 
-((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v) H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C 
-d c)).(let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O 
-H8) in (let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v 
-H12) in (eq_ind B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead 
-(Bind b0) v t))) (pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) 
-(pr0_delta v v (pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) 
-(\lambda (H7: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: 
-nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq 
-nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c 
-(THead (Bind b) v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: 
-(eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 
-\def (f_equal nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def 
-(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in 
-(pr2_head_2 c v t3 t (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) 
-(getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind 
-Abbr) u) x H9) t3 t4 H2 t H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) 
-H)))))).
-
-lemma pr2_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
-(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c 
-(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) 
-(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 
-c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 
-H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
-(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def 
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
-c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) 
-v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) 
-H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma
deleted file mode 100644 (file)
index 1bcb0b1..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/defs.ma
+++ /dev/null
@@ -1,28 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/defs.ma".
-
-include "basic_1/getl/defs.ma".
-
-inductive pr2: C \to (T \to (T \to Prop)) \def
-| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
-(pr2 c t1 t2))))
-| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: 
-T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 
-t)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
deleted file mode 100644 (file)
index 1c19f4a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
+++ /dev/null
@@ -1,2801 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/defs.ma".
-
-include "basic_1/pr0/fwd.ma".
-
-include "basic_1/getl/clear.ma".
-
-include "basic_1/getl/drop.ma".
-
-implied lemma pr2_ind:
- \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c: 
-C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
-(Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
-(\forall (t: T).((subst0 i u t2 t) \to (P c t1 t)))))))))))) \to (\forall (c: 
-C).(\forall (t: T).(\forall (t0: T).((pr2 c t t0) \to (P c t t0)))))))
-\def
- \lambda (P: ((C \to (T \to (T \to Prop))))).(\lambda (f: ((\forall (c: 
-C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 
-t2))))))).(\lambda (f0: ((\forall (c: C).(\forall (d: C).(\forall (u: 
-T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: 
-T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to 
-(P c t1 t))))))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
-T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2) 
-\Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8) 
-\Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))).
-
-lemma pr2_gen_sort:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to 
-(eq T x (TSort n)))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort 
-n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
-T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda 
-(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 
-t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 
-t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda 
-(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: 
-T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) 
-(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl 
-i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda 
-(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 
-t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) 
-(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) 
-(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) 
-t1 H4))))))))))))) c y x H0))) H)))).
-
-lemma pr2_gen_lref:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to 
-(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c 
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S 
-n) O u)))))))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef 
-n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
-T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead 
-d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O 
-u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) 
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) 
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 
-t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda 
-(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: 
-T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 
-(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
-n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) 
-(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) 
-(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
-T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
-T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref 
-t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda 
-(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef 
-n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) 
-in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda 
-(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
-C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 
-(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in 
-(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) 
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) 
-u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) 
-(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O 
-u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) 
-(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) 
-u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let 
-H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) 
-u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T 
-(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) 
-(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) 
-(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind 
-Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S 
-n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) 
-(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
-
-lemma pr2_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
-(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
-T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
-(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in 
-(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 
-x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) 
-(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 
-(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: 
-B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) 
-(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda 
-(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind 
-Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 
-(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead 
-(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 
-x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead 
-(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead 
-(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda 
-(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
-Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind 
-Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda 
-(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 
-x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 
-x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d 
-u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 
-(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: 
-T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
-Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) 
-x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 
-x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead 
-(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 
-x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
-(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 
-H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind 
-Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
-x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 
-t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t 
-(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: 
-(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) 
-(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 
-(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 
-H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
-(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 
-H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) 
-(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-lemma pr2_gen_cast:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
-(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c 
-t1 x))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 
-t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) 
-u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
-(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) 
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 
-t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind 
-T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) 
-x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T 
-(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead 
-(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 
-H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 
-H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: 
-(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 
-\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) 
-H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) 
-x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def 
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) 
-H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead 
-(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: 
-T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 
-c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
-x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 
-(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) 
-(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda 
-(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat 
-Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda 
-(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 
-(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) 
-(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
-x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat 
-Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 
-H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) 
-(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
-t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
-(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 
-H5)))))))))))))) c y x H0))) H))))).
-
-lemma pr2_gen_csort:
- \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) 
-\to (pr0 t1 t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort 
-n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: 
-C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind 
-(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 
-t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
-t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c 
-(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d 
-(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) 
-u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
-
-lemma pr2_gen_appl:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
-(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T 
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead 
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 
-t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
-u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 
-\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) 
-H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
-(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) 
-x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat 
-Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 
-H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 
-(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 
-x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind 
-Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: 
-B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 
-H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
-(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) 
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat 
-Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift 
-(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
-t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 
-Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T 
-t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda 
-(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 
-x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) 
-x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
-Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
-Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
-x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 
-u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 
-H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
-t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T 
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: 
-T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t 
-(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T 
-(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O 
-u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i 
-H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: 
-(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: 
-T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t 
-(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
-(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat 
-Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat 
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) 
-(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat 
-Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) 
-u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 
-(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) 
-x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t 
-H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
-x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: 
-(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 
-x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 
-(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) 
-x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 
-H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i 
-H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind 
-Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda 
-(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 
-(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in 
-(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 
-u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda 
-(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead 
-(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 
-i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda 
-(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 
-x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
-H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 
-H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t 
-(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) 
-x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T 
-(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
-(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d 
-(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
-O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t 
-(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: 
-(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) 
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) 
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) 
-(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
-H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
-(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) 
-(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) 
-(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda 
-(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) 
-t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda 
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T 
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
-(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind 
-x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) 
-(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 
-x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
-x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
-(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t 
-(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead 
-(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) 
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead 
-(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: 
-T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0) 
-u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u 
-x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
-(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) 
-x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 
-(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
-O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead 
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
-(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
-H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda 
-(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: 
-T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda 
-(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
-(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0) 
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind 
-x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) 
-x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat 
-Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
-u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) 
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
-x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T 
-(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
-(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
-(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
-x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift 
-(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H20: (lt (s (Bind x0) i) (S 
-O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
-x0) i)) H20 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
-(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
-(S O) O H19 Hle) in (let H20 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H21: (eq T x7 (lift (S O) O 
-x8))).(\lambda (H22: (subst0 (minus i O) u x3 x8)).(eq_ind_r T (lift (S O) O 
-x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H23 \def (eq_ind_r 
-nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x8)) H22 i (minus_n_O i)) in 
-(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
-(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
-Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
-(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))) 
-(pr2_delta c0 d u i H1 u1 x3 H10 x8 H23) (pr2_free c0 x1 x4 H11) (pr2_free 
-(CHead c0 (Bind x0) x4) x2 x5 H12)))) x7 H21)))) H20))))) x6 H18)))) H17)) 
-(\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S 
-O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O 
-x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 
-x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
-x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O 
-x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
-Appl) (lift (S O) O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
-x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 
-u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S 
-i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) 
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda 
-(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u 
-(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda 
-(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: 
-T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 
-(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) 
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
-(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
-O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
-x0) i)) H21 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
-(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
-(S O) O H19 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x7 (lift (S O) O 
-x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
-x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
-nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
-(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
-(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
-Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
-(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8))) 
-(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_free c0 x1 x4 H11) (pr2_delta 
-(CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) 
-c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 
-H20)))) x7 H22)))) H21))))) x6 H18)))))) H17)) (subst0_gen_head (Flat Appl) u 
-(lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) (\lambda (H14: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind x0) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) 
-O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat 
-Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
-x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t 
-(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: 
-(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat 
-Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
-u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) 
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
-x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T 
-(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
-T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
-(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
-(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T 
-x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift 
-(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
-O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
-x0) i)) H21 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
-(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
-(S O) O H20 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x8 (lift (S O) O 
-x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
-x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
-nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
-(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
-(S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
-(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat 
-Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
-(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))) 
-(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
-H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12)))) x8 H22)))) H21))))) x7 
-H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat 
-Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
-(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat 
-Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
-(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
-(H19: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x8))).(\lambda (H20: 
-(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat 
-Appl) (lift (S O) O x3) x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
-O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
-x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 
-u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 
-(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 
-(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 
-H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s 
-(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) 
-i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
-x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
-(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: 
-(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s 
-(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) 
-(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
-(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (H22: (lt (s (Bind x0) i) (S 
-O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
-x0) i)) H22 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
-(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
-(S O) O H20 Hle) in (let H22 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
-x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x10: T).(\lambda (H23: (eq T x8 (lift (S O) O 
-x10))).(\lambda (H24: (subst0 (minus i O) u x3 x10)).(eq_ind_r T (lift (S O) 
-O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq 
-T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H25 \def (eq_ind_r 
-nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x10)) H24 i (minus_n_O i)) 
-in (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat Appl) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
-O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) 
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T 
-(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) 
-(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H25) (pr2_delta 
-c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) 
-(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) 
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21)))) x8 H23)))) H22))))) x7 
-H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s 
-(Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 (THead 
-(Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
-(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-lemma pr2_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
-(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
-(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 
-t2))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
-(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
-\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e 
-t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: 
-C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) 
-(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h 
-d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T 
-t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda 
-(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T 
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 
-x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T 
-t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda 
-(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) 
-x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) 
-(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind 
-Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
-t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 
-(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def 
-(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T 
-(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 
-T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 
-x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h 
-d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) 
-(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def 
-(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i 
-(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d 
-(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) 
-(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
-C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: 
-C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl 
-i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 
-x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus 
-i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind 
-T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) 
-(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h 
-d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq 
-T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 
-x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: 
-nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T 
-(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 
-H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 
-t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h 
-(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 
-T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 
-H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 
-t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t 
-(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T 
-(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
-(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: 
-(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d 
-t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) 
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 
-H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 
-t2 h d H6)))))))))))))))) c y x H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
deleted file mode 100644 (file)
index 3151fc5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/pr2.ma
+++ /dev/null
@@ -1,236 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/defs.ma".
-
-include "basic_1/pr0/pr0.ma".
-
-include "basic_1/getl/fwd.ma".
-
-fact pr2_confluence__pr2_free_free:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 
-t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 
-t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) 
-(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 
-x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
-(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) 
-(pr0_confluence t0 t2 H0 t1 H))))))).
-
-fact pr2_confluence__pr2_free_delta:
- \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall 
-(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to 
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) 
-\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
-t))))))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 
-t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 
-t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 
-t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
-(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: 
-(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
-(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 
-c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 
-x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: 
-T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
-(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: 
-(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 
-H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) 
-(pr0_confluence t0 t4 H1 t1 H))))))))))))).
-
-fact pr2_confluence__pr2_delta_delta:
- \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall 
-(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: 
-T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d 
-(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c 
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to 
-(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
-t))))))))))))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: 
-T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i 
-c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 
-i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda 
-(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: 
-T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 
-x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 
-t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 
-x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
-(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
-(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda 
-(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
-w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 
-u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
-(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x 
-x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) 
-x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) 
-(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T 
-(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind 
-T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T 
-(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: 
-T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 
-t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
-w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
-(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
-(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H 
-t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) 
-(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 
-w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 
-t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 
-x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: 
-T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i 
-i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 
-i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
-t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: 
-(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d 
-u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 
-(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def 
-(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 
-\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) 
-H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: 
-C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
-Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) 
-(getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in 
-((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead 
-d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind 
-Abbr) u0) H15)) in (\lambda (H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 
-(\lambda (t: T).(subst0 i t x x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 
-(\lambda (t: T).(getl i c (CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 
-\def (eq_ind_r C d0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 
-d H19) in (or4_ind (eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) 
-(\lambda (t: T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) 
-(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda 
-(H23: (eq T x1 x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) 
-H11 x0 H23) in (ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda 
-(H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u 
-x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: 
-T).(subst0 i u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t))) (\lambda (x2: T).(\lambda (H24: (subst0 i u x1 
-x2)).(\lambda (H25: (subst0 i u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c 
-t1 t)) (\lambda (t: T).(pr2 c t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 
-H25) (pr2_delta c d u i H22 t2 x1 H11 x2 H24))))) H23)) (\lambda (H23: 
-(subst0 i u x1 x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 
-H23))) (\lambda (H23: (subst0 i u x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c 
-t1 t)) (\lambda (t: T).(pr2 c t2 t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 
-H23) (pr2_free c t2 x1 H11))) (subst0_confluence_eq x x1 u i H20 x0 H9))))))) 
-H17)))))))))) H10)) (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) 
-H7)) (pr0_subst0 t3 x H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 
-H3 t3 H0))))))))))))))))))).
-
-theorem pr2_confluence:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
-(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
-T).(pr2 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
-t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H with 
-[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: 
-(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T 
-t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c 
-t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind 
-T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: 
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 
-t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: 
-T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 
-t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda 
-(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 
-t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 
-t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
-t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 
-t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
-T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: 
-T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
-T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 
-t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 
-H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | 
-(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 
-c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c 
-(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d 
-(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda 
-(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: 
-(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c 
-(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T 
-(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) 
-(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d 
-(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda 
-(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: 
-(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda 
-(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i 
-H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 
-(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) 
-(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T 
-t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 
-H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 
-t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) 
-\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) 
-\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda 
-(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 
-(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
-((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 
-t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind 
-T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) 
-\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda 
-(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 
-\def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C 
-c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c 
-(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
-(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda 
-(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 
-t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
-t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 
-t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
-t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) 
-(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 
-(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 
-H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow 
-(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T 
-t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to 
-((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 
-t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
-t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T 
-t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to 
-((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
-(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 
-(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to 
-((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
-(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 
-t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 
-H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 
-(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) 
-(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T 
-t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C 
-c) (refl_equal T t0) (refl_equal T t1)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma
deleted file mode 100644 (file)
index 5677ca5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/props.ma
+++ /dev/null
@@ -1,998 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/fwd.ma".
-
-include "basic_1/pr0/subst0.ma".
-
-lemma pr2_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) 
-(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 
-t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u 
-(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 
-t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i 
-H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 
-t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 
-u)))))))))))) c t1 t2 H)))))).
-
-lemma pr2_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
-u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: 
-T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: 
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 
-(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 
-t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 
-t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c 
-u1 u2 H)))))).
-
-lemma pr2_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) 
-(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead 
-k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c 
-(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k 
-u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 
-t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: 
-C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) 
-\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: 
-T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u 
-t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
-(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
-\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) 
-\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: 
-(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
-t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 
-(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) 
-u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) 
-(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u 
-(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H7 
-\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
-_) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) 
-(clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) 
-u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) 
-(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) 
-H5))) in (\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def 
-(eq_ind T u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr 
-(\lambda (b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) 
-(pr2_free c (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u 
-(pr0_refl u) t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: 
-nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: 
-T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to 
-((eq C c0 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind 
-b) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) 
-u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
-(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c 
-(Bind b) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 
-(CHead d (Bind Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def 
-(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to 
-(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 
-n u0 t6 t0) \to ((eq C c1 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u 
-t5) (THead (Bind b) u t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta 
-c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) 
-(THead (Bind b) u t3) (THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 
-H3 (Bind b)) (THead (Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) 
-H4 u))))))))))))) i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
-(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
-\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) 
-\to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: 
-(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
-t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 
-(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) 
-u) H4) in (pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c 
-(drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O 
-(CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) 
-(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead 
-(Flat f) u t) (subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: 
-nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: 
-T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to 
-((eq C c0 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat 
-f) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) 
-u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
-(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c 
-(Flat f) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 
-(CHead d (Bind Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def 
-(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to 
-(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 
-n u0 t6 t0) \to ((eq C c1 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u 
-t5) (THead (Flat f) u t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta 
-c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) 
-(THead (Flat f) u t3) (THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 
-H3 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) 
-H4 u))))))))))))) i)))))) k) y t1 t2 H0))) H)))))).
-
-lemma clear_pr2_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to 
-(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 
-t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: 
-C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: 
-(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: 
-C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c 
-(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
-
-lemma pr2_cflat:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free 
-(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda 
-(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u 
-i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c 
-t1 t2 H)))))).
-
-lemma pr2_ctail:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) 
-t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: 
-(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail 
-Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
-
-lemma pr2_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
-T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to 
-(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
-(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind 
-b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda 
-(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) 
-(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 
-(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) 
-v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: 
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: 
-(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: 
-C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in 
-(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind 
-Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) 
-(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) 
-u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) 
-H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 
-(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 
-\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) 
-v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind 
-b) v1) (CHead d (Bind Abbr) u) H7))) in (\lambda (H12: (eq B Abbr 
-b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind T u (\lambda (t0: 
-T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def (eq_ind_r B b (\lambda 
-(b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B Abbr (\lambda (b0: 
-B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match (H15 (refl_equal B 
-Abbr)) in False with []) in H16) b H12)))))) H10)) H9)))) (\lambda (i0: 
-nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) 
-\to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda 
-(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda 
-(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) 
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
-(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) 
-y t1 t2 H1))) H0)))))))).
-
-lemma pr2_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift 
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 
-t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c 
-(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) 
-\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 
-e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) 
-(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 
-t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: 
-C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c 
-(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def 
-(drop_getl_trans_le i d (le_S_n i d (le_S_n (S i) (S d) (le_S (S (S i)) (S d) 
-(le_n_S (S i) d H7)))) c e h H (CHead d0 (Bind Abbr) u) H6) in (ex3_2_ind C C 
-(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda 
-(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear 
-e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t3) (lift h d t)) (\lambda 
-(x0: C).(\lambda (x1: C).(\lambda (H9: (drop i O c x0)).(\lambda (H10: (drop 
-h (minus d i) x0 x1)).(\lambda (H11: (clear x1 (CHead d0 (Bind Abbr) 
-u))).(let H12 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 
-x1)) H10 (S (minus d (S i))) (minus_x_Sy d i H7)) in (let H13 \def 
-(drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u H11) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) 
-u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t3) 
-(lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x (Bind Abbr) 
-(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x 
-d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x 
-(Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d 
-t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 
-h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) 
-(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) 
-(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h 
-H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
-
-lemma pr2_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
-(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T 
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) 
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
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-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda 
-(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 
-O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 
-(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) 
-u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl 
-Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S 
-i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl 
-x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda 
-(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t 
-(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: 
-(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 
-x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 
-(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T 
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O 
-(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) 
-u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 
-x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) 
-H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) 
-H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
-(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) 
-H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) 
-(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-
-lemma pr2_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
-(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
-t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift 
-(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 
-t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind 
-Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
-b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) 
-t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) 
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 
-x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 
-H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 
-(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
-t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T 
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t 
-(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind 
-Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 
-x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 
-x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead 
-(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda 
-(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: 
-T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 
-(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t 
-(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: 
-(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) 
-(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
-t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S 
-O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 
-t2 H5)))))))))))))) c y x H0))) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma
deleted file mode 100644 (file)
index 8f1063f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/subst1.ma
+++ /dev/null
@@ -1,265 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/fwd.ma".
-
-include "basic_1/pr0/subst1.ma".
-
-include "basic_1/csubst1/getl.ma".
-
-include "basic_1/subst1/subst1.ma".
-
-lemma pr2_delta1:
- \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) 
-\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 
-t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) 
-(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 
-H0 t0 H2))) t H1)))))))))).
-
-lemma pr2_subst1:
- \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
-\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c 
-w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 
-t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T 
-(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) 
-(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: 
-T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda 
-(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: 
-T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T 
-(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) 
-(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) 
-(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) 
-(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 
-x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v 
-t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v 
-(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
-(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda 
-(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda 
-(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c 
-(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: 
-T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
-(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda 
-(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 
-x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: 
-T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not 
-(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: 
-T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t 
-x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c 
-w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x 
-H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t 
-H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let 
-H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in 
-(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind 
-Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) 
-(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead 
-e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal 
-C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono 
-c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in ((let H15 \def 
-(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow v | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) 
-u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in 
-(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl 
-i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u 
-(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C 
-d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in 
-(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x 
-t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t 
-w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: 
-(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
-T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) 
-(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) 
-H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 
-H5))))))))))))))) y t1 t2 H1))) H0)))))))).
-
-lemma pr2_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
-(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
-x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: 
-C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to 
-(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 
-a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda 
-(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 
-x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
-C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
-d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d 
-x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: 
-T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
-x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 
-(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda 
-(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T 
-(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
-x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda 
-(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) 
-H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 
-(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 
-H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S 
-O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
-(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
-a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: 
-T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda 
-(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) 
-(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: 
-T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) 
-x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x 
-(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: 
-T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) 
-(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: 
-(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) 
-H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 
-d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: 
-(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: 
-T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 
-t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: 
-T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) 
-d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d 
-(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: 
-T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) 
-(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) 
-u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) 
-(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 
-(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) 
-d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
-(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
-C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 
-t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: 
-T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) 
-x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 
-(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl 
-i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 
-(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) 
-(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
-C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 
-u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: 
-T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) 
-x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop 
-(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: 
-T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) 
-H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) 
-x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S 
-i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 
-x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: 
-(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda 
-(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind 
-nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S 
-O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in 
-(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S 
-i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: 
-T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) 
-(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S 
-i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 
-(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift 
-(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat 
-(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S 
-i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let 
-H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: 
-nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) 
-in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) 
-(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) 
-d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) 
-(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) 
-(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S 
-i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i 
-H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 
-c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i 
-(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) 
-(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: 
-nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def 
-(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 
-\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in 
-(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind 
-Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: 
-T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let 
-H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) 
-H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead 
-e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match 
-e0 with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d 
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) 
-i H0 (CHead e (Bind Abbr) u0) H16)) in ((let H19 \def (f_equal C T (\lambda 
-(e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow 
-t0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d 
-(Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d 
-e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind 
-Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: 
-T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r 
-T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u 
-(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) 
-(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: 
-C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda 
-(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) 
-t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: 
-T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t 
-x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind 
-T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) 
-(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) 
-(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i 
-(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i 
-x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) 
-(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) 
-H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T 
-(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S 
-O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) 
-(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 
-u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T 
-(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 
-(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) 
-d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq 
-T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 
-x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 
-(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t 
-(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u 
-(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 
-(csubst1_getl_ge d0 i (le_S_n d0 i (le_S_n (S d0) (S i) (le_S (S (S d0)) (S 
-i) (le_n_S (S d0) i H12)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) 
-d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 
-(S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 H16)))))) (subst1_gen_lift_ge u x0 
-x2 i (S O) d0 H14 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) 
-H12 (plus d0 (S O)) (plus_sym d0 (S O)))))))) (subst1_confluence_neq t4 t u i 
-(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i 
-(lt_neq d0 i H12)))))))))) (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 
-t4 H1 u0 (lift (S O) d0 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 
-t2 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma
deleted file mode 100644 (file)
index 7430c89..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr2/defs.ma".
-
-inductive pr3 (c: C): T \to (T \to Prop) \def
-| pr3_refl: \forall (t: T).(pr3 c t t)
-| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: 
-T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
deleted file mode 100644 (file)
index 7f9e464..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma
+++ /dev/null
@@ -1,334 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/defs.ma".
-
-include "basic_1/pr2/fwd.ma".
-
-implied rec lemma pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: 
-T).(P t t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to 
-(\forall (t3: T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) 
-(t0: T) (p: pr3 c t t0) on p: P t t0 \def match p with [(pr3_refl t1) 
-\Rightarrow (f t1) | (pr3_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 
-((pr3_ind c P f f0) t2 t3 p1))].
-
-lemma pr3_gen_sort:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
-(eq T x (TSort n)))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort 
-n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
-T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda 
-(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: 
-T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
-(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 
-t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda 
-(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: 
-T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) 
-\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T 
-(TSort n)))) t1 H4))))))))) y x H0))) H)))).
-
-lemma pr3_gen_abst:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
-T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead 
-(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: 
-T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c 
-(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
-(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 
-x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind 
-Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl 
-(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
-(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
-(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) 
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 
-t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 
-x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: 
-T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 
-t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 
-(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) 
-x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) 
-x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: 
-T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) 
-H11)))))))) H6)))))))))))) y x H0))))) H))))).
-
-lemma pr3_gen_cast:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c 
-t1 x))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
-t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T 
-t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall 
-(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 
-t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 
-x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) 
-x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
-x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T 
-(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) 
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
-T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: 
-(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: 
-T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def 
-(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c 
-x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 
-c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda 
-(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) 
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) 
-x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 
-x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T 
-t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: 
-(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat 
-Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead 
-(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 
-(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) 
-(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 
-t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c 
-x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 
-H2))) H6)))))))))))) y x H0))))) H))))).
-
-lemma pr3_gen_lift:
- \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
-(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
-(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 
-t2))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
-(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
-\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e 
-t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
-(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T 
-(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) 
-(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h 
-d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T 
-t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: 
-T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: 
-C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h 
-d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
-T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 
-(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: 
-T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda 
-(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: 
-(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 
-(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in 
-(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e 
-x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: 
-T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d 
-x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift 
-h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 
-(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda 
-(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T 
-(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 
-H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x 
-H0)))) H)))))).
-
-lemma pr3_gen_lref:
- \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to 
-(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T x (lift (S n) O v))))))))))
-\def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef 
-n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
-T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
-x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or 
-(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T 
-t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) 
-(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: 
-T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or 
-(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef 
-n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) 
-in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T 
-(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
-Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef 
-n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T 
-t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef 
-n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or 
-(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def 
-(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 
-(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: 
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) 
-(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead 
-d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
-v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 
-(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def 
-(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C 
-T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
-Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
-v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda 
-(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift 
-c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda 
-(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq 
-T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T 
-t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 
-(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
-T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 
-(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) 
-H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
deleted file mode 100644 (file)
index c56ee4c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma
+++ /dev/null
@@ -1,1125 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/props.ma".
-
-include "basic_1/iso/props.ma".
-
-include "basic_1/tlist/fwd.ma".
-
-lemma pr3_iso_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
-Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) 
-u2))))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall 
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
-(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: 
-T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to 
-(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq 
-T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso 
-(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T 
-(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef 
-i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T 
-(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind 
-Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O 
-v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
-T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
-T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 
-(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift 
-(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) 
-\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S 
-i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
-Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
-H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
-(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 
-(Bind Abbr) x1) H4)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
-Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H 
-(CHead x0 (Bind Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
-(eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) 
-in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in 
-(let H14 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
-w))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w 
-x2)) H13 d H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 
-H15))))))) H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
-i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
-u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 
-\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda 
-(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t 
-x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) 
-H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t 
-x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) 
-u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
-w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t 
-x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
-Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c 
-(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t 
-(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
-(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead 
-(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso 
-(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead 
-(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
-(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) 
-x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 
-(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t 
-x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) 
-H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
-(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda 
-(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) 
-x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads 
-(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind 
-x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 
-x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 
-(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat 
-Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift 
-(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 
-H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 
-(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead 
-(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift 
-(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O 
-(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 
-(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) 
-u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 
-c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
-Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
-(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
-
-lemma pr3_iso_appls_cast:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
-\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: 
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THeads (Flat Appl) vs t) u2))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 
-(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) 
-(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda 
-(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 
-\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
-t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 
-x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def 
-(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall 
-(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat 
-Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat 
-Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
-(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: 
-((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) 
-\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall 
-(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: 
-T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead 
-(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: 
-Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead 
-(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat 
-Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
-Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
-(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) 
-x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) 
-t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: 
-T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v 
-t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in 
-(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat 
-Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat 
-Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) 
-t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) 
-(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
-T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) 
-(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: 
-(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 
-x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
-u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads 
-(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 
-(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead 
-(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast 
-Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c 
-(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) 
-(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind 
-Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c 
-t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) 
-y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat 
-Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
-Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) 
-(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda 
-(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
-Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat 
-Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) 
-c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead 
-(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads 
-(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda 
-(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 
-Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 
-c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead 
-(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead 
-(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
-x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 
-(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) 
-x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift 
-(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead 
-c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind 
-x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) 
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
-
-lemma pr3_iso_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
-T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
-(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c 
-(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead 
-(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let 
-H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T 
-T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) 
-(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 
-z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 
-(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq 
-T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: 
-(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: 
-T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S 
-O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) 
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
-(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda 
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
-t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead 
-(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) 
-x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) 
-v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def 
-(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in 
-(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind 
-Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
-(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
-x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
-t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
-(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 
-x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
-x5) H10) in (\lambda (H16: (eq T x0 x4)).(\lambda (H17: (eq B Abst b)).(let 
-H18 \def (eq_ind_r T x5 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) 
-H12 x1 H15) in (let H19 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) 
-H11 x0 H16) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c 
-(Bind b0) v2) t x1)) H18 Abst H17) in (let H21 \def (eq_ind_r B b (\lambda 
-(b0: B).(not (eq B b0 Abst))) H Abst H17) in (eq_ind B Abst (\lambda (b0: 
-B).(pr3 c (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
-(Bind Abbr) x2 x3))) (let H22 \def (match (H21 (refl_equal B Abst)) in False 
-with []) in H22) b H17)))))))) H14)) H13))))))) H9)) (\lambda (H9: (pr3 
-(CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1)))).(pr3_t 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
-(Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
-Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) (Bind b) 
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x2) (pr3_lift 
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
-v1 x2 H5) t (lift (S O) O (THead (Bind Abst) x0 x1)) H9 Appl)) (THead (Bind 
-Abbr) x2 x3) (eq_ind T (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
-x0 x1))) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 t0) (THead (Bind Abbr) x2 
-x3))) (pr3_sing c (THead (Bind Abbr) x2 x1) (THead (Bind b) v2 (lift (S O) O 
-(THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (pr2_free c (THead (Bind 
-b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (THead 
-(Bind Abbr) x2 x1) (pr0_zeta b H (THead (Flat Appl) x2 (THead (Bind Abst) x0 
-x1)) (THead (Bind Abbr) x2 x1) (pr0_beta x0 x2 x2 (pr0_refl x2) x1 x1 
-(pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) (pr3_head_12 c x2 x2 (pr3_refl 
-c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead (Flat Appl) (lift (S O) O x2) 
-(lift (S O) O (THead (Bind Abst) x0 x1))) (lift_flat Appl x2 (THead (Bind 
-Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) H3)) (\lambda (H3: (ex6_6 B T T 
-T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: 
-T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) 
-O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: 
-B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 Abst))).(\lambda (H5: (pr3 c 
-(THead (Bind b) v2 t) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: 
-(pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind 
-x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-x4) x3)) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let 
-H_x \def (pr3_gen_bind b H c v2 t (THead (Bind x0) x1 x2) H5) in (let H10 
-\def H_x in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
-(Bind x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) 
-v2) t t2)))) (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 
-x2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (H11: 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) 
-(THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind 
-x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c 
-v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))) 
-(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H12: (eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) x6 x7))).(\lambda (H13: (pr3 c v2 x6)).(\lambda (H14: (pr3 (CHead c (Bind 
-b) v2) t x7)).(let H15 \def (f_equal T B (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
-\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
-x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in ((let H16 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x1 | (TLRef 
-_) \Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
-(THead (Bind b) x6 x7) H12) in ((let H17 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 
-x7) H12) in (\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 
-\def (eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 
-H17) in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 
-H18) in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) 
-x5) x2 x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq 
-B b0 Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind 
-b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead 
-(Flat Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 
-x5 H8) (Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) 
-(lift (S O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift 
-(S O) O x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c 
-c (drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
-(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
-(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
-(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
-t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
-Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) 
-(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift 
-(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
-v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind 
-x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O 
-(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c 
-(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
-x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind 
-x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) 
-x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) 
-(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 
-H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead 
-(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 
-H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) 
-(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S 
-O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
-(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
-H10))) u2 H6))))))))))))) H3)) H2)))))))))).
-
-lemma pr3_iso_appls_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
-T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
-(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
-T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t))) u2)))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
-TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) 
-u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) 
-u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) 
-(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall 
-(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda 
-(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda 
-(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v 
-(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def 
-(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
-b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) 
-t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: 
-Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 
-(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat 
-Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T 
-T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
-B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) 
-u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 
-(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 
-x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind 
-b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 
-(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
-t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat 
-Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S 
-O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c 
-(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) 
-t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) 
-t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead 
-(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta 
-x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) 
-x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 
-(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
-t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
-(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
-Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
-(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat 
-Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u 
-(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead 
-(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) 
-t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t 
-(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads 
-(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c 
-(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) 
-H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
-(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: 
-Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 
-H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
-x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind 
-x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat 
-Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl 
-x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat 
-Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) 
-(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c 
-(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind 
-x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat 
-Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat 
-Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
-(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
-H7)))))))))))))) H4)) H3))))))))) vs)))))).
-
-lemma pr3_iso_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
-T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
-in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) 
-(lifts (S O) O vs) t)) u2))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: 
-TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: 
-T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: 
-C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) 
-t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda 
-(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso 
-(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: 
-TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead 
-(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) 
-u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat 
-Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: 
-T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) 
-(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat 
-Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: 
-Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) 
-(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) 
-u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) 
-(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let 
-H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) 
-(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) 
-t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) 
-ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind 
-b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 
-(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) 
-(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda 
-(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall 
-(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to 
-((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: 
-Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O 
-t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat 
-Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c 
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) 
-(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: 
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead 
-(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 
-t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads 
-(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c 
-(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) 
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b 
-H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: 
-((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 
-c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads 
-(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to 
-(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) 
-u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
-(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t 
-(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead 
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift 
-(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: 
-T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 
-ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 
-ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 
-(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) 
-u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead 
-(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat 
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to 
-(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 
-(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: 
-(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat 
-Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads 
-(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O 
-t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
-(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) 
-(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads 
-(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
-Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) 
-(lifts_tapp (S O) O t ts))))))))))) vs))).
-
-lemma pr3_iso_beta:
- \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
-Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
-u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
-Abbr) v t) u2))))))))
-\def
- \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: 
-T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
-u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
-\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind 
-Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
-T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) 
-w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v 
-x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T 
-u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) 
-\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) 
-t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead 
-(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
-(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
-T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
-w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v 
-x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 
-x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
-u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) 
-in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
-(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) x4 x5) H8) in ((let H12 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) 
-x4 x5) H8) in (\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
-t0)))) H10 x1 H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w 
-t0)) H9 x0 H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
-(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
-(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
-(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda 
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind 
-Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 
-(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v 
-x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 
-x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in 
-(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 
-x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
-u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
-(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead k _ _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
-(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in 
-((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq 
-T x1 x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
-t0)))) H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w 
-t0)) H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c 
-(Bind b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
-B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
-H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
-Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
-False with []) in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) 
-H1)))))))).
-
-lemma pr3_iso_appls_beta:
- \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
-\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
-(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
-(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) 
-v t)) u2)))))))))
-\def
- \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall 
-(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) 
-v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 
-u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat 
-Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: 
-T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso 
-(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: 
-Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda 
-(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: 
-T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: 
-Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) 
-u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: 
-C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: 
-(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def 
-(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
-Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
-(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) 
-(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
-(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: 
-T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
-(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) 
-(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 
-u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat 
-Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def 
-(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
-t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: 
-Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
-x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat 
-Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T 
-T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
-(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat 
-Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
-(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 
-c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 
-(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: 
-T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 
-c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 
-(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 
-x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 
-(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) 
-(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) 
-t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead 
-(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 
-(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t 
-x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) 
-w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
-T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
-Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
-u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq 
-B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v 
-(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
-(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c 
-(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
-O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) 
-v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
-(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t 
-(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
-(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c 
-(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead 
-(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda 
-(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) 
-w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) 
-O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead 
-(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead 
-(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
-(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) 
-x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
-x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) 
-(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 
-(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c 
-(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) 
-(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
-(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
-O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
-(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
-(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
-x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
deleted file mode 100644 (file)
index 04878c2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/defs.ma".
-
-include "basic_1/pr1/fwd.ma".
-
-lemma pr3_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
-t2))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: 
-T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 
-t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: 
-C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
deleted file mode 100644 (file)
index d973e0e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma
+++ /dev/null
@@ -1,68 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/props.ma".
-
-include "basic_1/pr2/pr2.ma".
-
-lemma pr3_strip:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
-(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
-T).(pr3 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t 
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t 
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
-t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 
-t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: 
-T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda 
-(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: 
-T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c 
-t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c 
-x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) 
-t0 t1 H)))).
-
-theorem pr3_confluence:
- \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
-(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
-T).(pr3 c t2 t))))))))
-\def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
-t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t 
-t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
-t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t 
-t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
-t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
-(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
-(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 
-t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c 
-t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 
-t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) 
-(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 
-x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
-(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
-(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
-(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 
-c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 
-H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
deleted file mode 100644 (file)
index c9cf9c3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
+++ /dev/null
@@ -1,1604 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/fwd.ma".
-
-include "basic_1/pr3/pr1.ma".
-
-include "basic_1/pr2/props.ma".
-
-include "basic_1/pr1/props.ma".
-
-lemma clear_pr3_trans:
- \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
-(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
-\def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: 
-T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: 
-T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
-t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
-
-lemma pr3_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
-t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
-
-theorem pr3_t:
- \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall 
-(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 
-t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 
-c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall 
-(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
-(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-
-lemma pr3_thin_dx:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
-(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
-T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: 
-T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 
-t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u 
-t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
-t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
-
-lemma pr3_head_1:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall 
-(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda 
-(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
-(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c 
-(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing 
-c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
-(H2 k t)))))))))) u1 u2 H)))).
-
-lemma pr3_head_2:
- \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u 
-t2)))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) 
-(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) 
-(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: 
-(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u 
-t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) 
-(THead k u t4) H2))))))) t1 t2 H)))))).
-
-theorem pr3_head_21:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
-(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c 
-u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
-
-theorem pr3_head_12:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 
-c (THead k u1 t1) (THead k u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
-(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
-u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
-
-lemma pr3_cflat:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
-(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: 
-T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: 
-F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead 
-c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c 
-(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
-
-theorem pr3_flat:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead 
-(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
-(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
-u2))))))))).
-
-lemma pr3_pr0_pr2_t:
- \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: 
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
-(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
-t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda 
-(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: 
-(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k 
-u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: 
-(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def 
-(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead 
-c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d 
-(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) 
-(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: 
-(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d 
-(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda 
-(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
-c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
-(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) 
-u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
-(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d 
-(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) 
-H9))) in ((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda 
-(H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u 
-(\lambda (t0: T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda 
-(b0: B).(pr3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
-T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind 
-Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda 
-(H17: (pr0 x t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c 
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 
-(CHead c (Bind Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
-(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: 
-F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
-u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O 
-(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c 
-(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind 
-Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda 
-(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 
-t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k 
-u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind 
-(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to 
-((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) 
-\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda 
-(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
-u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) 
-\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 
-(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) 
-(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
-(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: 
-F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
-u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) 
-\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 
-(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) 
-(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
-u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
-
-lemma pr3_pr2_pr2_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
-u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
-T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 
-(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
-K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 
-t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
-(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
-(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: 
-C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: 
-C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
-T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 
-t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
-t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 
-(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
-C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
-(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
-t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
-c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
-(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: 
-nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
-t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
-(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 
-(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
-(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
-c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 
-(CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
-(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
-(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
-d0 (Bind Abbr) u0) t H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
-(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
-d0 (Bind Abbr) u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
-C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
-H11 t H15) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 
-t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: 
-T).(subst0 (S (plus i O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) 
-(\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S 
-(plus i O)) u x t6)).(let H21 \def (f_equal nat nat S (plus i O) i (sym_eq 
-nat i (plus i O) (plus_n_O i))) in (let H22 \def (eq_ind nat (S (plus i O)) 
-(\lambda (n: nat).(subst0 n u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda 
-(t7: T).(subst0 O t1 t5 t7)) (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 
-(Bind Abbr) t1) t4 t6) (\lambda (x0: T).(\lambda (H23: (subst0 O t1 t5 
-x0)).(\lambda (H24: (pr0 x0 x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 
-(pr2_delta (CHead c0 (Bind Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 
-x0 H23) t6 (pr3_pr2 (CHead c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 
-(Bind Abbr) t1) d u (S i) (getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 
-t1 (clear_bind Abbr c0 t1) (CHead d (Bind Abbr) u) i H0) x0 x H24 t6 
-H22)))))) (pr0_subst0_back t2 t5 x O H19 t1 H1))))))) (subst0_subst0 t5 t6 t 
-O H18 t2 u i H2)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: 
-(clear (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
-(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 O (getl_intro O c0 
-(CHead d0 (Bind Abbr) u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead d0 
-(Bind Abbr) u0) t H12)) t4 t5 H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k 
-t) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda (_: (((getl 
-i1 (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to 
-(pr3 (CHead c0 k t1) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) 
-(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind 
-(\lambda (k0: K).((getl (S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to 
-(pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) 
-(CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) 
-t1) t4 t6 (pr2_delta (CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) 
-i1 c0 (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) 
-u0) t i1 H12) t1) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S 
-i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
-(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) 
-(getl_gen_S (Flat f) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) 
-f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c 
-u1 u2 H)))).
-
-lemma pr3_pr2_pr3_t:
- \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
-(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to 
-(pr3 (CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
-(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 
-(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
-u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
-(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) 
-\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 
-u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
-u1 H3)))))))))) t1 t2 H)))))).
-
-theorem pr3_pr3_pr3_t:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 
-(CHead c k u1) t1 t2))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
-(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 
-t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
-K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
-(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
-t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 
-(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: 
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
-t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
-
-lemma pr3_lift:
- \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
-d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift 
-h d t1) (lift h d t2)))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 
-t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h 
-d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 
-t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d 
-t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
-H0)))))))).
-
-lemma pr3_eta:
- \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind 
-Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v 
-(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
-\def
- \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind 
-Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead 
-(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c 
-(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w 
-u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead 
-(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat 
-Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u 
-(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat 
-Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) 
-(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S 
-O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing 
-(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) 
-(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) 
-(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) 
-(lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u (pr1_pr0 (THead (Bind 
-Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u 
-(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead 
-(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
-
-lemma pr3_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
-x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 
-(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c 
-(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
-(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
-(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind 
-Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
-(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) 
-(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
-u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c 
-t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
-(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
-H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
-x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
-t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
-Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
-T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
-O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
-Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
-Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
-t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
-(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
-(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
-x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
-(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
-(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
-c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
-(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
-(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
-Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
-(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
-t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
-(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
-O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
-O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-
-lemma pr3_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
-(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) 
-O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind 
-Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 
-(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
-T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
-Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
-x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
-x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
-(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
-T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
-T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
-t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
-(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
-b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
-c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
-t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
-T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
-(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
-c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
-(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
-(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
-T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
-z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
-(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
-(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
-T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
-Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
-(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
-x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
-Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
-(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
-t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
-t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
-(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
-x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
-(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
-x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
-(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
-H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
-x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
-O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
-T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
-x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
-(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
-T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
-x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
-(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
-x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
-x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
-x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
-(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
-x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
-(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
-x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
-t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
-(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
-T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
-t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
-x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
-x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
-(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
-O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
-(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
-O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 
-x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
-x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
-x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
-O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
-(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
-x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
-H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
-Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
-(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
-(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
-t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
-(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
-x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
-(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
-(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
-(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
-H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
-x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
-(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
-(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
-(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
-
-lemma pr3_gen_appl:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
-(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
-t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
-t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
-(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda 
-(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
-T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
-(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
-(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
-(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
-(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
-Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
-(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
-Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
-c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
-x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
-t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
-Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
-(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
-t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
-t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead 
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
-T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
-x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
-(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
-T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
-(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
-t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
-Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
-x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
-(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
-t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
-x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
-x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
-x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
-(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
-x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
-(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
-(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
-B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
-T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
-T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
-t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
-(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T 
-T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
-(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
-T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
-c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
-(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
-x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
-x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
-x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
-(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
-(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
-Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
-T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
-x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
-x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
-t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
-\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
-(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
-T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
-T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
-c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
-T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
-(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
-Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
-x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
-T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
-t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
-c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
-y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
-(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
-x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
-(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
-x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
-x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
-(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
-(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
-(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
-x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
-(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
-(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
-c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
-(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
-H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
-(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
-(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
-Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
-z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
-T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
-Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
-(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
-t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
-y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
-H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
-x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
-
-lemma pr3_gen_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
-T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
-b) u1) t1 (lift (S O) O x)))))))))
-\def
- \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
-(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
-b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
-(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
-Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
-(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
-t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
-(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
-(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
-H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
-O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
-(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
-(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
-\def (match (H (refl_equal B Abst)) in False with []) in H1))))))) (\lambda 
-(_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
-\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
-(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
-u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
-(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
-u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
-(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
-c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
-x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
-(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
-O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
-(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
-H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
-(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
-c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
-t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
deleted file mode 100644 (file)
index 3c2e56c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
+++ /dev/null
@@ -1,89 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/fwd.ma".
-
-include "basic_1/pr2/subst1.ma".
-
-lemma pr3_subst1:
- \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) 
-\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c 
-w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
-\def
- \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
-T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 
-w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: 
-T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 
-w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda 
-(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: 
-T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v 
-t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i 
-v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T 
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T 
-(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) 
-(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 
-x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 
-w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 
-w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v 
-t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 
-i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
-e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
-
-lemma pr3_gen_cabbr:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
-(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
-\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
-a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
-(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a 
-x1 x2))))))))))))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
-(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 
-d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) 
-(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
-(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: 
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 
-a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d 
-x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) 
-(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
-T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
-(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: 
-T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 
-x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
-(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: 
-(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 
-a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d 
-x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda 
-(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
-x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 
-d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda 
-(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) 
-(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: 
-T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) 
-d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d 
-u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x 
-x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 
-e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
deleted file mode 100644 (file)
index fa49b98..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma
+++ /dev/null
@@ -1,63 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/props.ma".
-
-include "basic_1/wcpr0/getl.ma".
-
-lemma pr3_wcpr0_t:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
-T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 
-t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: 
-C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: 
-T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) 
-(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: 
-T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead 
-c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 
-(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) 
-(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: 
-(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq 
-C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: 
-C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: 
-(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 
-k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) 
-u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda 
-(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k 
-u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind 
-Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: 
-C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: 
-T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
-x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T 
-(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 
-(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 
-x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta 
-(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t 
-(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 
-H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
-H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) 
-H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 
-H2)))))))))))))) c2 c1 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/preamble.ma b/matita/matita/contribs/lambdadelta/basic_1/preamble.ma
deleted file mode 100644 (file)
index 2354bcc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/preamble.ma
+++ /dev/null
@@ -1,15 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "ground_1/theory.ma".

diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma
deleted file mode 100644 (file)
index 6108d3c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/r/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-definition r:
- K \to (nat \to nat)
-\def
- \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | 
-(Flat _) \Rightarrow (S i)])).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
deleted file mode 100644 (file)
index 6dc07a0..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/r/props.ma
+++ /dev/null
@@ -1,153 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/r/defs.ma".
-
-include "basic_1/s/defs.ma".
-
-lemma r_S:
- \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S 
-i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r 
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat 
-f) i))))) k).
-
-lemma r_plus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
-(plus (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda 
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) 
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r 
-(Flat f) i) j))))) k).
-
-lemma r_plus_sym:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
-(plus i (r k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda 
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: 
-F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
-
-lemma r_minus:
- \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat 
-(minus (r k i) (S n)) (r k (minus i (S n)))))))
-\def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: 
-K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S 
-n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: 
-F).(minus_x_Sy i n H)) k)))).
-
-lemma r_dis:
- \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) 
-\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: 
-nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) 
-(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: 
-((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: 
-((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: 
-nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: 
-((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: 
-((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda 
-(i: nat).(refl_equal nat (S i)))))))) k).
-
-lemma s_r:
- \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 
-i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) 
-(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
-
-lemma r_arith0:
- \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
-\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: 
-nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: 
-nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) 
-(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
-
-lemma r_arith1:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S 
-i)) (S j)) (minus (r k i) j))))
-\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) 
-(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat 
-(minus (r k i) j)) (r k (S i)) (r_S k i)))).
-
-lemma r_arith2:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to 
-(le (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda 
-(i: nat).(\lambda (j: nat).(\lambda (H: (le (S i) (S j))).(let H_y \def 
-(le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
-nat).(\lambda (H: (le (S i) j)).H)))) k).
-
-lemma r_arith3:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to 
-(le j (r k i)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda 
-(i: nat).(\lambda (j: nat).(\lambda (H: (le (S j) (S i))).(let H_y \def 
-(le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
-nat).(\lambda (H: (le j (S i))).H)))) k).
-
-lemma r_arith4:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k 
-j)) (minus (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (minus (S i) (s k0 j)) (minus (r k0 i) j))))) (\lambda (b: 
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i) 
-j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
-(minus (r (Flat f) i) j))))) k).
-
-lemma r_arith5:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to 
-(lt j (r k i)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda 
-(i: nat).(\lambda (j: nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) 
-(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S 
-i))).H)))) k).
-
-lemma r_arith6:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S 
-j)) (minus i (s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (minus (r k0 i) (S j)) (minus i (s k0 j)))))) (\lambda (b: 
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b) 
-j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
-(minus i (s (Flat f) j)))))) k).
-
-lemma r_arith7:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j)) 
-\to (eq nat (r k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((eq nat (S i) (s k0 j)) \to (eq nat (r k0 i) j))))) (\lambda (_: 
-B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (S i) (S 
-j))).(eq_add_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
-nat).(\lambda (H: (eq nat (S i) j)).H)))) k).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma
deleted file mode 100644 (file)
index 9a2f29d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/s/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-definition s:
- K \to (nat \to nat)
-\def
- \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | 
-(Flat _) \Rightarrow i])).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma
deleted file mode 100644 (file)
index deac2db..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/s/fwd.ma
+++ /dev/null
@@ -1,48 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/s/defs.ma".
-
-lemma s_inj:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) 
-\to (eq nat i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: 
-B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s 
-(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda 
-(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
-
-lemma s_le_gen:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k i) (s k j)) \to 
-(le i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((le (s k0 i) (s k0 j)) \to (le i j))))) (\lambda (b: B).(\lambda (i: 
-nat).(\lambda (j: nat).(\lambda (H: (le (s (Bind b) i) (s (Bind b) 
-j))).(le_S_n i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
-nat).(\lambda (H: (le (s (Flat f) i) (s (Flat f) j))).H)))) k).
-
-lemma s_lt_gen:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k i) (s k j)) \to 
-(lt i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((lt (s k0 i) (s k0 j)) \to (lt i j))))) (\lambda (b: B).(\lambda (i: 
-nat).(\lambda (j: nat).(\lambda (H: (lt (s (Bind b) i) (s (Bind b) 
-j))).(le_S_n (S i) j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
-nat).(\lambda (H: (lt (s (Flat f) i) (s (Flat f) j))).H)))) k).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/s/props.ma b/matita/matita/contribs/lambdadelta/basic_1/s/props.ma
deleted file mode 100644 (file)
index 75318f0..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/s/props.ma
+++ /dev/null
@@ -1,109 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/s/defs.ma".
-
-lemma s_S:
- \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S 
-i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s 
-(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat 
-f) i))))) k).
-
-lemma s_plus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
-(plus (s k i) j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda 
-(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) 
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s 
-(Flat f) i) j))))) k).
-
-lemma s_plus_sym:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
-(plus i (s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda 
-(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq 
-nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) 
-(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
-nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
-
-lemma s_minus:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s 
-k (minus i j)) (minus (s k i) j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) 
-(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j 
-i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) 
-j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) 
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j 
-i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
-
-lemma minus_s_s:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s 
-k j)) (minus i j))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: 
-B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) 
-(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i 
-j))))) k).
-
-lemma s_le:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) 
-(s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: 
-nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: 
-F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
-
-lemma s_lt:
- \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) 
-(s k j)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
-nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: 
-nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(lt_n_S i j H))))) (\lambda (_: 
-F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
-
-lemma s_inc:
- \forall (k: K).(\forall (i: nat).(le i (s k i)))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) 
-(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S_n (S i) (S 
-(s (Bind b) i)) (le_S_n (S (S i)) (S (S (s (Bind b) i))) (le_S (S (S (S i))) 
-(S (S (s (Bind b) i))) (le_n (S (S (s (Bind b) i)))))))))) (\lambda (f: 
-F).(\lambda (i: nat).(le_n (s (Flat f) i)))) k).
-
-lemma s_arith0:
- \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
-\def
- \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: 
-nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal 
-nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
-
-lemma s_arith1:
- \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
-\def
- \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n 
-i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma
deleted file mode 100644 (file)
index 373c816..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sc3/arity.ma
+++ /dev/null
@@ -1,313 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubc/arity.ma".
-
-include "basic_1/csubc/getl.ma".
-
-include "basic_1/csubc/drop1.ma".
-
-include "basic_1/csubc/props.ma".
-
-lemma sc3_arity_csubc:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
-t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall 
-(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
-A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
-C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: 
-C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: 
-(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T 
-(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) 
-(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 
-n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n 
-is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: 
-A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall 
-(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
-a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda 
-(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let 
-H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in 
-(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: 
-C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) 
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 
-(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) 
-(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x 
-(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def 
-H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: 
-C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 
-(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 
-x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) 
-x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind 
-Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 
-(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
-c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
-(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda 
-(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) 
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq 
-C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc 
-g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 
-(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is 
-(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) 
-(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind 
-C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) 
-(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in 
-(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y 
-(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O 
-u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) 
-O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) 
-(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) 
-(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans 
-is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef 
-i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: 
-C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda 
-(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
-w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
-K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is 
-(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: 
-(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) 
-x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 
-(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 
-(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) 
-in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee with [(Bind 
-b) \Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False 
-| Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) 
-in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) 
-(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: 
-B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) 
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq 
-K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: 
-(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is 
-i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind 
-Abbr) (\lambda (ee: K).(match ee with [(Bind b) \Rightarrow (match b with 
-[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 
-(lift1 is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) 
-H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: 
-A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: 
-C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 
-c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda 
-(is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: 
-(csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 
-H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 
-(ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind 
-Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda 
-(x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is 
-i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def 
-(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is 
-i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans 
-is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans 
-is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda 
-(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) 
-(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 
-(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C 
-(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) 
-(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
-w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
-x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
-T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 
-C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) 
-(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 
-(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
-c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C 
-x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x 
-x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) 
-H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def 
-(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: 
-T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef 
-(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 
-t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 
-H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 
-(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is 
-i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
-T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
-(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
-is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
-w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
-K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
-A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
-T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
-C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is 
-(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: 
-(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) 
-x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x 
-(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def 
-(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind 
-Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef 
-(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 
-(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let 
-H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in 
-(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S 
-(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g 
-x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) 
-H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) 
-(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
-T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
-C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: 
-B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind 
-Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
-Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) 
-(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq 
-K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: 
-(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is 
-i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind 
-Abst) (\lambda (ee: K).(match ee with [(Bind b) \Rightarrow (match b with 
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 
-(lift1 is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) 
-H6))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
-Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
-g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 
-c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is 
-u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
-(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: 
-PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 
-c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: 
-PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g 
-d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead 
-(Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) 
-(H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) 
-(Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) 
-(csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is 
-(THead (Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g 
-a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) 
-\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is 
-u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c 
-(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: 
-PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g 
-d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: 
-PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g 
-d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) 
-(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall 
-(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g 
-a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead 
-(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: 
-C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d 
-c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 
-is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind 
-Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 
-is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 
-(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) 
-(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: 
-C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: 
-PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 
-is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 
-g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 
-d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr 
-not_abbr_abst a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) 
-(let H_x \def (csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x 
-in (ex2_ind C (\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g 
-c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) 
-(\lambda (x: C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x 
-d)).(eq_ind_r T (lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 
-(CHead d (Bind Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: 
-PList).(sc3 g a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind 
-Abst) (lift1 (papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c 
-(papp is0 is) x u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) 
-(csubc_abst g x d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) 
-(drop1_trans is0 x d1 H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) 
-(Ss is)) (papp_ss is0 is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 
-(Ss is0) (Ss is) t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g 
-c2 (asucc g a1) is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 
-(THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 
-(lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) 
-(lift1_bind Abst is u t0)))))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
-(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g 
-d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: 
-A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: 
-C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 
-c2) \to (sc3 g (AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: 
-C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: 
-C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let 
-H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 \def H_y0 in (land_ind (arity g c2 
-(lift1 is t0) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) 
-\to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat 
-Appl) w (lift1 is0 (lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat 
-Appl) u t0))) (\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda 
-(H8: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: 
-PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 
-(lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in 
-(eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
-T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u 
-t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g 
-a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) 
-\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is 
-u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
-((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
-C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1: 
-C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: 
-C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in 
-(eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
-T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0) 
-(H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is 
-u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: 
-A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall 
-(is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
-a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
-a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1 
-c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1 
-is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
-
-lemma sc3_arity:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
-a) \to (sc3 g a c t)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
-(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y 
-(drop1_nil c) c (csubc_refl g c))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma
deleted file mode 100644 (file)
index 417868b..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sc3/defs.ma
+++ /dev/null
@@ -1,29 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sn3/defs.ma".
-
-include "basic_1/arity/defs.ma".
-
-include "basic_1/drop1/defs.ma".
-
-rec definition sc3 (g: G) (a: A) on a: C \to (T \to Prop) \def \lambda (c: 
-C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t 
-(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead 
-a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
-t)))))))))])).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma
deleted file mode 100644 (file)
index 80da46d..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma
+++ /dev/null
@@ -1,697 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sc3/defs.ma".
-
-include "basic_1/sn3/lift1.ma".
-
-include "basic_1/nf2/lift1.ma".
-
-include "basic_1/csuba/arity.ma".
-
-include "basic_1/arity/lift1.ma".
-
-include "basic_1/arity/aprem.ma".
-
-include "basic_1/llt/props.ma".
-
-include "basic_1/llt/fwd.ma".
-
-include "basic_1/drop1/getl.ma".
-
-include "basic_1/drop1/props.ma".
-
-include "basic_1/lift1/drop1.ma".
-
-lemma sc3_arity_gen:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c 
-t) \to (arity g c t a)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind 
-(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
-t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity 
-g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: 
-(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to 
-(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity 
-g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: 
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in 
-(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g 
-a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
-Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity 
-g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g 
-a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
-Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
-
-lemma sc3_repl:
- \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c 
-t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: 
-C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 
-g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: 
-A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to 
-(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: 
-C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 
-g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall 
-(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 
-c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda 
-(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c 
-t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 
-in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda 
-(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def 
-(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n 
-n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) 
-(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: 
-nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort 
-n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 
-x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in 
-(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) 
-H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity 
-g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) 
-(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: 
-C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
-(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to 
-(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: 
-A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: 
-C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
-(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) 
-\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: 
-((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: 
-T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c 
-t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t 
-(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall 
-(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is 
-t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 
-\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: 
-T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
-(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity 
-g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a 
-d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat 
-Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in 
-(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a 
-a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: 
-A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 
-x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A 
-(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) 
-(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall 
-(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d 
-c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t 
-(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: 
-C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: 
-PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: 
-(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 
-t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 
-(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat 
-Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) 
-(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 
-H11))))))) H7))))) H4)))))))))))) a2)) a1)).
-
-lemma sc3_lift:
- \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e 
-t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) 
-\to (sc3 g a c (lift h d t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: 
-C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda 
-(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in 
-(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) 
-(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n 
-n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) 
-(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e 
-t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: 
-C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d 
-t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: 
-T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: 
-C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall 
-(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d 
-e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: 
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 
-\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall 
-(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g 
-a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) 
-(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
-(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
-(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda 
-(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
-PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
-t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: 
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) 
-(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: 
-T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 
-is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 
-(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w 
-t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) 
-(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
-
-lemma sc3_lift1:
- \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: 
-PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) 
-\to (sc3 g a c (lift1 hds t)))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: 
-PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g 
-a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: 
-C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c 
-e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: 
-C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 
-g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: 
-C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n 
-n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x 
-in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 
-e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n 
-n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 
-H4) c n n0 H3)))) H2))))))))))) hds)))).
-
-lemma sc3_abbr:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: 
-nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads 
-(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to 
-(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
-TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
-C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
-(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef 
-i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: 
-TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: 
-C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) 
-(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda 
-(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g 
-c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat 
-Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef 
-i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: 
-(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda 
-(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c 
-(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs 
-(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) 
-(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda 
-(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: 
-T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to 
-((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs 
-(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: 
-TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
-C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
-(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef 
-i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda 
-(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs 
-(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 
-d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat 
-Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda 
-(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity 
-g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: 
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
-(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead 
-a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
-PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
-(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads 
-(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: 
-C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
-\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
-(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) 
-(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
-(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
-(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs 
-(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 
-w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def 
-(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C 
-(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is 
-i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead 
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: 
-C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) 
-d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w 
-(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is 
-(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r 
-T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w 
-(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans 
-is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 
-d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T 
-(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 
-d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 
-is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) 
-vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) 
-H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs 
-(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) 
-H3))))))))))))) a)).
-
-theorem sc3_cast:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
-(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: 
-T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) 
-vs (THead (Flat Cast) u t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
-TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat 
-Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to 
-(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda 
-(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
-T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | 
-(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: 
-T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) 
-(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g 
-(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow 
-(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads 
-(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land 
-(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) 
-(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: 
-(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: 
-(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat 
-Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs 
-u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads 
-(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads 
-(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat 
-Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs 
-t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads 
-(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat 
-Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat 
-Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs 
-t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort 
-O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) 
-(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t 
-H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with 
-[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c 
-(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) 
-(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads 
-(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 
-n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads 
-(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let 
-H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) 
-(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs 
-(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs 
-(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) 
-(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def 
-H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 
-c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead 
-(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead 
-(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort 
-(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g 
-c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs 
-(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n 
-H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall 
-(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to 
-(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: 
-A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 
-g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c 
-(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead 
-(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
-T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc 
-g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
-is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land 
-(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall 
-(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
-d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 
-\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g 
-a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
-is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs 
-(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 
-g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead 
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u 
-t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 
-(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d 
-w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead 
-(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 
-in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: 
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
-t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
-(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
-(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity 
-g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: 
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
-t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
-(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
-(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
-(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c 
-u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: 
-(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y 
-\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 
-is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d 
-(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 
-is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) 
-(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat 
-Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w 
-t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) 
-(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat 
-Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w 
-H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl 
-is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) 
-(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl 
-is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
-
-fact sc3_props__sc3_sn3_abst:
- \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g 
-a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def 
-(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to 
-((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: 
-C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: 
-TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in 
-(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
-(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall 
-(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 
-c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c 
-(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to 
-((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n 
-n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: 
-C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
-t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c 
-t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) 
-H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: 
-(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: 
-(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat 
-Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H 
-(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land 
-(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall 
-(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) 
-vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c 
-(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: 
-(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) 
-(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
-(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
-(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: 
-C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: 
-C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
-\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) 
-(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
-(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c 
-vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) 
-(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads 
-(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: 
-T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall 
-(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
-d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind 
-(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) 
-(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads 
-(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to 
-(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: 
-((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 
-t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: 
-C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) 
-\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef 
-i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: 
-T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: 
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to 
-((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs 
-(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: 
-T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: 
-TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs 
-(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 
-(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind 
-(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) 
-\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
-(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 
-a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to 
-(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
-(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) 
-in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: 
-C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 
-O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 
-(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) 
-(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 
-H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) 
-I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) 
-(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 
-(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil 
-(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) 
-x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 
-(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S 
-x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef 
-O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O 
-t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop 
-(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) 
-(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g 
-c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c 
-(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) 
-vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) 
-\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
-(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: 
-C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: 
-PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall 
-(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
-TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
-vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 
-c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) 
-w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: 
-C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: 
-((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 
-(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 
-c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 
-\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to 
-(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: 
-C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef 
-i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef 
-i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
-(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) 
-\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: 
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) 
-\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat 
-Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) 
-in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) 
-(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef 
-(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat 
-Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) 
-(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 
-is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) 
-(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 
-(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 
-(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) 
-(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is 
-(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is 
-(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) 
-(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is 
-vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) 
-(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat 
-Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
-
-lemma sc3_sn3:
- \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c 
-t) \to (sn3 c t)))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: 
-(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def 
-H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 
-c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g 
-c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 
-vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) 
-(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 
-t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: 
-C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) 
-\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef 
-i))))))))))).(H1 c t H))) H0))))))).
-
-lemma sc3_abst:
- \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
-(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef 
-i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
-\def
- \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda 
-(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) 
-a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def 
-(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: 
-C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
-TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
-vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a 
-c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) 
-vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) 
-\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: 
-nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to 
-((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) 
-vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
-
-theorem sc3_bind:
- \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
-A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
-T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
-(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
-(THead (Bind b) v t)))))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda 
-(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall 
-(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads 
-(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads 
-(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: 
-T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat 
-Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 
-in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O 
-vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S 
-O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) 
-(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda 
-(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) 
-(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind 
-b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) 
-(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) 
-H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) 
-(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall 
-(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) 
-vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall 
-(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead 
-c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) 
-\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v 
-t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda 
-(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
-(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a 
-d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g 
-a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) 
-t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind 
-(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) 
-(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall 
-(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat 
-Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land 
-(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) 
-(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
-(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c 
-(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda 
-(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
-PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) 
-w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity 
-g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: 
-C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) 
-\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead 
-(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 
-H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 
-g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def 
-(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is 
-vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead 
-(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) 
-t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) 
-(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList 
-(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d 
-(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat 
-Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) 
-(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is 
-v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is 
-v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S 
-O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) 
-(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts 
-(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O 
-vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is 
-d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is 
-(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead 
-(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
-
-theorem sc3_appl:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
-TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
-(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
-T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
-(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: 
-A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 
-g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) 
-\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat 
-Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda 
-(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: 
-T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs 
-(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead 
-(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda 
-(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads 
-(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat 
-Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs 
-(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads 
-(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: 
-(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n 
-n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v 
-t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead 
-(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat 
-Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen 
-g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) 
-(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) 
-H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall 
-(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs 
-(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g 
-(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v 
-(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall 
-(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c 
-(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to 
-(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) 
-vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: 
-TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land 
-(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) 
-(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
-(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c 
-v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 
-in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) 
-(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall 
-(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
-(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c 
-(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead 
-a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: 
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
-(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w 
-t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind 
-Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: 
-T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
-(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v 
-t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v 
-(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: 
-T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
-(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v 
-(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g 
-c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) 
-(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: 
-PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 
-is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda 
-(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat 
-Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 
-g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) 
-(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: 
-T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) 
-(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 
-is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead 
-(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads 
-(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs 
-(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 
-t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead 
-(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead 
-(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) 
-(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d 
-w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) 
-(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is 
-v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat 
-Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v 
-(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma
deleted file mode 100644 (file)
index d9ba609..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/defs.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr3/defs.ma".
-
-inductive sn3 (c: C): T \to Prop \def
-| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
-
-rec definition sns3 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
-\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
deleted file mode 100644 (file)
index fe603b5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/fwd.ma
+++ /dev/null
@@ -1,189 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sn3/defs.ma".
-
-include "basic_1/pr3/props.ma".
-
-implied rec lemma sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: 
-T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 
-c t1 t2) \to (sn3 c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to 
-(\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) 
-(t: T) (s0: sn3 c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) 
-\Rightarrow (f t1 s1 (\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to 
-(\forall (P0: Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).((sn3_ind c P f) t2 
-(s1 t2 p p0))))))].
-
-lemma sn3_gen_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
-(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) 
-(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T 
-y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) 
-(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) 
-\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda 
-(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to 
-(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda 
-(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall 
-(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T 
-t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 
-x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead 
-(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
-(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
-(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c 
-(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
-(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
-b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead 
-_ t0 _) \Rightarrow t0])) (THead (Bind b) x x0) (THead (Bind b) t2 x0) H8) in 
-(let H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let 
-H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
-Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 
-(Bind b) x0 x0 (pr3_refl (CHead c (Bind b) t2) x0)) t2 x0 (refl_equal T 
-(THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) (sn3 (CHead c (Bind b) t2) 
-x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 (CHead c (Bind b) 
-t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) x) x0 (\lambda (t2: 
-T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
-(pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 (THead (Bind b) x t2) 
-(\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind b) x t2))).(\lambda 
-(P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind b) x x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T 
-t2 (\lambda (t0: T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 
-\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
-Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x 
-(pr3_refl c x) (Bind b) x0 t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) 
-in (land_ind (sn3 c x) (sn3 (CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) 
-x) t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) 
-t2)).H10)) H8))))))))))))))) y H0))))) H))))).
-
-lemma sn3_gen_flat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
-(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: 
-T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead 
-(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: 
-T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 
-c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T 
-t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land 
-(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: 
-(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
-T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 
-x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in 
-(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead 
-(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: 
-T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
-(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T 
-(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) x 
-x0) (THead (Flat f) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
-T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
-T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
-x) P)))))) (pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl (CHead c (Flat f) 
-t2) x0)) t2 x0 (refl_equal T (THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) 
-(sn3 c x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) 
-H8)))))) (sn3_sing c x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to 
-(\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead 
-(Flat f) x t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
-t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
-\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) x t2) H8) in (let H10 
-\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 
-\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
-Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_thin_dx c x0 t2 
-H7 x f) x t2 (refl_equal T (THead (Flat f) x t2))) in (land_ind (sn3 c x) 
-(sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 c 
-t2)).H10)) H8))))))))))))))) y H0))))) H))))).
-
-lemma sn3_gen_head:
- \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
-(THead k u t)) \to (sn3 c u)))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: 
-T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: 
-B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in 
-(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 
-c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: 
-F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
-(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
-(sn3 c t)).H1)) H0)))))))) k).
-
-lemma sn3_gen_cflat:
- \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
-c (Flat f) u) t) \to (sn3 c t)))))
-\def
- \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
-(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: 
-T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T 
-t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
-(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
-\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
-(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
-
-lemma sn3_gen_lift:
- \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
-(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
-\def
- \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 
-t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) 
-(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq 
-T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) 
-(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to 
-(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: 
-T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to 
-(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 
-c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d 
-x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T 
-t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) 
-\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d 
-x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq 
-T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) 
-H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T 
-x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d 
-t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let 
-H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h 
-d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to 
-(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T 
-x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 
-H4)))))))))))))) y H0)))) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
deleted file mode 100644 (file)
index 71d9e93..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/lift1.ma
+++ /dev/null
@@ -1,43 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sn3/props.ma".
-
-include "basic_1/drop1/fwd.ma".
-
-include "basic_1/lift1/props.ma".
-
-lemma sns3_lifts1:
- \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
-(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
-\def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
-(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c 
-(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda 
-(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) 
-in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList 
-ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c 
-H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
-(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to 
-(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 
-p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def 
-(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
-(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 
-(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda 
-(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: 
-TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) 
-(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
deleted file mode 100644 (file)
index e007361..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
+++ /dev/null
@@ -1,60 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sn3/fwd.ma".
-
-include "basic_1/nf2/dec.ma".
-
-include "basic_1/nf2/pr3.ma".
-
-lemma sn3_nf2:
- \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
-(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 
-H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) 
-in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: 
-Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
-(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
-
-lemma nf2_sn3:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
-t u)) (\lambda (u: T).(nf2 c u)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda 
-(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let 
-H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 
-c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
-(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
-T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
-(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c 
-t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 
-x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 
-x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: 
-T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 
-u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x 
-x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
-(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) 
-H2)))))) t H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
deleted file mode 100644 (file)
index 1e14061..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/props.ma
+++ /dev/null
@@ -1,2403 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sn3/nf2.ma".
-
-include "basic_1/nf2/iso.ma".
-
-include "basic_1/pr3/iso.ma".
-
-lemma sn3_pr3_trans:
- \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
-t2) \to (sn3 c t2)))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda 
-(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: 
-T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall 
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
-(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: 
-T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: 
-(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let 
-H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T 
-t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let 
-H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 
-\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) 
-H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 
-H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: 
-Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
-
-lemma sn3_pr2_intro:
- \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to 
-(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c 
-t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to 
-(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in 
-((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall 
-(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 
-c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) 
-(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall 
-(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t 
-t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: 
-T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 
-t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to 
-((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: 
-((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 
-t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: 
-Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq 
-T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: 
-(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to 
-(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: 
-T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t 
-t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: 
-T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) 
-\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) 
-H9))))))))))) t1 t2 H1 H3)) H2)))))))).
-
-theorem sn3_cast:
- \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to 
-(sn3 c (THead (Flat Cast) u t))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda 
-(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 
-t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: 
-T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: 
-((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 
-t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 
-t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: 
-T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def 
-(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c 
-t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 
-x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 
-\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to 
-(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T 
-(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def 
-(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) 
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: 
-(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat 
-Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 
-H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 
-H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) 
-(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) 
-((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) 
-(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: 
-T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall 
-(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: 
-T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead 
-(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c 
-(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to 
-(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) 
-(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda 
-(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda 
-(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 
-\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead 
-(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 
-\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 
-(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec 
-t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to 
-(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def 
-(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat 
-Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def 
-(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 
-(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: 
-(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 
-H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 
-t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
-H7))))))))) t H2)))))) u H))).
-
-lemma sn3_cflat:
- \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
-T).(sn3 (CHead c (Flat f) u) t)))))
-\def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: 
-F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 
-(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
-(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
-
-lemma sn3_shift:
- \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
-(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let 
-H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c 
-(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
-v) t)).H2)) H0))))))).
-
-lemma sn3_change:
- \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
-T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 
-(CHead c (Bind b) v2) t)))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
-(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda 
-(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind 
-b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 
-(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to 
-(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 
-(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: 
-Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 
-(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
-v1)))))))))) t H0))))))).
-
-lemma sn3_gen_def:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef 
-i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) 
-(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef 
-i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
-Abbr c d v i H))))))).
-
-lemma sn3_cdelta:
- \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
-(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: 
-C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
-\def
- \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: 
-T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 
-\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: 
-C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to 
-(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind 
-(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall 
-(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) 
-\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) 
-v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) 
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
-(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda 
-(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 
-c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 
-H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda 
-(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda 
-(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) 
-v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) 
-v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 
-(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) 
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 
-c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s 
-(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: 
-C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to 
-(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def 
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 
-(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: 
-(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b 
-(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) 
-H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 
-t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) 
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda 
-(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) 
-in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda 
-(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 
-H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
-(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: 
-C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d 
-v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d 
-(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def 
-(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
-H0)))))).
-
-lemma sn3_cpr3_trans:
- \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
-(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
-t)))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
-u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) 
-t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) 
-(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) 
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) 
-t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
-t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
-t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
-
-theorem sn3_bind:
- \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: 
-T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
-\def
- \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c 
-u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) 
-t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: 
-((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 
-t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c 
-(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: 
-T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) 
-t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: 
-T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) 
-t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) 
-t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda 
-(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda 
-(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) 
-in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) 
-(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c 
-(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b 
-(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: 
-Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall 
-(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
-b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let 
-H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 
-(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b 
-(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to 
-(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def 
-(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall 
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 
-x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall 
-(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 
-(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: 
-Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind 
-Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in 
-(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let 
-H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) 
-(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let 
-H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in 
-(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 
-\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda 
-(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T 
-(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: 
-Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: 
-T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) 
-H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 
-t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) 
-t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: 
-Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: 
-(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) 
-in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let 
-H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: 
-T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda 
-(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans 
-c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 
-H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 
-H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst 
-t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b 
-Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 
-in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind 
-b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) 
-t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq 
-T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: 
-(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: 
-T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead 
-(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
-T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in 
-(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 
-(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to 
-(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda 
-(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def 
-H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r 
-T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) 
-\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T 
-t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead 
-(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: 
-(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 
-H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 
-\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 
-x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: 
-(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c 
-(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c 
-(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind 
-b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq 
-T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 
-x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) 
-(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O 
-t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c 
-(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) 
-c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t 
-H2)))))) u H)))).
-
-theorem sn3_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
-t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
-(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 
-c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat 
-Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c 
-y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) 
-w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead 
-(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat 
-Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall 
-(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: 
-T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w 
-x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda 
-(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x 
-x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: 
-(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c 
-w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: 
-T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: 
-T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 
-x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 
-(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in 
-(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 
-x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: 
-((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 
-t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 
-x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
-(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead 
-(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x 
-(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
-(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
-(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) 
-t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq 
-T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: 
-(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda 
-(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to 
-(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T 
-(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def 
-(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) 
-(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) 
-x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind 
-T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
-(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) 
-x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def 
-H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 
-x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 
-x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 
-(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: 
-T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def 
-(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
-(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 
-x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) 
-t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let 
-H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind 
-T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 
-x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T 
-x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
-(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall 
-(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 
-H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead 
-(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind 
-Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 
-H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
-Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x x4) H32) in (let H34 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
-(\forall (P0: Prop).P0))) H31 x0 H33) in (let H35 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H20 x0 H33) in (H34 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 
-Abbr x))) x x4 (refl_equal T (THead (Bind Abbr) x x4)) t2 (sn3_sing c t2 
-H7))) H30))) x1 H27)))) (\lambda (H27: (((eq T x x1) \to (\forall (P: 
-Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) (\lambda (H28: (eq T (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x1 x4))).(\lambda (P: Prop).(let H29 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
-_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x1 x4) H28) in (\lambda (H31: (eq T x x1)).(let H32 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H20 x0 H30) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x 
-t0) \to (\forall (P0: Prop).P0))) H27 x H31) in (let H34 \def (eq_ind_r T x1 
-(\lambda (t0: T).(pr2 c x t0)) H14 x H31) in (H33 (refl_equal T x) P)))))) 
-H29)))) (pr3_head_12 c x x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 
-(CHead c (Bind Abbr) x1) x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead 
-(Bind Abbr) x1 x4)) t2 (sn3_sing c t2 H7))) H26))) x3 H23)))) (\lambda (H23: 
-(((eq T t2 x3) \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec x x1) in 
-(let H24 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda 
-(H25: (eq T x x1)).(let H26 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
-t0)) H14 x H25) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 
-(THead (Bind Abst) x3 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H27 \def 
-H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 
-x4)).(let H29 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (eq_ind T x0 
-(\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) x3 t0)))) (H8 
-x3 H23 (pr3_pr2 c t2 x3 H19)) x4 H28))) (\lambda (H28: (((eq T x0 x4) \to 
-(\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x x4) (\lambda (H29: (eq T 
-(THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4))).(\lambda (P: Prop).(let 
-H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 
-| (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
-(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) 
-P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
-(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
-(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) 
-(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
-Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
-\Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in 
-((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
-(THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq 
-T x x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let 
-H31 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
-Prop).P0))) H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 
-c x t0)) H14 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x 
-x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) 
-x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 
-x3 H23 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 
-H13))))))) H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 
-x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
-\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
-(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
-T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t2 | (TLRef _) 
-\Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) t2 x0) 
-(THead (Bind Abst) x1 x2) H13) in ((let H19 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) 
-x1 x2) H13) in (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 
-x4)))) H16 x0 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in 
-(or_ind (eq T x x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Bind Abbr) x3 x4)) (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 
-(\lambda (t0: T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: 
-T).(sn3 c (THead (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let 
-H25 \def H_x0 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let 
-H27 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c (Bind b) u) x0 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: 
-T).(sn3 c (THead (Bind Abbr) x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) 
-x4 H26))) (\lambda (H26: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 
-(THead (Bind Abbr) x x4) (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead 
-(Bind Abbr) x x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
-(\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H21 
-Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) \to (\forall (P: 
-Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq T (THead (Bind 
-Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: Prop).(let H25 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
-_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
-(THead (Bind Abbr) x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
-x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def (eq_ind_r T x4 
-(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
-t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T x 
-t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 \def (eq_ind_r T x3 
-(\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 (refl_equal T x) P)))))) 
-H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) (Bind Abbr) x0 x4 (pr3_pr2 
-(CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) H22)))))) H18)) t3 
-H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: 
-(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq 
-T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c 
-(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
-Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | 
-(THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
-in ((let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H22 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 
-x0) (THead (Bind x1) x2 x3) H14) in (\lambda (H23: (eq T t2 x2)).(\lambda 
-(H24: (eq B Abst x1)).(let H25 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 
-(CHead c (Bind x1) x6) t0 x4)) H18 x0 H22) in (let H26 \def (eq_ind_r T x2 
-(\lambda (t0: T).(pr2 c t0 x6)) H17 t2 H23) in (let H27 \def (eq_ind_r B x1 
-(\lambda (b: B).(pr2 (CHead c (Bind b) x6) x0 x4)) H25 Abst H24) in (let H28 
-\def (eq_ind_r B x1 (\lambda (b: B).(not (eq B b Abst))) H13 Abst H24) in 
-(eq_ind B Abst (\lambda (b: B).(sn3 c (THead (Bind b) x6 (THead (Flat Appl) 
-(lift (S O) O x5) x4)))) (let H29 \def (match (H28 (refl_equal B Abst)) in 
-False with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
-H11))))))))) w H4))))))))))) y H0))))) H)))).
-
-lemma sn3_appl_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
-T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead 
-(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef 
-i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: 
-T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall 
-(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) 
-t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
-(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 
-x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda 
-(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) 
-H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) 
-(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq 
-T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
-Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda 
-(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 
-x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall 
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T 
-t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat 
-Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: 
-Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c 
-t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t 
-(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c 
-(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) 
-\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H 
-x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T 
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
-(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: 
-(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: 
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
-H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) 
-t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r 
-T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind 
-T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Bind Abst) x0 x1) H7) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H12)) 
-t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H8: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda (H9: 
-(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def (eq_ind T t2 (\lambda (t: 
-T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) H3 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H9) in 
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H8) in 
-(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3))) H14)) t2 H9)))))))))))))) H6)) H5))))))))) v H0))))).
-
-lemma sn3_appl_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
-(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c 
-(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v 
-(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead 
-(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro 
-T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 
-c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: 
-T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat 
-Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: 
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
-t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat 
-Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef 
-i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift 
-(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: 
-T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
-(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) 
-H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t 
-t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 
-(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat 
-Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) 
-x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead 
-(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) 
-in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
-(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c 
-x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 
-(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: 
-Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat 
-Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i 
-H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: 
-T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
-T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: 
-(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead 
-(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
-Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x 
-in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def 
-(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead 
-(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T 
-(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 
-H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
-(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
-(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
-Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) H20) in (let H22 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
-(\forall (P0: Prop).P0))) H19 x H21) in (let H23 \def (eq_ind_r T x0 (\lambda 
-(t: T).(pr2 c x t)) H12 x H21) in (H22 (refl_equal T x) P)))))) (pr3_pr2 c 
-(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
-w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))) x0 (refl_equal T 
-(THead (Flat Appl) x0 (lift (S i) O w))))) H18))) x1 H16))) (\lambda (H16: 
-(ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) 
-u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
-u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
-u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) x3))).(\lambda (H18: (eq T 
-x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 (\lambda (t: T).((eq T 
-(THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
-Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T (lift (S i) O x3) 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 \def (eq_ind C 
-(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x2 (Bind 
-Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) 
-H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
-(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
-(Bind Abbr) x3) H17)) in ((let H22 \def (f_equal C T (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
-Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H 
-(CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 \def 
-(eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 w 
-H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S i) 
-O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
-(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
-H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
-(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
-\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x 
-(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c 
-(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x 
-x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
-(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
-Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef _) 
-\Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S 
-i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) H28) in (let H30 \def 
-(eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H27 
-x H29) in (let H31 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x 
-H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift 
-(S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 
-(Flat Appl) (lift (S i) O w))))) H26)))) x3 H22)))) H21))) x1 H18)))))) H16)) 
-H15)) t2 H11))))))) H10)) (\lambda (H10: (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
-(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H12: 
-(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c x x2)).(\lambda (_: 
-((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
-H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) 
-t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 x3) H12) in (eq_ind_r 
-T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H16 \def (eq_ind 
-T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Bind Abst) x0 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) 
-t2 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
-B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
-T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
-(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
-Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda 
-(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
-(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: 
-T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 
-(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
-(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
-(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H12) in 
-(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
-x3))) H18)) t2 H13)))))))))))))) H10)) H9))))))))))))) y H1)))) H0))))))).
-
-theorem sn3_appl_cast:
- \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v 
-u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
-(Flat Appl) v (THead (Flat Cast) u t))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead 
-(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 
-c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to 
-(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: 
-T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead 
-(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to 
-(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v 
-(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to 
-(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat 
-Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall 
-(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: 
-T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead 
-(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: 
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
-t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 
-(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x 
-t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 
-t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead 
-(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x 
-t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) 
-(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) 
-(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq 
-T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat 
-Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 
-(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
-x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda 
-(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 
-c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c 
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: 
-T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T 
-t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat 
-Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 
-x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 
-(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let 
-H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: 
-T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead 
-(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 
-t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 
-(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in 
-(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda 
-(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 
-x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat 
-Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x 
-(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T 
-T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
-(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) 
-x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq 
-T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: 
-(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda 
-(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to 
-(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T 
-(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def 
-(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c 
-x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead 
-(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 
-x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat 
-Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 
-x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: 
-Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat 
-Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x 
-\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let 
-H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
-x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall 
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) 
-(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
-x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) 
-(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28) in ((let H30 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) 
-(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def 
-(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat 
-Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall 
-(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: 
-T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead 
-(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 
-(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) 
-(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) 
-H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 
-x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead 
-(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat 
-Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead 
-(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
-(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) 
-\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x x5) H37) in 
-(let H39 \def (eq_ind_r T x5 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
-(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x (THead (Flat Cast) x0 t3))) 
-\to (\forall (P: Prop).P))) H34 x1 H38) in (let H40 \def (eq_ind_r T x5 
-(\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in (eq_ind T x1 (\lambda (t3: 
-T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t3)))) (H39 (refl_equal 
-T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) (sn3 c (THead (Flat Appl) 
-x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda (H37: (((eq T (THead 
-(Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall (P: Prop).P)))).(H9 
-(THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat Appl) x x1) (THead (Flat 
-Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 (refl_equal T (THead (Flat 
-Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) (\lambda (H28: (((eq T 
-(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall (P: 
-Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x x1) (THead (Flat 
-Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead (Flat Appl) x x1) 
-(THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) 
-x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat 
-Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) x x1) (THead (Flat 
-Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
-\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in 
-((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
-(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq 
-T x x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 
-H32) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead 
-(Flat Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
-(THead (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: 
-Prop).P))) H28 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c 
-x t3)) H18 x H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) 
-t3 (THead (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c 
-(THead (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x 
-Appl)) x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead 
-(Flat Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T 
-(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: 
-Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x 
-x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat 
-Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 
-c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) 
-H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat 
-Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T 
-(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) 
-x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead 
-(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead 
-(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
-\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in 
-((let H26 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
-(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq 
-T x x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 
-H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat 
-Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall 
-(P: Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead 
-(Flat Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) 
-\to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 
-(\lambda (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: 
-T).(sn3 c (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) 
-H10) x2 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x 
-x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat 
-Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 
-H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
-(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda 
-(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x 
-x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
-u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead 
-(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 
-(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) 
-(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 
-x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 
-x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 
-H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat 
-Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) 
-(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
-(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: 
-(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq 
-T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda 
-(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c 
-(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T 
-(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: 
-Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) 
-H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) 
-x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) 
-x0 x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) 
-H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) 
-O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) 
-H4))))))))) y H0))))) H)))).
-
-theorem sn3_appl_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) 
-(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v 
-(THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
-T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) 
-O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) 
-(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
-(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
-(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat 
-Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 
-t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead 
-(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
-t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) 
-(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t 
-(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c 
-(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: 
-T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 
-c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) 
-t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat 
-Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) 
-t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) 
-\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 
-(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 
-t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to 
-(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O 
-x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 
-x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead 
-(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: 
-T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 
-(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 
-(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 
-(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) 
-H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T 
-t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to 
-(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) 
-H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda 
-(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) 
-t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x 
-(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) 
-t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
-t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) 
-(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c 
-t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat 
-Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead 
-(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) 
-(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in 
-(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind 
-b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
-(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
-(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 
-x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) 
-x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: 
-Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 
-x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def 
-(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) 
-\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 
-x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: 
-T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 
-(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 
-\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T 
-t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) 
-(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) 
-((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead 
-(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
-(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 
-H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) 
-t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
-Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 
-\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 
-c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x 
-x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) 
-x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to 
-(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c 
-(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead 
-(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: 
-Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O 
-x) | (TLRef _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) 
-| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
-(THead (Flat Appl) (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 
-(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x 
-x1 (S O) O H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
-t0)) H15 x (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) 
-(pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift 
-(CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x 
-x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 
-x0 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 
-H29)))) (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead 
-(Flat Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) 
-(lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
-Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) 
-\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 
-_) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
-Appl) (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
-(THead (Flat Appl) (lift (S O) O x1) x4) H30) in (\lambda (H33: (eq T (lift 
-(S O) O x) (lift (S O) O x1))).(let H34 \def (eq_ind_r T x4 (\lambda (t0: 
-T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H29 x0 H32) in (let H35 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) 
-t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P0: 
-Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 
-(CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let H37 \def (eq_ind_r T x1 
-(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
-(Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall (P0: Prop).P0))) H35 x 
-(lift_inj x x1 (S O) O H33)) in (let H38 \def (eq_ind_r T x1 (\lambda (t0: 
-T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H33)) in (H34 (refl_equal T x0) 
-P)))))))) H31)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S 
-O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl) x1 x4 
-(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x4)))) H28))) x3 H25)))) 
-(\lambda (H25: (((eq T t1 x3) \to (\forall (P: Prop).P)))).(H2 x3 H25 H21 x4 
-x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead (Flat Appl) (lift (S O) O x1) 
-x4) (let H_x1 \def (term_dec x0 x4) in (let H26 \def H_x1 in (or_ind (eq T x0 
-x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
-(THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda (H27: (eq T x0 x4)).(let 
-H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) 
-H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
-(THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 \def (term_dec x x1) in 
-(let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
-Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) 
-x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T x1 (\lambda (t0: 
-T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) (sn3_sing (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) x1 H30))) (\lambda 
-(H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift 
-(S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
-(THead (Flat Appl) (lift (S O) O x1) x0))).(\lambda (P: Prop).(let H32 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
-(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
-(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to 
-(\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O H32)) in (let H34 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
-H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift 
-(S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O 
-(drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 
-(pr3_refl (CHead c (Bind b) t1) x0) Appl))) H29))) x4 H27))) (\lambda (H27: 
-(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S 
-O) O x1) x4) (\lambda (H28: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
-(THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: Prop).(let H29 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
-(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
-(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
-x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow 
-t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
-O x1) x4) H28) in (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O 
-x1))).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
-(\forall (P0: Prop).P0))) H27 x0 H30) in (let H33 \def (eq_ind_r T x4 
-(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 
-\def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) 
-O H31)) in (H32 (refl_equal T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) 
-t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
-O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
-x0 x4 H22 Appl))) H26)))))) H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 
-(CHead c (Bind b) t1) x0 (lift (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) 
-t1) (THead (Flat Appl) x1 x2) (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S 
-O) O x1) (lift (S O) (s (Flat Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c 
-(Bind b) t1) t0)) (sn3_pr3_trans (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O x1) x0) (let H_x0 \def (term_dec x x1) in (let H20 \def H_x0 in 
-(or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0)) (\lambda (H21: (eq T x 
-x1)).(let H22 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) 
-in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O t0) x0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) 
-(lift (S O) O x) x0) H9) x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall 
-(P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: 
-(eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
-O x1) x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 
-(S O))) O x) | (TLRef _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 (S 
-O))) O x) | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O 
-x) x0) (THead (Flat Appl) (lift (S O) O x1) x0) H22) in (let H24 \def 
-(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
-H21 x (lift_inj x x1 (S O) O H23)) in (let H25 \def (eq_ind_r T x1 (\lambda 
-(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal 
-T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O 
-x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind 
-b) t1) x0) Appl))) H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O 
-x2)) (pr3_thin_dx (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) 
-O x1) Appl)) (lift (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) 
-x1 x2 (S O) O)) c (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 
-H14))))))) H13)) (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
-T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
-b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
-T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) 
-(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 
-x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c 
-x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind 
-b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead 
-(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
-(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
-(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
-k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
-((let H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H21 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
-(THead (Bind Abst) x1 x2) H14) in (\lambda (_: (eq T t1 x1)).(\lambda (H23: 
-(eq B b Abst)).(let H24 \def (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: 
-B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let 
-H25 \def (eq_ind B b (\lambda (b0: B).((eq T (THead (Flat Appl) x (THead 
-(Bind b0) t1 x0)) (THead (Bind Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 
-Abst H23) in (let H26 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: 
-T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (P: 
-Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) (lift (S O) O 
-x) x0) t4) \to (sn3 (CHead c (Bind b0) t1) t4))))) H9 Abst H23) in (let H27 
-\def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat 
-Appl) (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c 
-(Bind b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (x5: 
-T).(\forall (x6: T).((eq T t4 (THead (Flat Appl) (lift (S O) O x5) x6)) \to 
-(sn3 c (THead (Flat Appl) x5 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in 
-(let H28 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) 
-\to (\forall (P: Prop).P))) \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall 
-(v0: T).((sn3 (CHead c (Bind b0) t4) (THead (Flat Appl) (lift (S O) O v0) 
-t0)) \to (sn3 c (THead (Flat Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 
-Abst H23) in (let H29 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) 
-H Abst H23) in (let H30 \def (match (H29 (refl_equal B Abst)) in False with 
-[]) in H30)))))))))) H20)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 
-B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda 
-(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S 
-O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) 
-y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
-b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
-(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) 
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: 
-(eq T (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T 
-t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
-c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
-(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
-H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
-(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
-k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
-((let H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
-(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H23 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
-_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
-(THead (Bind x1) x2 x3) H15) in (\lambda (H24: (eq T t1 x2)).(\lambda (H25: 
-(eq B b x1)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c 
-(Bind x1) x6) t0 x4)) H19 x0 H23) in (let H27 \def (eq_ind_r T x2 (\lambda 
-(t0: T).(pr2 c t0 x6)) H18 t1 H24) in (let H28 \def (eq_ind_r B x1 (\lambda 
-(b0: B).(pr2 (CHead c (Bind b0) x6) x0 x4)) H26 b H25) in (eq_ind B b 
-(\lambda (b0: B).(sn3 c (THead (Bind b0) x6 (THead (Flat Appl) (lift (S O) O 
-x5) x4)))) (sn3_pr3_trans c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
-O x5) x4)) (sn3_bind b c t1 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) 
-O x5) x4) (let H_x \def (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T 
-x x5) ((eq T x x5) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
-(THead (Flat Appl) (lift (S O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let 
-H31 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind 
-T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
-O) O t0) x4))) (let H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in 
-(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c 
-(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 
-x4)).(let H34 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) 
-x0 t0)) H28 x0 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) 
-t1) (THead (Flat Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) 
-(THead (Flat Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T 
-x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) 
-x4) (\lambda (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
-Appl) (lift (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in (let H36 \def 
-(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
-H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c 
-(Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
-(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
-x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
-(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
-(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) 
-Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: 
-Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
-x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O 
-x) | (TLRef _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O x) 
-| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
-(THead (Flat Appl) (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
-\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
-O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in (\lambda (H34: 
-(eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def (eq_ind_r T x5 
-(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x 
-x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x 
-t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def (eq_ind_r T x4 
-(\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 H33) in (H35 
-(refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) 
-(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
-x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) (lift (S O) O 
-x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind b) O c c 
-(drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c (Bind b) 
-x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat Appl) (lift 
-(S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
-O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-(pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O x5) x4)))) 
-x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))))) y 
-H4))))) H3))))))) u H0))))).
-
-theorem sn3_appl_appl:
- \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
-(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
-(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
-(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 
-u1)))))))))
-\def
- \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
-(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T 
-(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall 
-(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) 
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c 
-y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to 
-(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso 
-y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) 
-\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: 
-T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: 
-T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to 
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c 
-(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: 
-T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: 
-T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to 
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c 
-(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall 
-(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
-(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
-(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T 
-t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
-(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to 
-(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 
-t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 
-(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c 
-v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso 
-t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) 
-\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: 
-((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 
-t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to 
-(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 
-u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
-Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: 
-((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: 
-Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T 
-t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to 
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead 
-(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall 
-(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to 
-(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: 
-Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat 
-Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T 
-t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: 
-Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 
-(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall 
-(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to 
-(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 
-t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 
-(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) 
-\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in 
-(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat 
-Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x 
-x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead 
-(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c 
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def 
-(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda 
-(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
-x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T 
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
-T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
-z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
-y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
-(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) 
-x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T 
-t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: 
-(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: 
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
-Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat 
-Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 
-H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda 
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead 
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c 
-(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
-(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
-(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 
-x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat 
-Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 
-x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 
-(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
-Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat 
-Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def 
-(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 
-\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) 
-((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: 
-Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda 
-(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
-(TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x 
-x0) (THead (Flat Appl) x3 x4) H27) in ((let H29 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
-(THead _ _ t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 
-x4) H27) in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda 
-(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat 
-Appl) x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) 
-in (let H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in 
-(eq_ind T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) 
-x3 t)))) (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat 
-Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t 
-x0))) \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 
-(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c 
-(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 
-x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall 
-(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) 
-(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: 
-T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) 
-t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let 
-H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind 
-T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) 
-(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c 
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: 
-(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) 
-(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda 
-(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead 
-(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 
-u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 
-H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat 
-Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 
-(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 
-(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c 
-t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c 
-(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 
-x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 
-u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) 
-(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) 
-(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) 
-(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 
-(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x 
-x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead 
-(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat 
-Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
-T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat 
-Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
-(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: 
-(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda 
-(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 
-x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) 
-t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
-Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind 
-Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def 
-(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) 
-x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: 
-Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c 
-t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 
-(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
-(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall 
-(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall 
-(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) 
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) 
-H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c 
-(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to 
-(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead 
-(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: 
-T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso 
-(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
-(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
-t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat 
-Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing 
-c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) 
-(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind 
-Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind 
-Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 
-(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso 
-(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 
-x6))).(\lambda (P: Prop).(let H33 \def (match H32 with [(iso_sort n1 n2) 
-\Rightarrow (\lambda (H33: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
-Abst) x3 x4)))).(\lambda (H34: (eq T (TSort n2) (THead (Bind Abbr) x5 
-x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) 
-in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) 
-H34))) | (iso_lref i1 i2) \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead 
-(Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) 
-(THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) 
-x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to 
-P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H33: (eq T 
-(THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda 
-(H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 x6))).((let H35 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t4 | (TLRef 
-_) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead 
-(Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow v4 | (TLRef _) 
-\Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat 
-Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def (f_equal T K 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat 
-Appl) x (THead (Bind Abst) x3 x4)) H33) in (eq_ind K (Flat Appl) (\lambda 
-(k0: K).((eq T v4 x) \to ((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T 
-(THead k0 v5 t5) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H38: (eq T v4 
-x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq 
-T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda 
-(H39: (eq T t4 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 
-x4) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 
-x6)) \to P)) (\lambda (H40: (eq T (THead (Flat Appl) v5 t5) (THead (Bind 
-Abbr) x5 x6))).(let H41 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P 
-H41))) t4 (sym_eq T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x 
-H38))) k (sym_eq K k (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 
-(refl_equal T (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T 
-(THead (Bind Abbr) x5 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 
-x6)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat 
-Appl) x1 (THead (Bind Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) 
-(THead (Bind Abbr) x5 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: 
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
-(u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
-(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
-(x3: B).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: 
-T).(\lambda (x8: T).(\lambda (H22: (not (eq B x3 Abst))).(\lambda (H23: (eq T 
-x0 (THead (Bind x3) x4 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (H25: (pr2 c x 
-x7)).(\lambda (H26: (pr2 c x4 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) 
-x5 x6)).(let H28 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) 
-t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
-Prop).P))) H19 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
-H24) in (eq_ind_r T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind 
-T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) 
-(THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
-x7) x6)))) \to (\forall (P: Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in 
-(let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead 
-(Flat Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat 
-Appl) x t) t4) \to (sn3 c t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let 
-H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat 
-Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x 
-t) t4) \to (\forall (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 
-x10)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) 
-\to ((((iso t4 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) 
-v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind 
-x3) x4 x5) H23) in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: 
-T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) 
-u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 
-(THead (Bind x3) x4 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: 
-T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c 
-t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso 
-(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
-(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
-t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat 
-Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 
-(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c 
-(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat 
-Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead 
-(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) 
-x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl 
-x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
-(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift 
-(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c 
-(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) 
-x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x 
-x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) 
-(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl 
-(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind 
-x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
-x6)))).(\lambda (P: Prop).(let H35 \def (match H34 with [(iso_sort n1 n2) 
-\Rightarrow (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
-x3) x4 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead 
-(Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
-(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TSort n2) (THead (Bind 
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | 
-(iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef i1) (THead (Flat 
-Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (TLRef i2) (THead 
-(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def 
-(eq_ind T (TLRef i1) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T 
-(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
-P) H37)) H36))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T 
-(THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda 
-(H36: (eq T (THead k v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S 
-O) O x7) x6)))).((let H37 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
-\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) 
-\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
-x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T 
-t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t5) (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda (H40: (eq T v4 
-x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T 
-(THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) 
-O x7) x6))) \to P))) (\lambda (H41: (eq T t4 (THead (Bind x3) x4 
-x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead (Flat 
-Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) 
-\to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 \def (eq_ind T (THead 
-(Flat Appl) v5 t5) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
-| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
-x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H42) in (False_ind P H43))) 
-t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 (sym_eq T v4 x H40))) k 
-(sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) in (H35 (refl_equal T 
-(THead (Flat Appl) x (THead (Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) 
-x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))) (THead (Flat Appl) x1 
-(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr3_pr2 c 
-(THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
-x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift 
-(S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind x3) x8 
-(THead (Flat Appl) (lift (S O) O x7) x6)))))))))) x2 H24)))))))))))))) H21)) 
-H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T T T T (\lambda (y1: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
-x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T 
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
-z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (x4: T).(\lambda (H16: (eq T (THead (Flat Appl) x x0) (THead 
-(Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 (THead (Bind Abbr) x3 
-x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H20 \def (eq_ind T t3 (\lambda 
-(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall 
-(P: Prop).P))) H12 (THead (Bind Abbr) x3 x4) H17) in (eq_ind_r T (THead (Bind 
-Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat 
-Appl) x x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x1 
-x2) H16) in (False_ind (sn3 c (THead (Bind Abbr) x3 x4)) H21)) t3 
-H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat 
-Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
-t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
-T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
-z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
-Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead 
-(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
-b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
-(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: 
-(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T 
-t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
-(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c 
-(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T 
-(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
-Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
-H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
-x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x 
-x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) 
-H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) 
-O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 
-H4))))))))) y H0))))) H))))).
-
-theorem sn3_appl_beta:
- \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
-(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w 
-t))))))))))
-\def
- \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
-(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: 
-T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind 
-Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind 
-Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind 
-Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind 
-Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w 
-H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead 
-(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind 
-Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat 
-Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c 
-(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) 
-H1))))))))).
-
-theorem sn3_appl_appls:
- \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads 
-(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall 
-(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) 
-\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
-(sn3 c (THead (Flat Appl) v2 u1))))))))))
-\def
- \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads 
-(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead 
-(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: 
-(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 
-(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads 
-(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
-Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 
-H1))))))))).
-
-lemma sn3_appls_lref:
- \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: 
-TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
-(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads 
-(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) 
-(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 
-c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) 
-(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef 
-i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil 
-(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 
-in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) 
-TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref 
-c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: 
-(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land 
-(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 
-(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads 
-(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 
-c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) 
-(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) 
-(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c 
-t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda 
-(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) 
-(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat 
-Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) 
-(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 
-(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t 
-u2))))))))) H5))) H3))))))) t0))) us)))).
-
-theorem sn3_appls_cast:
- \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat 
-Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 
-c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
-\def
- \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
-(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads 
-(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u 
-t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda 
-(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: 
-TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads 
-(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) 
-\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to 
-(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to 
-(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) 
-\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u 
-t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil 
-u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c 
-(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: 
-T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil 
-u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: 
-T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat 
-Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to 
-(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall 
-(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall 
-(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c 
-(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u 
-t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) 
-(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 
-t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u 
-t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
-(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead 
-(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def 
-(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 
-\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat 
-Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) 
-(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c 
-(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let 
-H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in 
-(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads 
-(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 
-t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c 
-(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in 
-(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 
-(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) 
-(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) 
-(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: 
-Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons 
-t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) 
-(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t 
-Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
-
-theorem sn3_appls_bind:
- \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
-T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind 
-b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat 
-Appl) vs (THead (Bind b) u t))))))))))
-\def
- \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
-(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: 
-TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts 
-(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) 
-(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u 
-H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: 
-TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u 
-t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) 
-(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) 
-(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
-(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) 
-u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) 
-t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads 
-(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead 
-(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to 
-(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u 
-t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) 
-(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads 
-(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: 
-T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
-v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def 
-(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) 
-(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 
-(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat 
-Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u 
-t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda 
-(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads 
-(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c 
-(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c 
-(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: 
-T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u 
-t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind 
-b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b 
-H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v 
-(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) 
-(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) 
-t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead 
-(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 
-Appl)))))))))) H4))))))))) vs0))) vs)))))).
-
-theorem sn3_appls_beta:
- \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
-(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) 
-w t))))))))))
-\def
- \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: 
-TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead 
-(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: 
-(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c 
-w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: 
-TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 
-(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads 
-(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 
-c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to 
-(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat 
-Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: 
-T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: 
-(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
-t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead 
-(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u 
-(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c 
-w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) 
-v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat 
-Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
-\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
-(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads 
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: 
-T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads 
-(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in 
-(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) 
-(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: 
-(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead 
-(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 
-c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w 
-t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat 
-Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 
-\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c 
-(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
-t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
-t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
-
-lemma sn3_lift:
- \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
-nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
-\def
- \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda 
-(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) 
-\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
-T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d 
-t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
-Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall 
-(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: 
-C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c 
-d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T 
-(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i 
-t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T 
-(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) 
-(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda 
-(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h 
-i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T 
-(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 
-x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T 
-(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let 
-H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 
-(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) 
-H5))))))))))))) t H))).
-
-lemma sn3_abbr:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d 
-v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef 
-i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let 
-H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T 
-(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) 
-(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) 
-(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: 
-T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in 
-(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) 
-(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: 
-C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda 
-(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
-C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) 
-x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 
-(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S 
-i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let 
-H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H 
-(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 
-(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
-(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) 
-i H (CHead x0 (Bind Abbr) x1) H5)) in ((let H10 \def (f_equal C T (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow 
-t])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
-(Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d 
-x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind 
-Abbr) t))) H8 v H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) 
-(let H13 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
-v))) H12 d H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) 
-x1 H10)))) H9))) t2 H6)))))) H4)) H3))))))))))).
-
-lemma sn3_appls_abbr:
- \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
-vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
-(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
-(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 
-c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O 
-w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) 
-in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: 
-TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift 
-(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c 
-(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: 
-(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat 
-Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads 
-(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat 
-Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) 
-\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) 
-\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef 
-i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda 
-(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) 
-O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t 
-t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead 
-(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: 
-(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
-(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda 
-(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) 
-u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to 
-(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat 
-Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) 
-(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 
-(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) 
-H3)))))))) vs0))) vs)))))).
-
-lemma sns3_lifts:
- \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
-i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda 
-(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: 
-TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c 
-(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def 
-H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c 
-(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj 
-(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 
-H4)))) H2)))))) ts)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/spare.ma b/matita/matita/contribs/lambdadelta/basic_1/spare.ma
deleted file mode 100644 (file)
index 58a624e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/spare.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/theory.ma".
-
-axiom pc3_gen_appls_sort_abst:
- \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall 
-(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) 
-\to False)))))
-.
-
-axiom pc3_gen_appls_lref_abst:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall 
-(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to 
-False))))))))
-.
-
-axiom pc3_gen_appls_lref_sort:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: 
-TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads 
-(Flat Appl) ws (TSort n))) \to False))))))))
-.
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma
deleted file mode 100644 (file)
index e4e0ca8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/defs.ma
+++ /dev/null
@@ -1,39 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/G/defs.ma".
-
-include "basic_1/getl/defs.ma".
-
-inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def
-| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort 
-(next g n))))
-| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) 
-\to (sty0 g c (TLRef i) (lift (S i) O w))))))))
-| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) 
-\to (sty0 g c (TLRef i) (lift (S i) O v))))))))
-| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead 
-(Bind b) v t1) (THead (Bind b) v t2)))))))
-| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: 
-T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat 
-Appl) v t2))))))
-| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 
-v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c 
-(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
deleted file mode 100644 (file)
index 0d05ca1..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
+++ /dev/null
@@ -1,553 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty0/defs.ma".
-
-implied rec lemma sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
-(\forall (c: C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0: 
-(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w) 
-\to (P c (TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall 
-(d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) 
-\to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift 
-(S i) O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v: 
-T).(\forall (t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to 
-((P (CHead c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind 
-b) v t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
-Appl) v t1) (THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall 
-(v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1: 
-T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))))))) (c: C) (t: T) (t0: T) (s0: 
-sty0 g c t t0) on s0: P c t t0 \def match s0 with [(sty0_sort c0 n) 
-\Rightarrow (f c0 n) | (sty0_abbr c0 d v i g0 w s1) \Rightarrow (f0 c0 d v i 
-g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w s1)) | (sty0_abst c0 d v i g0 
-w s1) \Rightarrow (f1 c0 d v i g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w 
-s1)) | (sty0_bind b c0 v t1 t2 s1) \Rightarrow (f2 b c0 v t1 t2 s1 ((sty0_ind 
-g P f f0 f1 f2 f3 f4) (CHead c0 (Bind b) v) t1 t2 s1)) | (sty0_appl c0 v t1 
-t2 s1) \Rightarrow (f3 c0 v t1 t2 s1 ((sty0_ind g P f f0 f1 f2 f3 f4) c0 t1 
-t2 s1)) | (sty0_cast c0 v1 v2 s1 t1 t2 s2) \Rightarrow (f4 c0 v1 v2 s1 
-((sty0_ind g P f f0 f1 f2 f3 f4) c0 v1 v2 s1) t1 t2 s2 ((sty0_ind g P f f0 f1 
-f2 f3 f4) c0 t1 t2 s2))].
-
-lemma sty0_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
-(TSort n) x) \to (eq T x (TSort (next g n)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c 
-t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda 
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda 
-(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: 
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
-(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
-n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq T (TSort (next g n1)) (TSort 
-(next g n)))) (refl_equal T (TSort (next g n))) n0 H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
-n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
-(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
-H4) in (False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(_: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g 
-d v w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
-n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
-(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
-H4) in (False_ind (eq T (lift (S i) O v) (TSort (next g n))) H5))))))))))) 
-(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 t2)).(\lambda (_: (((eq 
-T t1 (TSort n)) \to (eq T t2 (TSort (next g n)))))).(\lambda (H3: (eq T 
-(THead (Bind b) v t1) (TSort n))).(let H4 \def (eq_ind T (THead (Bind b) v 
-t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
-(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) 
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort 
-(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let 
-H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Flat Appl) v 
-t2) (TSort (next g n))) H4))))))))) (\lambda (c0: C).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 
-(TSort n)) \to (eq T v2 (TSort (next g n)))))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
-T t2 (TSort (next g n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) 
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in (False_ind (eq T 
-(THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) c y x H0))) 
-H))))).
-
-lemma sty0_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
-(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda 
-(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c 
-t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u 
-t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O 
-u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda 
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u 
-t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) 
-O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: 
-nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort 
-n0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in 
-(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T (TSort (next g n0)) (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (TSort (next g n0)) (lift (S n) 
-O u))))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda 
-(i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
-T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
-e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
-e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
-(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
-(Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O w) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n0) O w) (lift (S n) O u)))))))) (or_introl (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n) O w) (lift (S n) O u)))))) (ex3_3_intro C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
-(lift (S n) O t))))) d v w H6 H2 (refl_equal T (lift (S n) O w)))) i 
-H5)))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: 
-T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
-e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
-e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
-(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
-(Bind Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) 
-(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i 
-H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda 
-(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T 
-(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v 
-t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in 
-(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T (THead (Bind b) v t2) (lift (S n) O t)))))) (ex3_3 C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Bind b) v 
-t2) (lift (S n) O u))))))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
-T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda 
-(_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
-u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H4 
-\def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Appl) v t2) (lift 
-(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O u))))))) H4))))))))) 
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
-v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 
-C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T v2 (lift (S n) 
-O u)))))))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 
-C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) 
-O u)))))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let 
-H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Cast) v2 t2) (lift 
-(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) H6)))))))))))) 
-c y x H0))) H))))).
-
-lemma sty0_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
-T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda 
-(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead 
-(Bind b) u t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) 
-x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) 
-(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) 
-(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda 
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g 
-(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u 
-t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) 
-(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in (False_ind 
-(ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: 
-T).(eq T (TSort (next g n)) (THead (Bind b) u t2)))) H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: 
-T).(sty0 g (CHead d (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind 
-b) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: 
-T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O 
-w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
-v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind 
-b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda 
-(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) 
-H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t2)) (\lambda (t2: T).(eq T (lift (S i) O v) (THead (Bind b) u t2)))) 
-H5))))))))))) (\lambda (b0: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H1: (sty0 g (CHead c0 (Bind b0) v) t0 
-t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: 
-T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) (\lambda (t3: 
-T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Bind b0) 
-v t0) (THead (Bind b) u t1))).(let H4 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | 
-(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b0])])) (THead (Bind b0) v t0) (THead (Bind b) u t1) H3) in ((let 
-H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v | 
-(TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v 
-t0) (THead (Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead (Bind b) u t1) 
-H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 b)).(let H9 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) 
-(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in (let H10 
-\def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t t2)) H1 t1 
-H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead (Bind b) u 
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) t) (Bind 
-b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H9 u H7) 
-in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) t) 
-t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
-T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind 
-b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 (\lambda (b1: 
-B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g 
-(CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 
-(THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B b0 (\lambda 
-(b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in (eq_ind_r B b 
-(\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead (Bind b) u t3))))) 
-(ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda 
-(t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u t3))) t2 H14 
-(refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) H4)))))))))) 
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to 
-(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: 
-T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Flat 
-Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T (THead (Flat Appl) 
-v t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
-H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u t3)))) 
-H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T 
-v2 (THead (Bind b) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
-(sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T 
-t2 (THead (Bind b) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) 
-(THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
-H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Bind b) u 
-t3)))) H6)))))))))))) c y x H0))) H))))))).
-
-lemma sty0_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: 
-T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g 
-c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead 
-(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T 
-(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat 
-Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
-u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T 
-t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: 
-nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Appl) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 
-t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Appl) u t2)))) 
-H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
-T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Appl) u 
-t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w 
-(THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat 
-Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in (False_ind (ex2 T 
-(\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O w) 
-(THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
-v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) 
-(\lambda (t2: T).(eq T w (THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T 
-(TLRef i) (THead (Flat Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u 
-t1) H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda 
-(t2: T).(eq T (lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) 
-(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq 
-T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 
-(Bind b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u 
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T 
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v 
-t2) (THead (Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
-T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 
-t2)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead (Flat Appl) u 
-t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v t0) 
-(THead (Flat Appl) u t1) H3) in (\lambda (H6: (eq T v u)).(let H7 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat 
-Appl) u t3)))))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: T).(sty0 
-g c0 t t2)) H1 t1 H5) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t2) (THead 
-(Flat Appl) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) 
-(\lambda (t3: T).(eq T (THead (Flat Appl) u t2) (THead (Flat Appl) u t3))) t2 
-H8 (refl_equal T (THead (Flat Appl) u t2))) v H6))))) H4))))))))) (\lambda 
-(c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T v2 (THead (Flat Appl) u 
-t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) u 
-t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-f) \Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) u t1) H5) in (False_ind (ex2 T (\lambda (t3: 
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead 
-(Flat Appl) u t3)))) H6)))))))))))) c y x H0))) H)))))).
-
-lemma sty0_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall 
-(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: 
-T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 
-g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 
-t2))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda 
-(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T 
-(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: 
-T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda 
-(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 
-g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq 
-T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: 
-T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) 
-(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) 
-(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat 
-Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H1) in (False_ind (ex3_2 
-T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq 
-T (TSort (next g n)) (THead (Flat Cast) v2 t2))))) H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda 
-(v2: T).(\lambda (_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: 
-T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat 
-Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 
-t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T 
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda 
-(t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S 
-i) O w) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda 
-(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d 
-(Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: 
-(((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda 
-(_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) 
-(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 
-t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda 
-(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: 
-T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O 
-v) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (b: B).(\lambda (c0: 
-C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g 
-(CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 
-t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind 
-b) v) v1 v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) 
-t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 
-t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T 
-T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T (THead (Bind b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda 
-(c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 
-g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T 
-T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) 
-v t0) (THead (Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v 
-t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow True 
-| Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind 
-(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 t3))))) H4))))))))) (\lambda 
-(c0: C).(\lambda (v0: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c0 v0 
-v2)).(\lambda (H2: (((eq T v0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T 
-(\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda 
-(t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead 
-(Flat Cast) v3 t2)))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
-(sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) v1 t1)) \to 
-(ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) v3 t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) 
-v0 t0) (THead (Flat Cast) v1 t1))).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | 
-(THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) 
-v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) v1 t1) H5) in 
-(\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T 
-t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: 
-T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) 
-(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 t3))))))) H4 
-t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H3 t1 
-H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t (THead (Flat Cast) 
-v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
-(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
-T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 v1 H8) in 
-(let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 H8) in 
-(ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
-(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 
-t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) 
-H6)))))))))))) c y x H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma
deleted file mode 100644 (file)
index f75c1c7..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/props.ma
+++ /dev/null
@@ -1,214 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty0/fwd.ma".
-
-include "basic_1/getl/drop.ma".
-
-lemma sty0_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e 
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
-e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) 
-\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda 
-(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: 
-(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d 
-(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g 
-c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort 
-(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v 
-w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: 
-nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 
-w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: 
-(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 
-(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le 
-i d0 (le_S_n i d0 (le_S_n (S i) (S d0) (le_S (S (S i)) (S d0) (le_n_S (S i) 
-d0 H4)))) c0 c h H3 (CHead d (Bind Abbr) v) H0) in (ex3_2_ind C C (\lambda 
-(e0: C).(\lambda (_: C).(drop i O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
-C).(drop h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
-(CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift 
-(S i) O w))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 
-x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
-(CHead d (Bind Abbr) v))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
-nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d v H8) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d0 (S i)) 
-v)))) (\lambda (c1: C).(drop h (minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 
-(TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x: C).(\lambda (H11: 
-(clear x0 (CHead x (Bind Abbr) (lift h (minus d0 (S i)) v)))).(\lambda (H12: 
-(drop h (minus d0 (S i)) x d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g 
-c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat (plus (S i) (minus d0 (S i))) 
-(\lambda (n: nat).(sty0 g c0 (TLRef i) (lift h n (lift (S i) O w)))) 
-(eq_ind_r T (lift (S i) O (lift h (minus d0 (S i)) w)) (\lambda (t: T).(sty0 
-g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: nat).(sty0 g c0 (TLRef i) 
-(lift (S i) O (lift h (minus d0 (S i)) w)))) (sty0_abbr g c0 x (lift h (minus 
-d0 (S i)) v) i (getl_intro i c0 (CHead x (Bind Abbr) (lift h (minus d0 (S i)) 
-v)) x0 H6 H11) (lift h (minus d0 (S i)) w) (H2 x h (minus d0 (S i)) H12)) 
-(plus (S i) (minus d0 (S i))) (le_plus_minus (S i) d0 H4)) (lift h (plus (S 
-i) (minus d0 (S i))) (lift (S i) O w)) (lift_d w h (S i) (minus d0 (S i)) O 
-(le_O_n (minus d0 (S i))))) d0 (le_plus_minus_r (S i) d0 H4)) (lift h d0 
-(TLRef i)) (lift_lref_lt i h d0 H4))))) H10)))))))) H5))) (\lambda (H4: (le 
-d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 
-(lift (S i) O w)))) (eq_ind nat (S i) (\lambda (_: nat).(sty0 g c0 (TLRef 
-(plus i h)) (lift h d0 (lift (S i) O w)))) (eq_ind_r T (lift (plus h (S i)) O 
-w) (\lambda (t: T).(sty0 g c0 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S 
-i) h) (\lambda (n: nat).(sty0 g c0 (TLRef (plus i h)) (lift n O w))) 
-(sty0_abbr g c0 d v (plus i h) (drop_getl_trans_ge i c0 c d0 h H3 (CHead d 
-(Bind Abbr) v) H0 H4) w H1) (plus h (S i)) (plus_sym h (S i))) (lift h d0 
-(lift (S i) O w)) (lift_free w (S i) h O d0 (le_S_n d0 (S i) (le_S (S d0) (S 
-i) (le_n_S d0 i H4))) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) 
-i) (\lambda (n: nat).(eq nat (S i) n)) (le_antisym (S i) (plus (S O) i) (le_n 
-(plus (S O) i)) (le_n (S i))) (plus i (S O)) (plus_sym i (S O)))) (lift h d0 
-(TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda 
-(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
-(Bind Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: 
-((\forall (c0: C).(\forall (h: nat).(\forall (d0: nat).((drop h d0 c0 d) \to 
-(sty0 g c0 (lift h d0 v) (lift h d0 w)))))))).(\lambda (c0: C).(\lambda (h: 
-nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g 
-c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (H4: (lt i 
-d0)).(let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S d0) 
-(le_S (S (S i)) (S d0) (le_n_S (S i) d0 H4)))) c0 c h H3 (CHead d (Bind Abst) 
-v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) 
-(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: 
-C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g c0 (lift h 
-d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: C).(\lambda (x1: 
-C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 
-x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let H9 \def (eq_ind 
-nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) 
-(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) 
-H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind 
-Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S 
-i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) 
-(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift h (minus 
-d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T 
-(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind 
-nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) 
-(lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S 
-i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: 
-nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) v)))) 
-(sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x 
-(Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) 
-w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) 
-(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S 
-i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 
-(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
-H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i 
-h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat 
-(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) 
-O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 
-(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g 
-c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) 
-(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus 
-h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) 
-h O d0 (le_S_n d0 (S i) (le_S (S d0) (S i) (le_n_S d0 i H4))) (le_O_n d0))) 
-(plus i (S O)) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(eq nat (S i) 
-n)) (le_antisym (S i) (plus (S O) i) (le_n (plus (S O) i)) (le_n (S i))) 
-(plus i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h 
-d0 H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: 
-T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) 
-v) t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h 
-d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s 
-(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v 
-t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) 
-(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) 
-d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) 
-t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 
-b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift 
-h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda 
-(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g 
-c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d 
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat 
-Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v 
-t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) 
-t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s 
-(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) 
-d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h 
-d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d 
-(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) 
-(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 
-v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d 
-v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 
-t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d 
-t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: 
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s 
-(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) 
-v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) 
-d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h 
-(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 
-h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h 
-(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat 
-Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) 
-v1 t3 h d))))))))))))))) e t1 t2 H))))).
-
-lemma sty0_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
-t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: 
-T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda 
-(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) 
-(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 
-in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g 
-c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w 
-x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) 
-O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) 
-H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: 
-T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g 
-d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex 
-T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda 
-(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) 
-t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v 
-i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: 
-T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) 
-v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) 
-t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 
-(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) 
-t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 
-x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead 
-(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: 
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 
-t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 
-\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: 
-T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: 
-(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) 
-v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) 
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
-v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: 
-T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T 
-(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda 
-(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat 
-Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 
-\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: 
-T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: 
-(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) 
-v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) 
-H6)))) H4))))))))))) c t1 t H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma
deleted file mode 100644 (file)
index 5435437..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/cnt.ma
+++ /dev/null
@@ -1,86 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty1/props.ma".
-
-include "basic_1/cnt/props.ma".
-
-lemma sty1_cnt:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
-t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt 
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
-T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) 
-(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 
-(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 
-(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 
-g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda 
-(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d 
-v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef 
-i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v 
-x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) 
-t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x 
-H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T 
-(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def 
-H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) 
-(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) 
-(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt 
-x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: 
-T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) 
-(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) 
-(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i 
-H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: 
-C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g 
-(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g 
-(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in 
-(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda 
-(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) 
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead 
-c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: 
-T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead 
-(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) 
-H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: 
-T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: 
-T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in 
-(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) 
-(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda 
-(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda 
-(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v 
-t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v 
-t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: 
-C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 
-v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: 
-T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 
-t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: 
-T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 
-t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead 
-(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda 
-(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 
-t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g 
-c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat 
-Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) 
-t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 
-x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 
-x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) 
-(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat 
-Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma
deleted file mode 100644 (file)
index 9b98d8a..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/defs.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty0/defs.ma".
-
-inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def
-| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2))
-| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c 
-t t2) \to (sty1 g c t1 t2)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma
deleted file mode 100644 (file)
index 7307b49..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/fwd.ma
+++ /dev/null
@@ -1,25 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty1/defs.ma".
-
-implied rec lemma sty1_ind (g: G) (c: C) (t1: T) (P: (T \to Prop)) (f: 
-(\forall (t2: T).((sty0 g c t1 t2) \to (P t2)))) (f0: (\forall (t: T).((sty1 
-g c t1 t) \to ((P t) \to (\forall (t2: T).((sty0 g c t t2) \to (P t2))))))) 
-(t: T) (s0: sty1 g c t1 t) on s0: P t \def match s0 with [(sty1_sty0 t2 s1) 
-\Rightarrow (f t2 s1) | (sty1_sing t0 s1 t2 s2) \Rightarrow (f0 t0 s1 
-((sty1_ind g c t1 P f f0) t0 s1) t2 s2)].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma
deleted file mode 100644 (file)
index 0780b7e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty1/props.ma
+++ /dev/null
@@ -1,142 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty1/fwd.ma".
-
-include "basic_1/sty0/props.ma".
-
-theorem sty1_trans:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c 
-t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g 
-c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g 
-c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g 
-c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 
-g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
-
-lemma sty1_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead 
-(Bind b) v t1) (THead (Bind b) v t2))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 
-t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead 
-(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g 
-(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead 
-(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: 
-(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v 
-t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c 
-(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) 
-H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
-
-lemma sty1_appl:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
-(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat 
-Appl) v t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 
-g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: 
-T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) 
-(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: 
-T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v 
-t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t 
-t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 
-(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
-
-lemma sty1_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e 
-t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
-e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: 
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h 
-d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 
-t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop 
-h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 
-c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: 
-((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to 
-(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: 
-(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) 
-(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
-
-lemma sty1_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c 
-t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: 
-T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 
-t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 
-t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: 
-T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
-
-lemma sty1_abbr:
- \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
-nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) 
-\to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: 
-T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c 
-(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v 
-t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 
-H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c 
-(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t 
-t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) 
-(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w 
-H0)))))))).
-
-lemma sty1_cast2:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c 
-t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T 
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: 
-T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c 
-v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
-Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda 
-(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T 
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) 
-(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast 
-g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 
-t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to 
-(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead 
-(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda 
-(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g 
-c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T 
-(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 
-v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) 
-v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 
-g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def 
-(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: 
-T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: 
-T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda 
-(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g 
-c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
-Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast 
-g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma
deleted file mode 100644 (file)
index 8d9ed5f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst/defs.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/defs.ma".
-
-rec definition subst (d: nat) (v: T) (t: T) on t: T \def match t with [(TSort 
-n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true 
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k 
-u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))].
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma
deleted file mode 100644 (file)
index 3ba9fc8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst/props.ma
+++ /dev/null
@@ -1,157 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst/defs.ma".
-
-include "basic_1/subst0/fwd.ma".
-
-lemma subst_sort:
- \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort 
-k)) (TSort k))))
-\def
- \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort 
-k)))).
-
-lemma subst_lref_lt:
- \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T 
-(subst d v (TLRef i)) (TLRef i)))))
-\def
- \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i 
-d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true 
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) 
-(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
-
-lemma subst_lref_eq:
- \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
-\def
- \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq 
-T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with 
-[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift 
-i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
-
-lemma subst_lref_gt:
- \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T 
-(subst d v (TLRef i)) (TLRef (pred i))))))
-\def
- \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d 
-i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true 
-\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef 
-(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true 
-\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred 
-i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) 
-(le_bge d i (lt_le_weak d i H)))))).
-
-lemma subst_head:
- \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: 
-nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w 
-t)))))))
-\def
- \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: 
-nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).
-
-lemma subst_lift_SO:
- \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S 
-O) d t)) t)))
-\def
- \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq 
-T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: 
-nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) 
-(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T 
-(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) 
-(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq 
-T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n 
-d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) 
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T 
-(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d 
-(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T 
-(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) 
-(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) 
-(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T 
-t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) 
-(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) 
-(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef 
-(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) 
-(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d 
-(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) 
-t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v 
-(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) 
-d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k 
-t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v 
-(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq 
-T (THead k t0 t1) (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v 
-(lift (S O) (s k d) t1))) (sym_eq T (THead k (subst d v (lift (S O) d t0)) 
-(subst (s k d) v (lift (S O) (s k d) t1))) (THead k t0 t1) (f_equal3 K T T T 
-THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v (lift (S O) (s k 
-d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))))) (subst d v (THead k (lift 
-(S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S O) d t0) (lift 
-(S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) (lift_head k t0 t1 (S O) 
-d)))))))) t)).
-
-lemma subst_subst0:
- \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 
-d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
-\def
- \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda 
-(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) 
-(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: 
-T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) 
-(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T 
-(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 
-t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) 
-(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) 
-(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 
-i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) 
-(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: 
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 
-u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: 
-T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) 
-(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k 
-(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst 
-i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: 
-T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s 
-k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) 
-(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) 
-(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: 
-K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: 
-nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k 
-i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst 
-i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u 
-t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda 
-(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T 
-(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead 
-k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 
-u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u 
-t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u 
-t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 
-u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) 
-v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s 
-k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r 
-T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T 
-(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 
-u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i 
-v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda 
-(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k 
-i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) 
-(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) 
-(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 
-t3 i))))))))))))) d v t1 t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
deleted file mode 100644 (file)
index 93aa073..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
+++ /dev/null
@@ -1,176 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/defs.ma".
-
-include "basic_1/lift/props.ma".
-
-lemma dnf_dec2:
- \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
-O) d v))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda 
-(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: 
-nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) 
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d 
-v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort 
-n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T 
-(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: 
-nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T 
-(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: 
-T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) 
-(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d 
-(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind 
-nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 
-w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift 
-(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w 
-(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S 
-O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) 
-(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) 
-(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n 
-(lift n O w)) (lift_free w n (S O) O n (le_plus_r O n) (le_O_n n)))))) d H)) 
-(\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) 
-(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred 
-n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda 
-(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
-t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda 
-(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) 
-in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 
-d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) 
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) 
-d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 
-(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in 
-(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S 
-O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) 
-(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift 
-(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d 
-v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w 
-t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
-T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) 
-in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift 
-(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S 
-O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s 
-k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda 
-(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w 
-(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d 
-w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 
-t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) 
-(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) 
-(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) 
-(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T 
-(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex 
-T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) 
-(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def 
-H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T 
-(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d 
-v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d 
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) 
-x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) 
-(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) 
-x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 
-H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex 
-T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: 
-T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in 
-(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s 
-k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S 
-O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
-k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) 
-(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) 
-d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
-(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 
-t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T 
-(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) 
-(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T 
-(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda 
-(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda 
-(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) 
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) 
-(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift 
-(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) 
-t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift 
-(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) 
-(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T 
-(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) 
-(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
-k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
-T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) 
-(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) 
-(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k 
-d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d 
-x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) 
-(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift 
-(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) 
-d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
-
-lemma dnf_dec:
- \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or 
-(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
-\def
- \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t 
-d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: 
-T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
-O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
-(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: 
-T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 
-\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T 
-(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
-v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d 
-x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
-(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t 
-(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t 
-(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex 
-T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
-v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T 
-(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 
-(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: 
-T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) 
-(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d 
-x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d 
-x)))) t H1))) H0)) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma
deleted file mode 100644 (file)
index fca79e9..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma
+++ /dev/null
@@ -1,32 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/lift/defs.ma".
-
-inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
-| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift 
-(S i) O v)))
-| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: 
-nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v 
-(THead k u1 t) (THead k u2 t))))))))
-| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: 
-T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i 
-v (THead k u t1) (THead k u t2))))))))
-| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: 
-nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: 
-T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 
-t2)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
deleted file mode 100644 (file)
index 930835c..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
+++ /dev/null
@@ -1,912 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/defs.ma".
-
-include "basic_1/lift/fwd.ma".
-
-implied rec lemma subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f: 
-(\forall (v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0: 
-(\forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 
-i v u1 u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v 
-(THead k u1 t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v: 
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 
-t2) \to ((P (s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead 
-k u t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2: 
-T).(\forall (i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k: 
-K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k 
-i) v t1 t2) \to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat) 
-(t: T) (t0: T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match 
-s0 with [(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2 
-k) \Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2 
-k) | (subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1 
-((subst0_ind P f f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1 
-k t2 t3 s2) \Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 
-s1) k t2 t3 s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
-
-lemma subst0_gen_sort:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
-i v (TSort n) x) \to (\forall (P: Prop).P)))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) 
-(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda 
-(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda 
-(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: 
-T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def 
-(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(TSort n) H1) in (False_ind P H2))))) (\lambda (v0: T).(\lambda (u2: 
-T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
-u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (t: T).(\lambda (k: 
-K).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(let H4 \def (eq_ind T 
-(THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H3) in (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda 
-(t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 
-t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
-(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
-(False_ind P H4))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T 
-u1 (TSort n)) \to P))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TSort 
-n)) \to P))).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let H6 \def 
-(eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) 
-H)))))).
-
-lemma subst0_gen_lref:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
-i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 
-i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) 
-(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: 
-nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) 
-\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: 
-T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef 
-n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(land (eq nat n n0) (eq T (lift 
-(S n0) O v0) (lift (S n) O v0)))) (conj (eq nat n n) (eq T (lift (S n) O v0) 
-(lift (S n) O v0)) (refl_equal nat n) (refl_equal T (lift (S n) O v0))) i0 
-H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) 
-\to (land (eq nat n i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (t: 
-T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) (TLRef n))).(let H4 
-\def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
-t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: 
-T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 
-(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s 
-k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T 
-(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in (False_ind (land 
-(eq nat n i0) (eq T (THead k u t2) (lift (S n) O v0))) H4))))))))))) (\lambda 
-(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n 
-i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 
-(TLRef n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O 
-v0)))))).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind 
-T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 t2) (lift 
-(S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
-
-lemma subst0_gen_head:
- \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
-(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T 
-(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: 
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) 
-x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda 
-(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) 
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
-t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda 
-(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k 
-u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
-(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 
-t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 
-t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef 
-i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead k u1 t1) H1) in (False_ind (or3 
-(ex2 T (\lambda (u2: T).(eq T (lift (S i0) O v0) (THead k u2 t1))) (\lambda 
-(u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t2: T).(eq T (lift (S i0) O 
-v0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u2 
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))) H2))))) (\lambda (v0: 
-T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 
-i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead k u1 t1)) \to (or3 (ex2 T 
-(\lambda (u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 
-u3))) (ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: 
-T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 
-u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 
-t2)))))))).(\lambda (t: T).(\lambda (k0: K).(\lambda (H3: (eq T (THead k0 u0 
-t) (THead k u1 t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
-_) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k 
-u1 t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) 
-\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T 
-u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
-(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) 
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda 
-(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead 
-k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k 
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
-(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind 
-T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda 
-(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
-(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 
-(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
-(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) 
-in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 
-H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
-(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 
-t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) 
-(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T 
-(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 
-H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: 
-T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 
-t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
-T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
-(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
-t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 
-t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) 
-(THead k0 u t0) (THead k u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
-(THead _ t _) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let 
-H6 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
-| (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u t0) 
-(THead k u1 t1) H3) in (\lambda (H7: (eq T u u1)).(\lambda (H8: (eq K k0 
-k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead 
-k0 t t2) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
-(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead k0 t t2) (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) 
-v0 t1 t3)))))) (let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 
-t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
-(u2: T).(subst0 (s k0 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead 
-k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H2 t1 H6) in (let H10 \def 
-(eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t t2)) H1 t1 H6) in (let 
-H11 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 
-(ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 
-(s k1 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 (s k1 i0) v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
-k (s k1 i0)) v0 t1 t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda 
-(k1: K).(subst0 (s k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: 
-K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead 
-k1 u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
-(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: 
-T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
-v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) 
-H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
-(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead 
-k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: 
-(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
-k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
-v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
-t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 
-\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) 
-(THead k u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0) (THead k 
-u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k0 k)).(let 
-H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 
-T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k0 
-i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda 
-(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 (s k0 i0) v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
-k (s k0 i0)) v0 t1 t3))))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda 
-(t: T).(subst0 (s k0 i0) v0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 
-(\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
-T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
-(s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
-T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 
-u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
-t3))))))) H11 k H10) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 
-(s k1 i0) v0 t1 t2)) H12 k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
-(\lambda (u3: T).(eq T (THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: 
-T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H15 \def (eq_ind T 
-u0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
-T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T 
-(\lambda (t3: T).(eq T u2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
-v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 
-\def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in 
-(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead 
-k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k 
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
-(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k 
-u2 t2)) H16 H14)))) k0 H10)))))))) H7)) H6)))))))))))))) i v y x H0))) 
-H))))))).
-
-lemma subst0_gen_lift_lt:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) 
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d 
-u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: 
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) 
-x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: 
-T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) 
-in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) 
-t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S 
-(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: 
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
-n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h 
-(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H 
-(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq 
-T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: 
-(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T 
-(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) 
-(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) 
-O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda 
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
-(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T 
-(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O 
-(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) 
-(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) 
-n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: 
-T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S 
-(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n 
-h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat 
-(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d 
-u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) 
-H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: 
-T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
-n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) 
-(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) 
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
-(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift 
-h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 
-t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t 
-t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) 
-(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i 
-d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) 
-in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus 
-i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) 
-t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) 
-t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
-d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k 
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S 
-(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h 
-d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
-t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k 
-(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S 
-(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
-(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h 
-(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
-T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) 
-t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) 
-x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T 
-x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T 
-(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
-(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda 
-(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) 
-t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
-t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
-(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T 
-(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h 
-(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k 
-(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 
-t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: 
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) 
-(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) 
-t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i 
-d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
-(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
-T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda 
-(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) 
-x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 
-T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: 
-T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i 
-d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S 
-(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus 
-i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) 
-H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
-(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h 
-(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
-(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) 
-x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s 
-k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h 
-(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i 
-u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T 
-(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) 
-(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
-t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda 
-(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind 
-T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda 
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
-(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i 
-d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) 
-(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i 
-d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) 
-(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 
-H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
-(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq 
-T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d 
-u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda 
-(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x 
-(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) 
-t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
-d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda 
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
-(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda 
-(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i 
-d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) 
-(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 
-(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 
-(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
-(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda 
-(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k 
-i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) 
-t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T 
-(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
-(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S 
-(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S 
-(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
-x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
-t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S 
-(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind 
-nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k 
-(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus 
-i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S 
-(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: 
-T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead 
-k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus 
-i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T 
-(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S 
-(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k 
-x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u 
-t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S 
-(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) 
-(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 
-i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k 
-(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
-H2))))))))))))) t1)).
-
-lemma subst0_gen_lift_false:
- \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
-(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u 
-(lift h d t) x) \to (\forall (P: Prop).P)))))))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: 
-T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i 
-(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
-Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda 
-(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) 
-x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda 
-(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in 
-(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: 
-T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: 
-nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: 
-(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P 
-(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
-(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind 
-(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda 
-(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: 
-nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n 
-H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) 
-(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d 
-H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P 
-(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n 
-h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d 
-h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h 
-n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: 
-K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall 
-(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) 
-\to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
-Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall 
-(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to 
-((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: 
-Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus 
-d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: 
-Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: 
-T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) 
-(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k 
-u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) 
-(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: 
-T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: 
-T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u 
-(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
-(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda 
-(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: 
-(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda 
-(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda 
-(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: 
-T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u 
-(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k 
-(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) 
-x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) 
-(\lambda (n: nat).(lt (s k i) n)) (s_lt k i (plus d h) H2) (plus (s k d) h) 
-(s_plus k d h)) H7 P)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
-(s k i) u (lift h (s k d) t1) t2))) P (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: (subst0 i u (lift h d 
-t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) t1) x1)).(H u x0 h d 
-i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d t0) (lift h (s k d) 
-t1) x i H4))))))))))))))))) t).
-
-lemma subst0_gen_lift_ge:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) 
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h 
-d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: 
-nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus 
-d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 
-i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 
-T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i 
-h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: 
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d 
-(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda 
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef 
-n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) 
-(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in 
-(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq 
-T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) 
-(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 
-\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus 
-d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) 
-(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind 
-T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) 
-(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S 
-(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n 
-h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n 
-h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) 
-(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S 
-(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) 
-(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S 
-(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) 
-t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h 
-d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) 
-(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n 
-h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O 
-u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: 
-nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift 
-(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n 
-h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d 
-(plus O (S n)) (le_plus_plus O O d (S n) (le_O_n O) (le_S d n H1))) (le_O_n 
-d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i 
-H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) 
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: 
-((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
-nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda 
-(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 
-t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda 
-(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) 
-(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) 
-t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x 
-(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) 
-u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda 
-(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
-k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
-(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) 
-(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: 
-T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u 
-(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
-(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u 
-(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
-(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 
-(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda 
-(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: 
-T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: 
-(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r 
-T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
-(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u 
-(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: 
-T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
-(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift 
-h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u 
-(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) 
-(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k 
-d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) 
-(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) 
-(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T 
-(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 
-(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
-t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
-(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: 
-(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) 
-x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) 
-(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T 
-(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 
-(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d 
-t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) 
-t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda 
-(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) 
-(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) 
-(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) 
-(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: 
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t 
-t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: 
-nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r 
-d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) 
-(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) 
-(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 
-(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t 
-x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) 
-(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) 
-(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
-k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
-(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
-u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift 
-h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u 
-(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) 
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq 
-T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) 
-t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda 
-(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T 
-(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead 
-k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) 
-x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda 
-(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t 
-t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda 
-(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t 
-x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T 
-(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead 
-k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T 
-(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda 
-(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d 
-(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d 
-t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 
-\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 
-(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T 
-(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda 
-(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal 
-T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 
-H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) 
-x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind 
-nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i 
-H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u 
-(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
-
-lemma subst0_gen_lift_rev_ge:
- \forall (t1: T).(\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst0 i v t1 (lift h d u2)) \to ((le (plus d h) 
-i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
-T).(eq T t1 (lift h d u1)))))))))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (v: T).(\forall (u2: 
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i v t (lift 
-h d u2)) \to ((le (plus d h) i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i 
-h) v u1 u2)) (\lambda (u1: T).(eq T t (lift h d u1)))))))))))) (\lambda (n: 
-nat).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (subst0 i v (TSort n) (lift h d 
-u2))).(\lambda (_: (le (plus d h) i)).(subst0_gen_sort v (lift h d u2) i n H 
-(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
-(TSort n) (lift h d u1))))))))))))) (\lambda (n: nat).(\lambda (v: 
-T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H: (subst0 i v (TLRef n) (lift h d u2))).(\lambda (H0: (le 
-(plus d h) i)).(land_ind (eq nat n i) (eq T (lift h d u2) (lift (S n) O v)) 
-(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
-(TLRef n) (lift h d u1)))) (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T 
-(lift h d u2) (lift (S n) O v))).(let H3 \def (eq_ind_r nat i (\lambda (n0: 
-nat).(le (plus d h) n0)) H0 n H1) in (eq_ind nat n (\lambda (n0: nat).(ex2 T 
-(\lambda (u1: T).(subst0 (minus n0 h) v u1 u2)) (\lambda (u1: T).(eq T (TLRef 
-n) (lift h d u1))))) (eq_ind_r nat (plus (minus n h) h) (\lambda (n0: 
-nat).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) (\lambda (u1: 
-T).(eq T (TLRef n0) (lift h d u1))))) (eq_ind T (lift h d (TLRef (minus n 
-h))) (\lambda (t: T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) 
-(\lambda (u1: T).(eq T t (lift h d u1))))) (let H4 \def (eq_ind nat n 
-(\lambda (n0: nat).(eq T (lift h d u2) (lift (S n0) O v))) H2 (plus h (minus 
-n h)) (le_plus_minus h n (le_trans h (plus d h) n (le_plus_r d h) H3))) in 
-(let H5 \def (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(eq T 
-(lift h d u2) (lift n0 O v))) H4 (plus h (S (minus n h))) (plus_n_Sm h (minus 
-n h))) in (let H6 \def (eq_ind_r T (lift (plus h (S (minus n h))) O v) 
-(\lambda (t: T).(eq T (lift h d u2) t)) H5 (lift h d (lift (S (minus n h)) O 
-v)) (lift_free v (S (minus n h)) h O d (le_S d (minus n h) (le_minus d n h 
-H3)) (le_O_n d))) in (eq_ind_r T (lift (S (minus n h)) O v) (\lambda (t: 
-T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 t)) (\lambda (u1: T).(eq 
-T (lift h d (TLRef (minus n h))) (lift h d u1))))) (ex_intro2 T (\lambda (u1: 
-T).(subst0 (minus n h) v u1 (lift (S (minus n h)) O v))) (\lambda (u1: T).(eq 
-T (lift h d (TLRef (minus n h))) (lift h d u1))) (TLRef (minus n h)) 
-(subst0_lref v (minus n h)) (refl_equal T (lift h d (TLRef (minus n h))))) u2 
-(lift_inj u2 (lift (S (minus n h)) O v) h d H6))))) (TLRef (plus (minus n h) 
-h)) (lift_lref_ge (minus n h) h d (le_minus d n h H3))) n (le_plus_minus_sym 
-h n (le_trans h (plus d h) n (le_plus_r d h) H3))) i H1)))) (subst0_gen_lref 
-v (lift h d u2) i n H)))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: 
-((\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: 
-nat).(\forall (d: nat).((subst0 i v t (lift h d u2)) \to ((le (plus d h) i) 
-\to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
-T).(eq T t (lift h d u1))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
-(v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
-nat).((subst0 i v t0 (lift h d u2)) \to ((le (plus d h) i) \to (ex2 T 
-(\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T t0 
-(lift h d u1))))))))))))).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: 
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (subst0 i v (THead k t 
-t0) (lift h d u2))).(\lambda (H2: (le (plus d h) i)).(or3_ind (ex2 T (\lambda 
-(u3: T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
-u3))) (ex2 T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) (\lambda 
-(t2: T).(subst0 (s k i) v t0 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
-T).(eq T (lift h d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
-T).(subst0 i v t u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 
-t2)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
-T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (H3: (ex2 T (\lambda (u3: 
-T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
-u3)))).(ex2_ind T (\lambda (u3: T).(eq T (lift h d u2) (THead k u3 t0))) 
-(\lambda (u3: T).(subst0 i v t u3)) (ex2 T (\lambda (u1: T).(subst0 (minus i 
-h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
-(x: T).(\lambda (H4: (eq T (lift h d u2) (THead k x t0))).(\lambda (H5: 
-(subst0 i v t x)).(let H6 \def (sym_eq T (lift h d u2) (THead k x t0) H4) in 
-(let H_x \def (lift_gen_head k x t0 u2 h d H6) in (let H7 \def H_x in 
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T x (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t0 (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
-T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
-h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
-x0 x1))).(\lambda (H9: (eq T x (lift h d x0))).(\lambda (H10: (eq T t0 (lift 
-h (s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 i v t t2)) 
-H5 (lift h d x0) H9) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T 
-(\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq T (THead 
-k t t0) (lift h d u1))))) (eq_ind_r T (lift h (s k d) x1) (\lambda (t2: 
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
-(\lambda (u1: T).(eq T (THead k t t2) (lift h d u1))))) (let H_x0 \def (H v 
-x0 i h d H11 H2) in (let H12 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
-(minus i h) v u1 x0)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
-(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: 
-T).(eq T (THead k t (lift h (s k d) x1)) (lift h d u1)))) (\lambda (x2: 
-T).(\lambda (H13: (subst0 (minus i h) v x2 x0)).(\lambda (H14: (eq T t (lift 
-h d x2))).(eq_ind_r T (lift h d x2) (\lambda (t2: T).(ex2 T (\lambda (u1: 
-T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k 
-t2 (lift h (s k d) x1)) (lift h d u1))))) (eq_ind T (lift h d (THead k x2 
-x1)) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead 
-k x0 x1))) (\lambda (u1: T).(eq T t2 (lift h d u1))))) (ex_intro2 T (\lambda 
-(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T 
-(lift h d (THead k x2 x1)) (lift h d u1))) (THead k x2 x1) (subst0_fst v x0 
-x2 (minus i h) H13 x1 k) (refl_equal T (lift h d (THead k x2 x1)))) (THead k 
-(lift h d x2) (lift h (s k d) x1)) (lift_head k x2 x1 h d)) t H14)))) H12))) 
-t0 H10) u2 H8))))))) H7))))))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq 
-T (lift h d u2) (THead k t t2))) (\lambda (t2: T).(subst0 (s k i) v t0 
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) 
-(\lambda (t2: T).(subst0 (s k i) v t0 t2)) (ex2 T (\lambda (u1: T).(subst0 
-(minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) 
-(\lambda (x: T).(\lambda (H4: (eq T (lift h d u2) (THead k t x))).(\lambda 
-(H5: (subst0 (s k i) v t0 x)).(let H6 \def (sym_eq T (lift h d u2) (THead k t 
-x) H4) in (let H_x \def (lift_gen_head k t x u2 h d H6) in (let H7 \def H_x 
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T t (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T x (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
-T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
-h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
-x0 x1))).(\lambda (H9: (eq T t (lift h d x0))).(\lambda (H10: (eq T x (lift h 
-(s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 (s k i) v t0 
-t2)) H5 (lift h (s k d) x1) H10) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: 
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq 
-T (THead k t t0) (lift h d u1))))) (eq_ind_r T (lift h d x0) (\lambda (t2: 
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
-(\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) (let H_y \def (H0 v 
-x1 (s k i) h (s k d) H11) in (let H12 \def (eq_ind_r nat (plus (s k d) h) 
-(\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: T).(subst0 (minus 
-(s k i) h) v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) u1)))))) H_y 
-(s k (plus d h)) (s_plus k d h)) in (let H13 \def (eq_ind_r nat (minus (s k 
-i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to (ex2 T (\lambda 
-(u1: T).(subst0 n v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) 
-u1)))))) H12 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) i 
-(le_plus_r d h) H2))) in (let H14 \def (H13 (s_le k (plus d h) i H2)) in 
-(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x1)) (\lambda (u1: 
-T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
-v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t0) 
-(lift h d u1)))) (\lambda (x2: T).(\lambda (H15: (subst0 (s k (minus i h)) v 
-x2 x1)).(\lambda (H16: (eq T t0 (lift h (s k d) x2))).(eq_ind_r T (lift h (s 
-k d) x2) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
-(THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t2) (lift h d 
-u1))))) (eq_ind T (lift h d (THead k x0 x2)) (\lambda (t2: T).(ex2 T (\lambda 
-(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T t2 
-(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
-(THead k x0 x1))) (\lambda (u1: T).(eq T (lift h d (THead k x0 x2)) (lift h d 
-u1))) (THead k x0 x2) (subst0_snd k v x1 x2 (minus i h) H15 x0) (refl_equal T 
-(lift h d (THead k x0 x2)))) (THead k (lift h d x0) (lift h (s k d) x2)) 
-(lift_head k x0 x2 h d)) t0 H16)))) H14))))) t H9) u2 H8))))))) H7))))))) 
-H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (lift h 
-d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 t2))))).(ex3_2_ind T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T (lift h d u2) (THead k u3 t2)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) (\lambda (_: T).(\lambda 
-(t2: T).(subst0 (s k i) v t0 t2))) (ex2 T (\lambda (u1: T).(subst0 (minus i 
-h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T (lift h d u2) (THead k x0 
-x1))).(\lambda (H5: (subst0 i v t x0)).(\lambda (H6: (subst0 (s k i) v t0 
-x1)).(let H7 \def (sym_eq T (lift h d u2) (THead k x0 x1) H4) in (let H_x 
-\def (lift_gen_head k x0 x1 u2 h d H7) in (let H8 \def H_x in (ex3_2_ind T T 
-(\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T x0 (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T x1 (lift h (s k d) z)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
-v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T u2 (THead k x2 x3))).(\lambda 
-(H10: (eq T x0 (lift h d x2))).(\lambda (H11: (eq T x1 (lift h (s k d) 
-x3))).(let H12 \def (eq_ind T x1 (\lambda (t2: T).(subst0 (s k i) v t0 t2)) 
-H6 (lift h (s k d) x3) H11) in (let H13 \def (eq_ind T x0 (\lambda (t2: 
-T).(subst0 i v t t2)) H5 (lift h d x2) H10) in (eq_ind_r T (THead k x2 x3) 
-(\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) 
-(\lambda (u1: T).(eq T (THead k t t0) (lift h d u1))))) (let H_x0 \def (H v 
-x2 i h d H13 H2) in (let H14 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
-(minus i h) v u1 x2)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
-(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: 
-T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (x: T).(\lambda (H15: 
-(subst0 (minus i h) v x x2)).(\lambda (H16: (eq T t (lift h d x))).(eq_ind_r 
-T (lift h d x) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v 
-u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) 
-(let H_y \def (H0 v x3 (s k i) h (s k d) H12) in (let H17 \def (eq_ind_r nat 
-(plus (s k d) h) (\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: 
-T).(subst0 (minus (s k i) h) v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h (s k 
-d) u1)))))) H_y (s k (plus d h)) (s_plus k d h)) in (let H18 \def (eq_ind_r 
-nat (minus (s k i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to 
-(ex2 T (\lambda (u1: T).(subst0 n v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h 
-(s k d) u1)))))) H17 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) 
-i (le_plus_r d h) H2))) in (let H19 \def (H18 (s_le k (plus d h) i H2)) in 
-(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x3)) (\lambda (u1: 
-T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
-v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t0) (lift 
-h d u1)))) (\lambda (x4: T).(\lambda (H20: (subst0 (s k (minus i h)) v x4 
-x3)).(\lambda (H21: (eq T t0 (lift h (s k d) x4))).(eq_ind_r T (lift h (s k 
-d) x4) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
-(THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t2) (lift h d 
-u1))))) (eq_ind T (lift h d (THead k x x4)) (\lambda (t2: T).(ex2 T (\lambda 
-(u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T t2 
-(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
-(THead k x2 x3))) (\lambda (u1: T).(eq T (lift h d (THead k x x4)) (lift h d 
-u1))) (THead k x x4) (subst0_both v x x2 (minus i h) H15 k x4 x3 H20) 
-(refl_equal T (lift h d (THead k x x4)))) (THead k (lift h d x) (lift h (s k 
-d) x4)) (lift_head k x x4 h d)) t0 H21)))) H19))))) t H16)))) H14))) u2 
-H9)))))))) H8))))))))) H3)) (subst0_gen_head k v t t0 (lift h d u2) i 
-H1)))))))))))))) t1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
deleted file mode 100644 (file)
index 762f857..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
+++ /dev/null
@@ -1,224 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/fwd.ma".
-
-lemma subst0_refl:
- \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: 
-nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: 
-nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort 
-n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda 
-(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef 
-n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O 
-u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O 
-u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 
-P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: 
-Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u 
-t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 
-d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead 
-k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T 
-(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T 
-(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 
-\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) (THead k t0 t1) 
-(THead k x t1) H3) in (let H6 \def (eq_ind_r T x (\lambda (t2: T).(subst0 d u 
-t0 t2)) H4 t0 H5) in (H d H6 P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: 
-T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u 
-t1 t2)))).(ex2_ind T (\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) 
-(\lambda (t2: T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: 
-(eq T (THead k t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 
-x)).(let H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) \Rightarrow t2])) 
-(THead k t0 t1) (THead k t0 x) H3) in (let H6 \def (eq_ind_r T x (\lambda 
-(t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s k d) H6 P)))))) H2)) 
-(\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: T).(\lambda 
-(x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 x1))).(\lambda (H4: 
-(subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 x1)).(let H6 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k 
-x0 x1) H3) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) 
-\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T 
-t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 
-t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u 
-t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 
-t1 (THead k t0 t1) d H1)))))))))) t)).
-
-lemma subst0_lift_lt:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
-i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i 
-(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall 
-(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d 
-t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda 
-(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: 
-T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) 
-(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) 
-(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O 
-v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: 
-T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S 
-i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d 
-v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r 
-(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda 
-(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall 
-(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d 
-u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt 
-i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) 
-t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d 
-(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) 
-(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d 
-u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h 
-d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k 
-u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h 
-d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: 
-((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) 
-(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda 
-(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 
-\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s 
-k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) 
-(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r 
-T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 
-(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead 
-k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h 
-(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind 
-nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) 
-(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s 
-k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s 
-k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d 
-u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) 
-(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h 
-d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: 
-nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) 
-(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: 
-nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d 
-(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda 
-(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) 
-(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: 
-nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 
-t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) 
-(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t 
-(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k 
-d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift 
-h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) 
-(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k 
-d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s 
-k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) 
-(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d 
-(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) 
-(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
-
-lemma subst0_lift_ge:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
-(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 
-(plus i h) u (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: 
-nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le 
-d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: 
-T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T 
-(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift 
-(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 
-(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda 
-(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r 
-nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) 
-O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S 
-i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d 
-(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d 
-H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le 
-d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: 
-T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T 
-(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 
-h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift 
-h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
-(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 
-h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 
-t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda 
-(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: 
-nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d 
-t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 
-\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: 
-nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 
-(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) 
-(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead 
-k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda 
-(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) 
-(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k 
-d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k 
-u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) 
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda 
-(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to 
-(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: 
-K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 
-t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k 
-i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le 
-d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: 
-nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) 
-(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T 
-(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 
-h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift 
-h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
-(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 
-h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d 
-i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead 
-k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
-
-lemma subst0_lift_ge_S:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
-i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d 
-t1) (lift (S O) d t2))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat 
-(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d 
-t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) 
-i) (\lambda (n: nat).(eq nat n (S i))) (le_antisym (plus (S O) i) (S i) (le_n 
-(S i)) (le_n (plus (S O) i))) (plus i (S O)) (plus_sym i (S O)))))))))).
-
-lemma subst0_lift_ge_s:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
-i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s 
-(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda 
-(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma
deleted file mode 100644 (file)
index 99caee5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma
+++ /dev/null
@@ -1,1389 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/props.ma".
-
-include "basic_1/s/fwd.ma".
-
-theorem subst0_subst0:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i 
-u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: 
-T).(subst0 (S (plus i j)) u t t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
-(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: 
-T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n 
-u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda 
-(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) 
-(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda 
-(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: 
-T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t 
-(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v 
-u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) 
-(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: 
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
-(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: 
-T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: 
-nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i 
-u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda 
-(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 
-i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 
-x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: 
-T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) 
-u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst 
-u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: 
-K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: 
-nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: 
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda 
-(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k 
-i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: 
-T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda 
-(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k 
-i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) 
-(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: 
-T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 
-(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: 
-nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in 
-(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n 
-u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T 
-(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S 
-(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 
-u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 
-H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: 
-T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda 
-(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t 
-u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: 
-(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: 
-T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 
-(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t 
-t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: 
-(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: 
-T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u 
-t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 
-x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda 
-(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) 
-(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: 
-T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda 
-(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 
-u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 
-(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def 
-(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 
-(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: 
-T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u 
-t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) 
-(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) 
-(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
-
-theorem subst0_subst0_back:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i 
-u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: 
-T).(subst0 (S (plus i j)) u t2 t)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
-(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: 
-T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n 
-u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda 
-(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) 
-(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda 
-(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: 
-T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift 
-(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 
-(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 
-(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda 
-(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: 
-((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to 
-(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus 
-i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: 
-T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v 
-u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: 
-T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 
-(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) 
-t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 
-(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k 
-u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) 
-(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 
-i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 
-(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall 
-(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 
-t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda 
-(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: 
-(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) 
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: 
-T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 
-(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 
-x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def 
-(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 
-(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k 
-(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) 
-(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u 
-t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead 
-k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 
-u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: 
-T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
-(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: 
-K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 
-t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
-nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
-(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: 
-T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v 
-u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: 
-T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 
-(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) 
-t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: 
-(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i 
-u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda 
-(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 
-i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 
-x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r 
-nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus 
-i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 
-i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k 
-(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) 
-t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 
-x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) 
-H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 
-t1 t2 H))))).
-
-theorem subst0_trans:
- \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 
-i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 
-t3)))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to 
-(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: 
-T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false 
-v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 
-(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: 
-T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: 
-((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda 
-(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k 
-u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) 
-(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 
-(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
-T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda 
-(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: 
-T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 
-t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) 
-(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 
-i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead 
-k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda 
-(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: 
-T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k 
-u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k 
-u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda 
-(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: 
-T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 
-H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T 
-t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) 
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T 
-(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
-T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: 
-(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T 
-(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) 
-(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) 
-(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: 
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 
-(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 
-t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: 
-T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda 
-(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) 
-(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s 
-k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
-(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 
-(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 
-t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq 
-T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 
-(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x 
-t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda 
-(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) 
-t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u 
-t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: 
-T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) 
-(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 
-(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k 
-u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 
-i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 
-t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k 
-u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) 
-t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 
-x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) 
-t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) 
-(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda 
-(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 
-u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 
-u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: 
-(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) 
-v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: 
-(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda 
-(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 
-t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
-t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 
-t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) 
-(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 
-(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead 
-k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda 
-(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 
-i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 
-H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) 
-(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: 
-T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) 
-(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 
-(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead 
-k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 
-u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
-(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda 
-(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 
-i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) 
-(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda 
-(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: 
-T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 
-x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 
-H4))))))))))))))) i v t1 t2 H))))).
-
-theorem subst0_confluence_neq:
- \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: 
-nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall 
-(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda 
-(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: 
-nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: 
-nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: 
-T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq 
-nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: 
-T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda 
-(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 
-(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq 
-T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) 
-t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda 
-(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: 
-nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda 
-(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda 
-(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in 
-False with []) in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) 
-(\lambda (v: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
-(H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: 
-T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 
-T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 
-t)))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: 
-T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) 
-t2)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq 
-T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda 
-(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t 
-t3))) (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
-t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: 
-T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) 
-(\lambda (H4: (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
-T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 
-t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 
-i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
-T).(\lambda (H5: (eq T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 
-x)).(eq_ind_r T (THead k x t) (\lambda (t3: T).(ex2 T (\lambda (t4: 
-T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
-(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i 
-v x t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i v (THead k x t) t3))) (\lambda (x0: T).(\lambda (H7: 
-(subst0 i2 u3 u2 x0)).(\lambda (H8: (subst0 i v x x0)).(ex_intro2 T (\lambda 
-(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead 
-k x t) t3)) (THead k x0 t) (subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x 
-i H8 t k))))) (H1 x u3 i2 H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T 
-(\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) 
-u3 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda 
-(t3: T).(subst0 (s k i2) u3 t t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 
-(THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
-T).(\lambda (H5: (eq T t2 (THead k u0 x))).(\lambda (H6: (subst0 (s k i2) u3 
-t x)).(eq_ind_r T (THead k u0 x) (\lambda (t3: T).(ex2 T (\lambda (t4: 
-T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
-(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i v (THead k u0 x) t3)) (THead k u2 x) (subst0_snd k u3 x t i2 H6 
-u2) (subst0_fst v u2 u0 i H0 x k)) t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T 
-(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: 
-T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i2) u3 t t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda 
-(t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 
-i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) 
-(ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i v t2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
-t2 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: 
-(subst0 (s k i2) u3 t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 
-T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 
-i v t3 t4)))) (ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: 
-T).(subst0 i v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) 
-t3)) (\lambda (t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: 
-T).(\lambda (H8: (subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 
-x)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i v (THead k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x 
-i2 H8 k t x1 H7) (subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 
-H5)))))) H4)) (subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: 
-K).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: 
-nat).(\lambda (H0: (subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: 
-T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq 
-nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: 
-T).(subst0 (s k i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda 
-(u2: T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) 
-t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq 
-T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda 
-(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 
-t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
-t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: 
-T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda 
-(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k 
-u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 
-i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
-T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u 
-x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: 
-T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
-(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: 
-T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 
-k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T 
-(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) 
-u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda 
-(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 
-(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
-T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 
-t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: 
-T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 
-(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) 
-(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: 
-(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 
-T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i 
-v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) 
-(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: 
-T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq 
-nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k 
-i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s 
-k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: 
-(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) 
-(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k 
-u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda 
-(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: 
-T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k 
-u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u 
-x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) 
-(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) 
-(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k 
-i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: 
-T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 
-x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda 
-(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 
-(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k 
-x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) 
-(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) 
-(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to 
-False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: 
-(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj 
-k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k 
-i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 
-t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: 
-((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) 
-\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) 
-(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: 
-T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: 
-((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) 
-\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 
-t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: 
-T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k 
-u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: 
-T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T 
-(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) 
-u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 
-t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
-T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: 
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda 
-(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k 
-u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: 
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 
-i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: 
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
-(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x 
-t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: 
-T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 
-u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: 
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x 
-t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 
-i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T 
-(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) 
-u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda 
-(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 
-(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
-T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 
-t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: 
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 
-(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) 
-(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda 
-(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x 
-x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda 
-(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 
-i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 
-(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 
-(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: 
-T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x 
-x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) 
-(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i 
-i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: 
-T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: 
-T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) 
-u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 
-(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) 
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda 
-(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 
-x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 
-t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: 
-T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
-(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v 
-x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: 
-T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 
-u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: 
-T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T 
-(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v 
-(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 
-x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: 
-T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 
-x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both 
-v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: 
-T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq 
-nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k 
-i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat 
-(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda 
-(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 
-H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 
-H4)))))))))))))))))) i1 u1 t0 t1 H))))).
-
-theorem subst0_confluence_eq:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
-i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) 
-(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) 
-(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
-T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to 
-(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: 
-T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda 
-(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef 
-i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T 
-(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) 
-t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) 
-(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda 
-(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) 
-(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: 
-T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 
-(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v 
-t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda 
-(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: 
-T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 
-i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 
-i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda 
-(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T 
-t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda 
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t 
-t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) 
-(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
-(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 
-t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
-T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead 
-k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
-(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x 
-t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda 
-(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v 
-(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v 
-(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) 
-(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x 
-t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead 
-k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k 
-x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 
-x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) 
-(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: 
-T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x 
-t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x 
-t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) 
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) 
-(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead 
-k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) 
-(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 
-v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) 
-(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
-(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: 
-T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x 
-x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: 
-T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x 
-t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x 
-t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) 
-t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) 
-(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) 
-(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x 
-t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x 
-t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t 
-k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) 
-(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
-(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
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-k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) 
-(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead 
-k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda 
-(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead 
-k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 
-i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) 
-(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
-
-theorem subst0_confluence_lift:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
-i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i 
-t2)) \to (eq T t1 t2)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 
-i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i 
-t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: 
-T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S 
-O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) 
-(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def 
-(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i 
-H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) 
-(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: 
-T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) 
-i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i 
-t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) 
-x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S 
-O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) 
-(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift 
-(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) 
-(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) 
-(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) 
-(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i 
-t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) 
-(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) 
-i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) 
-(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
deleted file mode 100644 (file)
index 2d32fca..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
+++ /dev/null
@@ -1,457 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/fwd.ma".
-
-include "basic_1/lift/tlt.ma".
-
-lemma subst0_weight_le:
- \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
-u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift 
-(S i) O v)) (weight_map g (TLRef i)) (le_S_n (S (weight_map f (lift (S i) O 
-v))) (S (weight_map g (TLRef i))) (le_S (S (S (weight_map f (lift (S i) O 
-v)))) (S (weight_map g (TLRef i))) (le_n_S (S (weight_map f (lift (S i) O 
-v))) (weight_map g (TLRef i)) H1)))))))))) (\lambda (v: T).(\lambda (u2: 
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda 
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S 
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
-(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
-g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S 
-(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd 
-g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_O_n O) n))))))))) (\lambda (f: 
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g 
-H2 O O (le_O_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
-u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) 
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g 
-t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
-((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
-f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) 
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
-\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
-T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
-(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S 
-(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
-(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
-(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) 
-(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
-g H2 O O (le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g 
-O) (\lambda (m: nat).(wadd_le f g H2 O O (le_O_n O) m)) (eq_ind nat 
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
-(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
-f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
-t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
-(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
-(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map 
-f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
-(le_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
-(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
-(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
-(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
-(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map 
-g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
-T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le 
-(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
-k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
-t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2) 
-(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
-(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) 
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f 
-g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) 
-(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) 
-(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat 
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 
-(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) 
-(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
-i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
-(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O 
-(le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
-t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
-(\lambda (m: nat).(wadd_le f g H4 O O (le_O_n O) m)) (eq_ind nat (weight_map 
-f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) 
-(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 
-t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
-(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
-t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: 
-(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) 
-(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus 
-(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 
-f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
-
-lemma subst0_weight_lt:
- \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
-u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: 
-T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
-v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda 
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S 
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
-(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
-g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S 
-(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
-O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) 
-(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd 
-f O n) (wadd g O n) (wadd_le f g H2 O O (le_O_n O) n))))))))))) (\lambda (f: 
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f 
-u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) 
-(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n 
-(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O 
-O (le_O_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
-u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) 
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g 
-t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
-((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
-f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) 
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
-\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
-T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
-(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
-((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
-nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
-(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
-(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
-(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f 
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f 
-u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
-g H2 O O (le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
-O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd 
-g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_O_n O) m)) (eq_ind nat 
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
-(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
-f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
-t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
-(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
-(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
-f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
-(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
-(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
-(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
-(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
-(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map 
-g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
-T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt 
-(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
-k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
-t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) 
-(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
-(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) 
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
-t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
-(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 
-f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) 
-(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S 
-(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) 
-(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O 
-v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f 
-u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) 
-f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
-t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
-t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
-(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) 
-(wadd g O m) (wadd_le f g H4 O O (le_O_n O) m)))) (eq_ind nat (weight_map f 
-(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
-i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
-(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) 
-(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_O_n O) 
-m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g 
-i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v 
-(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) 
-\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) 
-(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
-(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
-f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) 
-(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) 
-(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t 
-z H))))).
-
-lemma subst0_tlt_head:
- \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt 
-(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
-z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
-(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map 
-(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map 
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n 
-(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda 
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n 
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) 
-(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: 
-nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda 
-(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: 
-nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) 
-(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda 
-(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda 
-(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
-
-lemma subst0_tlt:
- \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z 
-(THead (Bind Abbr) u t)))))
-\def
- \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
-z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx 
-(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma
deleted file mode 100644 (file)
index e54fcfc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/defs.ma
+++ /dev/null
@@ -1,22 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/defs.ma".
-
-inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
-| subst1_refl: subst1 i v t1 t1
-| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma
deleted file mode 100644 (file)
index 48d11a4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/fwd.ma
+++ /dev/null
@@ -1,175 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst1/defs.ma".
-
-include "basic_1/subst0/fwd.ma".
-
-implied lemma subst1_ind:
- \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (P: ((T \to 
-Prop))).((P t1) \to (((\forall (t2: T).((subst0 i v t1 t2) \to (P t2)))) \to 
-(\forall (t: T).((subst1 i v t1 t) \to (P t))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (P: ((T \to 
-Prop))).(\lambda (f: (P t1)).(\lambda (f0: ((\forall (t2: T).((subst0 i v t1 
-t2) \to (P t2))))).(\lambda (t: T).(\lambda (s0: (subst1 i v t1 t)).(match s0 
-with [subst1_refl \Rightarrow f | (subst1_single x x0) \Rightarrow (f0 x 
-x0)])))))))).
-
-lemma subst1_gen_sort:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
-i v (TSort n) x) \to (eq T x (TSort n))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T 
-t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 
-i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x 
-H))))).
-
-lemma subst1_gen_lref:
- \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
-i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift 
-(S n) O v))))))))
-\def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
-(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or 
-(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl 
-(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O 
-v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v 
-(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq 
-nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 
-(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) 
-(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x 
-H))))).
-
-lemma subst1_gen_head:
- \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
-(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst1 (s k i) v t1 t2))))))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) 
-x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
-T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 
-t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal 
-T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda 
-(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
-u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 
-(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda 
-(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda 
-(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 
-x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 
-x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: 
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
-T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: 
-T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v 
-t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) 
-(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda 
-(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 
-i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 
-x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) 
-H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
-
-lemma subst1_gen_lift_lt:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) 
-x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
-(t2: T).(subst1 i u t1 t2)))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S 
-(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) 
-(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) 
-(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
-(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: 
-T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) 
-(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) 
-(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h 
-(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda 
-(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 
-t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) 
-x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T 
-t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 
-(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x 
-H))))))).
-
-lemma subst1_gen_lift_eq:
- \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
-(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u 
-(lift h d t) x) \to (eq T x (lift h d t))))))))))
-\def
- \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
-(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d 
-h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) 
-(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda 
-(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t 
-u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
-
-lemma subst1_gen_lift_ge:
- \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
-(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) 
-i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
-T).(subst1 (minus i h) u t1 t2))))))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) 
-x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda 
-(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: 
-T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift 
-h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 
-(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: 
-T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: 
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) 
-(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 
-(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d 
-x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: 
-T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 
-H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h 
-d H1 H0)))) x H)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma
deleted file mode 100644 (file)
index f7bf1d1..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/props.ma
+++ /dev/null
@@ -1,165 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst1/fwd.ma".
-
-include "basic_1/subst0/props.ma".
-
-theorem subst1_head:
- \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 
-i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s 
-k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
-\def
- \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: 
-K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i 
-v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k 
-i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) 
-(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k 
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k 
-v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 
-t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 
-(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead 
-k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) 
-(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k 
-i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both 
-v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
-
-lemma subst1_lift_lt:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 
-i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i 
-(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: 
-nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) 
-(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i 
-d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d 
-t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: 
-nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h 
-(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d 
-H1 h))))))) t2 H))))).
-
-lemma subst1_lift_ge:
- \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
-(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 
-(plus i h) u (lift h d t1) (lift h d t2)))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: 
-T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h 
-d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u 
-(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda 
-(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) 
-(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
-
-lemma subst1_ex:
- \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: 
-T).(subst1 d u t1 (lift (S O) d t2))))))
-\def
- \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex 
-T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: 
-nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) 
-(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d 
-u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) 
-(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d 
-(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda 
-(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) 
-d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) 
-t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S 
-O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T 
-(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T 
-(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) 
-(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) 
-t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S 
-O) n (lift n O u)) (lift_free u n (S O) O n (le_plus_r O n) (le_O_n n)))) d 
-H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) 
-(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: 
-T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef 
-(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: 
-T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift 
-(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T 
-(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: 
-nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: 
-T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u 
-(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u 
-t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in 
-(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex 
-T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda 
-(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) 
-x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d 
-t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k 
-d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t 
-(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k 
-x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
-
-lemma subst1_lift_S:
- \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i 
-(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
-\def
- \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: 
-nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i 
-t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: 
-(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift 
-(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef 
-h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) 
-(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S 
-i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: 
-(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) 
-(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) 
-(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T 
-(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i 
-(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) 
-(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S_n (S n) (S i) 
-(le_S (S (S n)) (S i) (le_n_S (S n) i H0)))))) (\lambda (H0: (eq nat n 
-i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: nat).(le h n0)) H n H0) in 
-(eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef h) (lift (S h) (S n0) 
-(TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t: 
-T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) (eq_ind_r T (TLRef (plus 
-n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (eq_ind nat (S 
-(plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) 
-(eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n) 
-(TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: nat).(subst1 n 
-(TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O (TLRef h)) (\lambda 
-(t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n (TLRef h) (TLRef n) 
-(lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) (TLRef (plus h (S n))) 
-(lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) (sym_eq nat (S (plus h 
-n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) (plus_sym n h)) (plus n (S 
-h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) (lift_lref_ge n (S h) n (le_n 
-n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt n (S h) (S n) (le_n (S n)))) 
-i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T (TLRef (plus n (S h))) (\lambda 
-(t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T (TLRef 
-(plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) (TLRef (plus n (S h))) 
-t)) (subst1_refl i (TLRef h) (TLRef (plus n (S h)))) (lift (S h) i (TLRef n)) 
-(lift_lref_ge n (S h) i (le_S_n i n (le_S_n (S i) (S n) (le_S (S (S i)) (S n) 
-(le_n_S (S i) n H0)))))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) 
-(S i) H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: 
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) 
-(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: 
-nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) 
-t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: 
-(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) 
-t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) 
-(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: 
-T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) 
-t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i 
-h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S 
-(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift 
-(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k 
-(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) 
-(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma
deleted file mode 100644 (file)
index 9558c84..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/subst1/subst1.ma
+++ /dev/null
@@ -1,196 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst1/fwd.ma".
-
-include "basic_1/subst0/subst0.ma".
-
-theorem subst1_subst1:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i 
-u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: 
-T).(subst1 (S (plus i j)) u t t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
-(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: 
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda 
-(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 
-t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: 
-(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda 
-(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl 
-(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 
-t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 
-i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: 
-T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S 
-(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 
-y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: 
-T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) 
-(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda 
-(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 
-t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: 
-T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) 
-(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: 
-(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 
-(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: 
-T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T 
-(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u 
-t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: 
-(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 
-t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 
-x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j 
-H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))).
-
-theorem subst1_subst1_back:
- \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 
-j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i 
-u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: 
-T).(subst1 (S (plus i j)) u t2 t)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
-(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: 
-T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda 
-(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t 
-t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: 
-(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda 
-(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl 
-(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 
-t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 
-i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: 
-T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) 
-(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S 
-(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i 
-j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T 
-(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u 
-t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S 
-(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 
-x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: 
-T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x 
-(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) 
-(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
-
-theorem subst1_trans:
- \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 
-i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 
-t3)))))))
-\def
- \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: 
-T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda 
-(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 
-t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 
-(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: 
-T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans 
-t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
-
-theorem subst1_confluence_neq:
- \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: 
-nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall 
-(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda 
-(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: 
-nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: 
-T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) 
-\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) 
-(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: 
-T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not 
-(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda 
-(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: 
-T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: 
-T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not 
-(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: 
-T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T 
-(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 
-(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: 
-T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 
-u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 
-i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda 
-(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T 
-(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x 
-(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) 
-(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 
-H2))))) t3 H1)))))))) t1 H))))).
-
-theorem subst1_confluence_eq:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 
-i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: 
-T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: 
-T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) 
-(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: 
-(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda 
-(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: 
-T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i 
-u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 
-i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: 
-T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u 
-t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u 
-t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) 
-(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) 
-(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) 
-(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda 
-(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 
-T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 
-(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T 
-(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 
-t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i 
-u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i 
-u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: 
-(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda 
-(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 
-x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: 
-T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u 
-t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 
-t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 
-i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) 
-(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
-
-theorem subst1_confluence_lift:
- \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 
-i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i 
-t2)) \to (eq T t1 t2)))))))
-\def
- \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) 
-(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 
-i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: 
-(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i 
-t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 
-t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda 
-(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda 
-(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 
-\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u 
-(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda 
-(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) 
-H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: 
-(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: 
-T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 
-(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T 
-(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 
-t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 
-(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: 
-(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: 
-T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in 
-(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r 
-nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i 
-(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: 
-(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def 
-(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in 
-(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) 
-H3))))))) y H0))) H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/theory.ma b/matita/matita/contribs/lambdadelta/basic_1/theory.ma
deleted file mode 100644 (file)
index 771c60e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/theory.ma
+++ /dev/null
@@ -1,40 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/subst0/tlt.ma".
-
-include "basic_1/subst/props.ma".
-
-include "basic_1/sty1/cnt.ma".
-
-include "basic_1/ex0/props.ma".
-
-include "basic_1/pr3/wcpr0.ma".
-
-include "basic_1/ex2/props.ma".
-
-include "basic_1/ex1/props.ma".
-
-include "basic_1/ty3/sty0.ma".
-
-include "basic_1/csubt/csuba.ma".
-
-include "basic_1/ty3/fwd_nf2.ma".
-
-include "basic_1/ty3/nf2.ma".
-
-include "basic_1/wf3/props.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma
deleted file mode 100644 (file)
index d0eed7f..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlist/defs.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-inductive TList: Type[0] \def
-| TNil: TList
-| TCons: T \to (TList \to TList).
-
-rec definition THeads (k: K) (us: TList) on us: T \to T \def \lambda (t: 
-T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u 
-(THeads k ul t))]).
-
-rec definition TApp (ts: TList) on ts: T \to TList \def \lambda (v: T).(match 
-ts with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t 
-(TApp ts0 v))]).
-
-rec definition tslen (ts: TList) on ts: nat \def match ts with [TNil 
-\Rightarrow O | (TCons _ ts0) \Rightarrow (S (tslen ts0))].
-
-definition tslt:
- TList \to (TList \to Prop)
-\def
- \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma
deleted file mode 100644 (file)
index 65b3b60..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlist/fwd.ma
+++ /dev/null
@@ -1,70 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/tlist/props.ma".
-
-fact tslt_wf__q_ind:
- \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
-\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
-\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen 
-ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
-(tslen ts)))))).
-
-lemma tslt_wf_ind:
- \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
-TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
-TList).(P ts)))
-\def
- let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
-TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
-Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt 
-(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: 
-TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: 
-nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda 
-(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) 
-m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 
-\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to 
-(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) 
-H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
-ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
-
-lemma tlist_ind_rev:
- \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
-TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
-TList).(P ts))))
-\def
- \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: 
-((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts 
-t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) 
-(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: 
-TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: 
-TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: 
-TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) 
-\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) 
-\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in 
-(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t 
-t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) 
-(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: 
-T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat 
-(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P 
-t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen 
-(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) 
-H4))))) H3))))))) ts2)) ts)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma
deleted file mode 100644 (file)
index 267f9d9..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlist/props.ma
+++ /dev/null
@@ -1,64 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/tlist/defs.ma".
-
-lemma theads_tapp:
- \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T 
-(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
-\def
- \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: 
-TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads 
-k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda 
-(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k 
-v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 
-(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads 
-k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
-
-lemma tcons_tapp_ex:
- \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
-\def
- \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 
-TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp 
-ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen 
-ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda 
-(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal 
-TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T 
-(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 
-t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen 
-ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in 
-(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t 
-t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) 
-(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq 
-TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda 
-(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda 
-(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq 
-nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: 
-TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons 
-t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S 
-(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: 
-nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons 
-t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq 
-nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: 
-TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) 
-(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) 
-(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat 
-(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma
deleted file mode 100644 (file)
index 1747400..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlt/defs.ma
+++ /dev/null
@@ -1,43 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/defs.ma".
-
-definition wadd:
- ((nat \to nat)) \to (nat \to (nat \to nat))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n 
-with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
-
-rec definition weight_map (f: (nat \to nat)) (t: T) on t: nat \def match t 
-with [(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f 
-u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f 
-O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) 
-t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f 
-t0)))])].
-
-definition weight:
- T \to nat
-\def
- weight_map (\lambda (_: nat).O).
-
-definition tlt:
- T \to (T \to Prop)
-\def
- \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma
deleted file mode 100644 (file)
index f53eb19..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlt/fwd.ma
+++ /dev/null
@@ -1,46 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/tlt/defs.ma".
-
-fact tlt_wf__q_ind:
- \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to 
-Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 
-t))))) P n))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: 
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to 
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) 
-n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight 
-t)))))).
-
-lemma tlt_wf_ind:
- \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) 
-\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
-\def
- let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: 
-T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to 
-Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) 
-(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind 
-(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: 
-T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
-\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat 
-(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
-(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P 
-t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt 
-(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight 
-v))))))))))))) t)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma
deleted file mode 100644 (file)
index 6df71b3..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/tlt/props.ma
+++ /dev/null
@@ -1,238 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/T/fwd.ma".
-
-include "basic_1/tlt/defs.ma".
-
-lemma wadd_le:
- \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
-nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to 
-(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: 
-((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: 
-nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: 
-nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le 
-(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
-
-lemma wadd_lt:
- \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
-nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to 
-(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
-\def
- \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: 
-((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: 
-nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: 
-nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S_n (S v) (S w) (le_S 
-(S (S v)) (S w) (le_n_S (S v) w H0)))) (\lambda (n0: nat).(\lambda (_: (le 
-(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
-
-lemma wadd_O:
- \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: 
-nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat 
-(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
-
-lemma weight_le:
- \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) 
-(weight_map g t)))))
-\def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
-\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda 
-(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall 
-(n0: nat).(le (f n0) (g n0))))).(le_O_n (weight_map g (TSort n))))))) 
-(\lambda (n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) 
-(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat 
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g 
-n)))) \to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: 
-T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) 
-\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) 
-(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: 
-B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) 
-(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
-\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
-\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) 
-(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus 
-(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus 
-(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr 
-\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g 
-t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g 
-O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) 
-t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
-\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda 
-(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
-(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g 
-t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus 
-(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus 
-(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) 
-(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S 
-(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g 
-H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) 
-(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) 
-(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) 
-(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: 
-((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f 
-t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) 
-(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat 
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le 
-(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) 
-(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map 
-f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) 
-(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O 
-(le_O_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat 
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g 
-n)))) \to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: 
-T).(\lambda (H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) 
-(weight_map g t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus 
-(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) 
-(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) 
-(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f 
-O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_O_n O) n)))))))))))) 
-b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) 
-\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda 
-(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
-(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g 
-t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus 
-(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g 
-t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) 
-(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
-
-lemma weight_eq:
- \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f 
-t) (weight_map g t)))))
-\def
- \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym 
-(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: 
-nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) 
-(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: 
-nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
-
-lemma weight_add_O:
- \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) 
-(weight_map (\lambda (_: nat).O) t))
-\def
- \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: 
-nat).O) (\lambda (n: nat).(wadd_O n))).
-
-lemma weight_add_S:
- \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) 
-O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
-\def
- \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) 
-(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: 
-nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_O_n O)) O (S m) (le_O_n (S 
-m)) n)))).
-
-theorem tlt_trans:
- \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to 
-(tlt u t)))))
-\def
- \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) 
-(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) 
-(weight v) (weight t) H H0))))).
-
-lemma tlt_head_sx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead 
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall 
-(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr 
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst 
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda 
-(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
-(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: 
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: 
-nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map 
-(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map 
-(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) 
-u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda 
-(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l 
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) 
-t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S 
-(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) 
-(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: 
-nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
-
-lemma tlt_head_dx:
- \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
-\def
- \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
-(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead 
-k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall 
-(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr 
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst 
-\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda 
-(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda 
-(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S 
-(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: 
-nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: 
-nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S 
-(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) 
-(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) 
-u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
-(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd 
-(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda 
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t 
-(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) 
-(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map 
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) 
-(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: 
-nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus 
-(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: 
-nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: 
-nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map 
-(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) 
-(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map 
-(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: 
-nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S 
-(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) 
-(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: 
-nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda 
-(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: 
-T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) 
-(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: 
-nat).O) t)))))) k).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma
deleted file mode 100644 (file)
index 4380212..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity.ma
+++ /dev/null
@@ -1,182 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/pr3_props.ma".
-
-include "basic_1/arity/pr3.ma".
-
-include "basic_1/asucc/fwd.ma".
-
-lemma ty3_arity:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
-t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity 
-g c t2 (asucc g a1))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g 
-c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity 
-g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: 
-T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A 
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
-a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda 
-(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) 
-(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 
-(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: 
-(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A 
-(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g 
-a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: 
-(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: 
-T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: 
-A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) 
-(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 
-x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
-g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g 
-(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 
-g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) 
-H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A 
-(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 
-(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort 
-g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
-(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
-a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
-(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g 
-a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g 
-d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) 
-(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g 
-c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr 
-c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
-(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
-a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
-(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
-a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d 
-t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind 
-A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g 
-c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
-a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A 
-(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 
-(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g 
-d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g 
-d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) 
-H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity 
-g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: 
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
-u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) 
-u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g 
-a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) 
-(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
-g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) 
-u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u 
-x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A 
-(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: 
-A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: 
-A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead 
-(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead 
-c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc 
-g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A 
-(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 
-(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) 
-(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g 
-x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to 
-((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: 
-A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 
-(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 
-(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 
-(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u 
-t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) 
-x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr 
-not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g 
-(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind 
-Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead 
-(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) 
-(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 
-(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead 
-x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) 
-H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead 
-x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 
-x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g 
-x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) 
-(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 
-(arity_bind g Void not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void 
-not_void_abst c0 u x H5 t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) 
-H4)))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: 
-(ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) 
-(\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A 
-(\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind 
-Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: 
-A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A 
-(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: 
-A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) 
-(\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u 
-(asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v 
-a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 
-A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: 
-A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) 
-(\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 
-(THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t 
-(asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A 
-(asucc g x0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u 
-(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind 
-Abst) u) t a2))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) 
-a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u 
-t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A 
-(asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g 
-x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def 
-(sym_eq A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g 
-x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) 
-(\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 
-(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) 
-w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: 
-(eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def 
-(eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 
-(asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v 
-a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead 
-(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w 
-(THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl 
-g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g 
-x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind 
-Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) 
-H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A 
-(\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
-a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A 
-(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
-a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) 
-(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
-g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat 
-Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 
-x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A 
-(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
-a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) 
-(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) 
-(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 
-t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
-Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) 
-(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g 
-x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 
-(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) 
-H4)))))))))) c t1 t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma
deleted file mode 100644 (file)
index fb0df9e..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/arity_props.ma
+++ /dev/null
@@ -1,105 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/arity.ma".
-
-include "basic_1/sc3/arity.ma".
-
-lemma ty3_predicative:
- \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: 
-T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: 
-Prop).P)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: 
-T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u 
-v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: 
-T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: 
-T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v 
-x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def 
-(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def 
-(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind 
-A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: 
-A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c 
-(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 
-\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: 
-A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
-A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
-(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda 
-(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g 
-x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def 
-(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) 
-in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead 
-x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g 
-Abst c v t u H1)))))))))).
-
-theorem ty3_repellent:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: 
-T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead 
-c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: 
-Prop).P)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: 
-T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: 
-T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O 
-u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: 
-T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: 
-T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 
-\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop 
-(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 
-(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 
-t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) 
-O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 
-(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind 
-Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda 
-(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g 
-(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: 
-A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity 
-g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def 
-(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in 
-(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda 
-(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity 
-g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g 
-x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g 
-c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) 
-(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) 
-H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) 
-t (lift (S O) O u2) H0))))))))))).
-
-lemma ty3_acyclic:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
-u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
-(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def 
-(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in 
-(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda 
-(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g 
-c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x 
-(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
-
-lemma ty3_sn3:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
-u) \to (sn3 c t)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
-(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in 
-(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u 
-(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t 
-x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t 
-x H1))))) H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma
deleted file mode 100644 (file)
index 7b2c899..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/dec.ma
+++ /dev/null
@@ -1,431 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/dec.ma".
-
-include "basic_1/getl/flt.ma".
-
-include "basic_1/getl/dec.ma".
-
-include "basic_1/flt/fwd.ma".
-
-lemma ty3_inference:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
-T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: 
-C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: 
-T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind 
-(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or 
-(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) 
-\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall 
-(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: 
-((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T 
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
-False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) 
-(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda 
-(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) 
-(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 
-c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: 
-T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let 
-H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
-(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) 
-\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) 
-t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: 
-(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead 
-e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
-(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g 
-c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) 
-(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 
-(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) 
-in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g 
-x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) 
-(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T 
-(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: 
-T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g 
-x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or 
-(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
-(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) 
-x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall 
-(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: 
-T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x 
-H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T 
-(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef 
-n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) 
-(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 
-(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 
-(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) 
-(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 
-(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind 
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O 
-t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: 
-(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind 
-Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 
-(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) 
-(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) 
-in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
-with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
-\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) 
-I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 
-(CHead x3 (Bind Abbr) x4) H10)) in (False_ind False H13))))))))) H8)) 
-(\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False 
-(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift 
-(S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst) 
-x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind 
-Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) 
-(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) 
-in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
-with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
-\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) 
-I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 
-(CHead x3 (Bind Abst) x4) H10)) in (False_ind False H13))))))))) H8)) 
-(ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda (H4: ((\forall (t3: 
-T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g 
-c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) 
-(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 
-(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t))))) False (\lambda (H6: (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind 
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O 
-t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: 
-(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) 
-x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind 
-x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) 
-(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in 
-(let H11 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2) 
-(CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead 
-x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k with 
-[(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) 
-x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 
-(CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) 
-(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 
-(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda (_: (eq B x1 
-Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: 
-T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 \def 
-(eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def 
-(eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind Abbr) x2))) H16 x0 
-H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 
-x0 H15) in (H4 x5 H19)))))))) H12)) H11))))))))) H6)) (\lambda (H6: (ex3_3 C 
-T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 
-c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t)))) False (\lambda (x3: C).(\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) t3)).(\lambda 
-(H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 g x3 x4 
-x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c0: C).(getl n 
-c2 c0)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n 
-H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 
-(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal 
-C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow x1 | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
-x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 
-(CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow x2 | (CHead 
-_ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) 
-(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in 
-(\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def 
-(eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 
-H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 
-H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) 
-t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 
-(CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 
-(\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H20))))))))) H12)) 
-H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) H3)))))) H1)) (\lambda 
-(H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))))).(or_intror 
-(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
-(TLRef n) t3) \to False)) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) 
-t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
-T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False 
-(\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
-T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift 
-(S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) 
-x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 
-False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
-(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
-t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: 
-(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) 
-x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 
-False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 
-t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: 
-T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g 
-c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: 
-T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to 
-(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 
-t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) 
-(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall 
-(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T 
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
-False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: 
-T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 
-t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall 
-(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T 
-(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
-False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in 
-(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 
-t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) 
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) 
-(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda 
-(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) 
-t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to 
-False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 
-(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T 
-(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 
-g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g 
-c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t 
-t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 
-(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) 
-t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) 
-(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda 
-(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 
-g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 
-g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b 
-t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) 
-t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead 
-(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) 
-\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) 
-t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) 
-t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: 
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda 
-(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 
-x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: 
-((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda 
-(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g 
-c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: 
-T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 
-g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 
-t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead 
-(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead 
-c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) 
-H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt 
-c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda 
-(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t 
-t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 
-g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
-(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) 
-\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 
-c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 
-t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 
-c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: 
-T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T 
-(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t 
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
-(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) 
-(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 
-(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g 
-c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda 
-(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: 
-T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: 
-T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
-(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) 
-(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: 
-T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
-False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def 
-(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def 
-H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) 
-(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall 
-(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: 
-T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def 
-(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 
-x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: 
-T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: 
-T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: 
-T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 
-v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or 
-(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 
-T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) 
-(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) 
-(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda 
-(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 
-c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 
-(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 
-v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: 
-T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 
-g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda 
-(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 
-\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 
-\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) 
-H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) 
-t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
-False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) 
-(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred 
-g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 
-x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 
-(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: 
-T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 
-g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda 
-(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda 
-(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: 
-T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) 
-t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: 
-(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 
-\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 
-(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) 
-H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 
-(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 
-H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 
-t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to 
-False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
-False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) 
-t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat 
-Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 
-g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 
-t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead 
-(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 
-(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind 
-Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) 
-(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex 
-T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: 
-T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T 
-(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind 
-Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind 
-Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t 
-(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) 
-x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t 
-t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt 
-c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 
-t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def 
-(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: 
-T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T 
-(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t 
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) 
-(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
-(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 
-(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g 
-c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda 
-(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
-(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: 
-T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: 
-T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
-t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) 
-(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
-(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 
-x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 
-t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) 
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
-(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
-(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
-t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
-Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t 
-x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to 
-False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) 
-t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) 
-(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) 
-t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) 
-(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False 
-(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda 
-(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def 
-(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 
-x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g 
-c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda 
-(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T 
-(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
-T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: 
-T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda 
-(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 
-t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: 
-(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 
-t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 
-H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to 
-False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
-t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
-False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) 
-t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) 
-(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False 
-(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda 
-(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: 
-T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 
-x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) 
-H4))) f H2))) k H1))))))) t2))) c t1))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma
deleted file mode 100644 (file)
index 2531108..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/defs.ma
+++ /dev/null
@@ -1,49 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/G/defs.ma".
-
-include "basic_1/pc3/defs.ma".
-
-inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
-| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) 
-\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to 
-(ty3 g c u t2))))))))
-| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next 
-g m))))
-| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
-T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to 
-(ty3 g c (TLRef n) (lift (S n) O t))))))))
-| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
-T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to 
-(ty3 g c (TLRef n) (lift (S n) O u))))))))
-| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to 
-(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) 
-u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))
-| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to 
-(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 
-g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u 
-t)))))))))
-| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) 
-\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) 
-(THead (Flat Cast) t0 t2))))))).
-
-inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
-| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c 
-TNil u)))
-| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
-TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma
deleted file mode 100644 (file)
index d6d6beb..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fsubst0.ma
+++ /dev/null
@@ -1,975 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/props.ma".
-
-include "basic_1/pc3/fsubst0.ma".
-
-include "basic_1/getl/getl.ma".
-
-lemma ty3_fsubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
-t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
-Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
-(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
-(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: 
-T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: 
-T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall 
-(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 
-c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 
-t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u 
-t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: 
-C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c 
-(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c 
-t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
-T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda 
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) 
-\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u 
-t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
-u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) 
-e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda 
-(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 
-t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u 
-(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 
-(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: 
-(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c 
-c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) 
-u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) 
-t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c 
-t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 
-H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: 
-nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 
-i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: 
-C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to 
-(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i 
-u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) 
-u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) 
-(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda 
-(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: 
-T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: 
-(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind 
-Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g 
-m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
-(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: 
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 
-t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
-t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda 
-(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c 
-(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead 
-e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: 
-T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: 
-(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S 
-n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda 
-(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: 
-T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: 
-nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C 
-(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind 
-Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) 
-H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
-(Bind Abbr) u0) H8)) in ((let H11 \def (f_equal C T (\lambda (e0: C).(match 
-e0 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
-n H0 (CHead e (Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 
-\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 
-d H12) in (let H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d 
-(Bind Abbr) t4))) H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift 
-(S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop 
-Abbr c d u n H14)) u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n 
-H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: 
-C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 
-(TLRef n) (lift (S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def 
-(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
-(getl n c3 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) 
-u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T 
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
-(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
-n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) 
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S 
-n) O t0)) (\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 
-d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 
-w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 
-(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) 
-(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 
-(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda 
-(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda 
-(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
-c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
-((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
-(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i 
-(S n)) u0 t3 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: 
-C).(getl n c3 (CHead c0 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl n c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let 
-H20 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind 
-Abbr) x3) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) 
-c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) 
-(CHead d (Bind Abbr) x3) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S 
-n)) (S i) (le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in 
-(ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus 
-i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) 
-x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 
-B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq 
-C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: 
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 
-(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
-(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
-u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: 
-B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C 
-(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 
-(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 
-x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
-\Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
-(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
-(eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u 
-H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S 
-n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
-B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abbr H15) in (let H20 \def (eq_ind 
-nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e 
-(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 
-(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead 
-x2 (Bind Abbr) u) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) 
-(le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr 
-g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) 
-u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n 
-(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) 
-x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) 
-H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
-Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) 
-(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda 
-(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
-(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 
-x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
-d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
-\Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
-(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
-(eq_ind_r T x3 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 
-c0 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 
-(CHead x2 (Bind b) x4))) H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i 
-n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) 
-u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
-i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 
-(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
-(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr g n c3 x2 x4 H20 t0 (H2 
-(minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) 
-e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S 
-n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) x2 (CHead e 
-(Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) H14)) H13))))))))))) H8)) 
-H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u (csubst0_getl_ge i n H6 c 
-c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda 
-(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 
-c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift 
-(S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O 
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) 
-O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
-(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: 
-nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind 
-Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) 
-(getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
-(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
-(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
-(Bind Abbr) u0) H9)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match 
-e0 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
-n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 
-\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 
-d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d 
-(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: 
-T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 
-(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) 
-(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d 
-(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i 
-n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
-Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: 
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind 
-Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: 
-T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) 
-c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: 
-T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 
-(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) 
-t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) 
-u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S 
-n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O 
-u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) 
-O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
-(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) 
-(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c 
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def 
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
-Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) 
-H8)) in (False_ind (ty3 g c (lift (S n) O u0) (lift (S n) O u)) H10)))) t3 
-H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda (c3: C).(\lambda (H4: 
-(csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind 
-Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (H6: 
-(lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind 
-Abst) u) H0) in (or4_ind (getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) 
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) 
-u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i (S n)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) 
-(ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind 
-Abst) u))).(ty3_abst g n c3 d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T 
-(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) 
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
-(minus i (S n)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead 
-e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: 
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) 
-u0 u1 w))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda 
-(x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind 
-Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind 
-x0) x3))).(\lambda (H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def 
-(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
-(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
-x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
-[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind 
-b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) 
-(CHead x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) 
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B 
-Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda 
-(t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) in (let H18 \def 
-(eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind x0) x3))) H10 d 
-H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead d 
-(Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus i n) 
-(\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind Abbr) 
-u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
-i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n H19 
-(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
-(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u) 
-(lift (S n) O t0) (ty3_lift g d u t0 H1 c3 O (S n) (getl_drop Abst c3 d x3 n 
-H19)) (TLRef n) (lift (S n) O x3) (ty3_abst g n c3 d x3 H19 t0 (H2 (minus i 
-(S n)) u0 d x3 (fsubst0_snd (minus i (S n)) u0 d u x3 H17) e (getl_gen_S 
-(Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20))) (pc3_lift c3 
-d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u (pc3_pr2_x d x3 u (pr2_delta d 
-e u0 (r (Bind Abst) (minus i (S n))) (getl_gen_S (Bind Abst) d (CHead e (Bind 
-Abbr) u0) x3 (minus i (S n)) H20) u u (pr0_refl u) x3 H17))))))))))) H13)) 
-H12))))))))) H8)) (\lambda (H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) 
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 
-(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2))))) (ty3 g c3 (TLRef n) 
-(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x3))).(\lambda (H11: 
-(csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def (f_equal C C (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) 
-(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
-(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abst | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
-\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
-((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) 
-(CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: 
-(eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead 
-x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: 
-C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r 
-B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let 
-H20 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind 
-Abst) u) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) 
-c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) 
-(CHead x2 (Bind Abst) u) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S 
-n)) (S i) (le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in 
-(ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus 
-i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S 
-n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S 
-(Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) 
-H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: 
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
-(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
-(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda 
-(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
-x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
-(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 
-x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) 
-(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
-C).(match e0 with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead 
-d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
-(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
-\Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
-(\lambda (H16: (eq B Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
-(eq_ind_r T x3 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) 
-in (let H19 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 
-c0 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 
-(CHead x2 (Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i 
-n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) 
-u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
-i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 
-(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
-(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u) 
-(lift (S n) O t0) (ty3_lift g x2 u t0 (H2 (minus i (S n)) u0 x2 u 
-(fsubst0_fst (minus i (S n)) u0 d u x2 H19) e (csubst0_getl_ge_back (minus i 
-(S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind 
-Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S 
-n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 n H20)) (TLRef n) (lift (S n) 
-O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 
-(fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back 
-(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e 
-(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus 
-i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop Abst c3 x2 x4 n H20) x4 u 
-(pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) u0 x2 x4 (fsubst0_both 
-(minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) 
-(minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) 
-(getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) 
-H21)))))))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i 
-n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind 
-Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef 
-n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: 
-C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) 
-(eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S n) O u)) (\lambda (H7: (eq 
-nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) 
-O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) O u))) (let H9 \def (eq_ind_r 
-nat i (\lambda (n0: nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in 
-(let H10 \def (eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n 
-H7) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl 
-n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n 
-H0 (CHead e (Bind Abbr) u0) H9)) in (let H12 \def (eq_ind C (CHead d (Bind 
-Abst) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
-(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c 
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 
-g c3 (lift (S n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref 
-u0 t3 i n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: 
-T).(\lambda (t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: 
-nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 
-t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
-t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: 
-(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall 
-(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) 
-t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind 
-Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: 
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead 
-(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda 
-(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) 
-\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: 
-(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i 
-c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead 
-(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: 
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) 
-u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c 
-t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 
-(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
-(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i 
-u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: 
-(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T 
-(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) 
-(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c 
-(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: 
-(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead 
-c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) 
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g 
-c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 
-H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 
-c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) 
-x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) 
-(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind 
-Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) 
-(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd 
-i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 
-(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 
-(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead 
-c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c 
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 
-H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda 
-(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) 
-(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: 
-(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda 
-(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g 
-(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u 
-t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 
-x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x 
-(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c 
-(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 
-(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) 
-u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 
-H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T 
-t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u 
-u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c 
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq 
-T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda 
-(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) 
-(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: 
-T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead 
-(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 
-x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c 
-(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: 
-(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) 
-(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) 
-(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c 
-u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S 
-i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind 
-b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) 
-(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead 
-(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead 
-(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) 
-e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c 
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead 
-c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c 
-(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 
-H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) 
-(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda 
-(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g 
-(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) 
-u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 
-x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 
-t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind 
-b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e 
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g 
-(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 
-(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) 
-(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind 
-Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead 
-(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c 
-c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) 
-(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
-T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead 
-(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind 
-b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
-T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) 
-(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 
-(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead 
-(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind 
-T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead 
-(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g 
-(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 
-(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) 
-(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T 
-(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind 
-b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead 
-c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u 
-x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 
-x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i 
-u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead 
-c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 
-(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) 
-i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) 
-(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind 
-b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) 
-(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead 
-c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) 
-u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u 
-x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) 
-(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 
-(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 
-(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 
-H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) 
-H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) 
-(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: 
-T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) 
-u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: 
-(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 
-x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead 
-(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 
-t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: 
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 
-i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 
-(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 
-(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c 
-(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 
-(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) 
-u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e 
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) 
-(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 
-t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq 
-T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda 
-(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) 
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: 
-T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) 
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind 
-b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) 
-(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: 
-T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: 
-T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) 
-(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind 
-b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) 
-(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x 
-H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i 
-u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 
-(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 
-H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e 
-(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c 
-(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u 
-t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u 
-t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e 
-H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 
-(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead 
-c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head 
-(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 
-(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) 
-u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 
-(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind 
-b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head 
-(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: 
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: 
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: 
-T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: 
-((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
-T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: 
-nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: 
-(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead 
-(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c 
-(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat 
-Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
-u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
-(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: 
-T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
-(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) 
-x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) 
-(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) 
-(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c 
-(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
-(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) 
-x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind 
-Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda 
-(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c 
-(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
-(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u 
-x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind 
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-(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda 
-(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) 
-(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst 
-t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u 
-t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 
-H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead 
-(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u 
-t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead 
-(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x 
-(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 
-H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) 
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-(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda 
-(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat 
-Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w 
-x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead 
-(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x 
-(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda 
-(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) 
-u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c t3 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i 
-u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind 
-Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead 
-(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c 
-(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u 
-t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 
-t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead 
-(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g 
-(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) 
-(ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x4: T).(\lambda (H15: (ty3 g c u x4)).(ty3_conv g c (THead 
-(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind 
-Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind g c u 
-x4 H15 Abst t0 x2 H14)) (THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 
-(THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 (fsubst0_snd i u0 
-c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 (fsubst0_snd (s (Flat 
-Appl) i) u0 c v x1 H10) e H6)) (pc3_fsubst0 c (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c 
-(THead (Flat Appl) w (THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x0 
-(THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) 
-(subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl))) e H6)))) 
-(ty3_correct g c w u H0))))))) (ty3_gen_bind g Abst c u t0 x H11)))) 
-(ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7)) 
-(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda 
-(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e 
-(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 
-H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda 
-(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\lambda (c3: 
-C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H7: (getl i c 
-(CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead 
-(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda 
-(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat 
-Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w 
-u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) 
-(ty3 g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: 
-(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: 
-T).(subst0 i u0 w u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat 
-Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c3 t3 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T t3 
-(THead (Flat Appl) x v))).(\lambda (H10: (subst0 i u0 w x)).(eq_ind_r T 
-(THead (Flat Appl) x v) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind 
-Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 (THead 
-(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 
-c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 
-u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 
-t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind 
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-(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g 
-(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g 
-c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u 
-t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v) 
-(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3 
-x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i 
-u0 c v c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u 
-t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))) i u0 c3 (THead (Flat Appl) x (THead (Bind 
-Abst) u t0)) (fsubst0_both i u0 c (THead (Flat Appl) w (THead (Bind Abst) u 
-t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 
-(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst 
-c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v 
-(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T 
-(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 
-(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead 
-(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 
-g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: 
-T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s 
-(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: 
-T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g 
-c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x 
-(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
-T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c3 t3 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 
-i u0 w x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T 
-(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind 
-Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead 
-(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c3 
-(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 u 
-t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 
-t4))) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c3 (THead 
-(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15: (ty3 g 
-(CHead c3 (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4)) 
-(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x4: T).(\lambda (H16: (ty3 g c3 u x4)).(ty3_conv g c3 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind 
-Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e 
-H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15)) 
-(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) 
-(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) 
-x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 
-c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) i u0 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i 
-u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 
-(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) 
-(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst 
-i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) 
-(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 
-c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i 
-H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda 
-(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: 
-nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 
-t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4 
-t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3: 
-((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: 
-T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: 
-T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead 
-(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) 
-(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind 
-Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: 
-T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: 
-C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda 
-(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 
-u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda 
-(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
-T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 
-t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 
-t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq 
-T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g 
-c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead 
-(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat 
-Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind 
-T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead 
-(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 
-x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) 
-(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 
-x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x 
-H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x 
-(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x 
-H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) 
-(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) 
-(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) 
-(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i 
-u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T 
-(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
-T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 
-(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
-t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq 
-T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 
-x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead 
-(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x 
-(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) 
-(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
-t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
-(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
-(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g 
-c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 
-x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead 
-(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) 
-(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 
-x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 
-x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
-(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 
-x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c 
-t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat 
-Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c 
-t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 
-(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) 
-(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead 
-(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat 
-Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct 
-g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) 
-(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda 
-(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e 
-(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 
-H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda 
-(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda 
-(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl 
-i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 
-(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T 
-(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
-T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat 
-Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: 
-(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: 
-T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat 
-Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat 
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x 
-t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) 
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat 
-Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g 
-c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x 
-t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u 
-c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 
-(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 
-(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i 
-u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat 
-Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) 
-i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) 
-(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e 
-H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e 
-H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead 
-(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
-t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) 
-(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead 
-(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) 
-t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead 
-(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) 
-(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 
-(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda 
-(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) 
-u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
-T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead 
-(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 
-(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: 
-(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) 
-(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
-(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat 
-Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i 
-u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) 
-(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u 
-c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 
-(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 
-t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 
-(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c 
-(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat 
-Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead 
-(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 
-i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst 
-i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 
-t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
-
-lemma ty3_csubst0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
-(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g 
-c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
-nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: 
-C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 
-(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
-
-lemma ty3_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 
-t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e 
-(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 
-t)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
-(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: 
-(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 
-H1) e H0))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
deleted file mode 100644 (file)
index 318a979..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
+++ /dev/null
@@ -1,923 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/defs.ma".
-
-include "basic_1/pc3/props.ma".
-
-implied rec lemma ty3_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
-(\forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to ((P c t2 
-t) \to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((P c u t1) \to 
-((pc3 c t1 t2) \to (P c u t2)))))))))))) (f0: (\forall (c: C).(\forall (m: 
-nat).(P c (TSort m) (TSort (next g m)))))) (f1: (\forall (n: nat).(\forall 
-(c: C).(\forall (d: C).(\forall (u: T).((getl n c (CHead d (Bind Abbr) u)) 
-\to (\forall (t: T).((ty3 g d u t) \to ((P d u t) \to (P c (TLRef n) (lift (S 
-n) O t))))))))))) (f2: (\forall (n: nat).(\forall (c: C).(\forall (d: 
-C).(\forall (u: T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: 
-T).((ty3 g d u t) \to ((P d u t) \to (P c (TLRef n) (lift (S n) O 
-u))))))))))) (f3: (\forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u 
-t) \to ((P c u t) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 
-g (CHead c (Bind b) u) t1 t2) \to ((P (CHead c (Bind b) u) t1 t2) \to (P c 
-(THead (Bind b) u t1) (THead (Bind b) u t2))))))))))))) (f4: (\forall (c: 
-C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to ((P c w u) \to (\forall 
-(v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to ((P c v 
-(THead (Bind Abst) u t)) \to (P c (THead (Flat Appl) w v) (THead (Flat Appl) 
-w (THead (Bind Abst) u t))))))))))))) (f5: (\forall (c: C).(\forall (t1: 
-T).(\forall (t2: T).((ty3 g c t1 t2) \to ((P c t1 t2) \to (\forall (t0: 
-T).((ty3 g c t2 t0) \to ((P c t2 t0) \to (P c (THead (Flat Cast) t2 t1) 
-(THead (Flat Cast) t0 t2))))))))))) (c: C) (t: T) (t0: T) (t1: ty3 g c t t0) 
-on t1: P c t t0 \def match t1 with [(ty3_conv c0 t2 t3 t4 u t5 t6 p) 
-\Rightarrow (f c0 t2 t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t2 t3 t4) u 
-t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t5 t6) p) | (ty3_sort c0 m) 
-\Rightarrow (f0 c0 m) | (ty3_abbr n c0 d u g0 t2 t3) \Rightarrow (f1 n c0 d u 
-g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) d u t2 t3)) | (ty3_abst n c0 d u 
-g0 t2 t3) \Rightarrow (f2 n c0 d u g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 
-f5) d u t2 t3)) | (ty3_bind c0 u t2 t3 b t4 t5 t6) \Rightarrow (f3 c0 u t2 t3 
-((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t2 t3) b t4 t5 t6 ((ty3_ind g P f f0 
-f1 f2 f3 f4 f5) (CHead c0 (Bind b) u) t4 t5 t6)) | (ty3_appl c0 w u t2 v t3 
-t4) \Rightarrow (f4 c0 w u t2 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 w u t2) v 
-t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 v (THead (Bind Abst) u t3) t4)) | 
-(ty3_cast c0 t2 t3 t4 t5 t6) \Rightarrow (f5 c0 t2 t3 t4 ((ty3_ind g P f f0 
-f1 f2 f3 f4 f5) c0 t2 t3 t4) t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t3 
-t5 t6))].
-
-implied rec lemma tys3_ind (g: G) (c: C) (P: (TList \to (T \to Prop))) (f: 
-(\forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (P TNil u))))) (f0: 
-(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
-TList).((tys3 g c ts u) \to ((P ts u) \to (P (TCons t ts) u)))))))) (t: 
-TList) (t0: T) (t1: tys3 g c t t0) on t1: P t t0 \def match t1 with 
-[(tys3_nil u u0 t2) \Rightarrow (f u u0 t2) | (tys3_cons t2 u t3 ts t4) 
-\Rightarrow (f0 t2 u t3 ts t4 ((tys3_ind g c P f f0) ts u t4))].
-
-lemma ty3_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
-(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t 
-x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda 
-(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
-(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: 
-T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u 
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 
-t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: 
-T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 
-\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in 
-(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 
-H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
-(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e 
-with [(TSort n0) \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) 
-\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0: 
-nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort 
-(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
-(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
-(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
-(lift (S n0) O t)) H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
-(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
-(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
-(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
-(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to 
-(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq 
-T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n)) 
-t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let H6 \def 
-(eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Bind 
-b) u t2)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 
-(TSort (next g n)) u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
-c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 
-(TSort (next g n)) (THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead 
-(Flat Appl) w v) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
-(False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w (THead (Bind Abst) 
-u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 
-c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
-t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) 
-t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort n))).(let H6 \def 
-(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) 
-(THead (Flat Cast) t0 t2)) H6))))))))))) c y x H0))) H))))).
-
-lemma ty3_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
-(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda 
-(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t 
-x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) 
-(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: 
-C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) 
-t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e 
-u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 
-c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u 
-t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u 
-t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
-(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: 
-(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) 
-H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift 
-(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) 
-in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) 
-H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
-t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 
-(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
-(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
-t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 
-H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
-t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 
-(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) 
-x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
-(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 
-(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) 
-(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef 
-n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (TSort (next g 
-m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 
-(lift (S n) O u) (TSort (next g m)))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) H2))))) (\lambda (n0: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H1: (getl n0 
-c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d u 
-t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
-n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
-(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
-(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
-(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
-T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
-t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
-u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O 
-t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) 
-(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 
-g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
-n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
-(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
-(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
-(Bind Abst) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
-T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
-t0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O u))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))))) (or_intror (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O u))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
-u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O 
-u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O u)) H6 H2)) n0 H5)))))))))))) 
-(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
-t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda 
-(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
-(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
-(S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 
-(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O u0) t2)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef n))).(let 
-H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
-(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
-n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: 
-T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u 
-t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind 
-Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 
-\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
-(Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T 
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
-(THead (Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) H6)))))))))))) 
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)))) (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T 
-T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) 
-t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq 
-T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(pc3 c0 (lift (S n) O t) t0)))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t))))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TLRef n))).(let H6 
-\def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (THead (Flat 
-Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) 
-H6))))))))))) c y x H0))) H))))).
-
-lemma ty3_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
-T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda 
-(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: 
-T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind b) u) t1 t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq 
-T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 
-T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) 
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: 
-(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda 
-(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: 
-T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
-b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: 
-T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 
-c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u 
-t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind 
-b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u 
-t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind 
-b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) 
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 
-(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 
-g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T 
-(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: 
-T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 
-g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) 
-t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) 
-u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) 
-(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 
-(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) 
-H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
-(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
-False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in 
-(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
-b) u t2) (TSort (next g m))))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u 
-t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
-T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
-t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
-T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
-T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
-(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
-b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
-t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: 
-T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
-t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
-T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
-T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
-(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
-b) u t2) (lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
-t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
-H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: 
-(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T 
-T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) 
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda 
-(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 
-t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) 
-t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) 
-(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind 
-b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind 
-b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match 
-k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind 
-b0) u0 t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda 
-(e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
-(THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u 
-t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow 
-t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq 
-T u0 u)).(\lambda (H10: (eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: 
-T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda 
-(_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: 
-T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: 
-T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 
-t4)))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g 
-(CHead c0 (Bind b0) u0) t3 t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 
-(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead (Bind b) u t3) 
-t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) 
-(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind 
-b) u) t1 t3)))))) H11 b H10) in (let H14 \def (eq_ind B b0 (\lambda (b1: 
-B).(ty3 g (CHead c0 (Bind b1) u0) t1 t2)) H12 b H10) in (eq_ind_r B b 
-(\lambda (b1: B).(ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t3) (THead (Bind b1) u0 t2)))) (\lambda (_: T).(\lambda (t4: 
-T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
-b) u) t1 t3))))) (let H15 \def (eq_ind T u0 (\lambda (t3: T).((eq T t1 (THead 
-(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead 
-c0 (Bind b) t3) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: 
-T).(ty3 g (CHead c0 (Bind b) t3) u t5))) (\lambda (t4: T).(\lambda (_: 
-T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind b) u) t1 t4)))))) H13 u H9) in 
-(let H16 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b) t3) t1 
-t2)) H14 u H9) in (let H17 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 
-c0 (THead (Bind b) u t4) t))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u 
-t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g c0 t3 
-t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: T).(ex3_2 T T (\lambda (t4: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind b) t3 t2)))) 
-(\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) (ex3_2_intro T T (\lambda (t3: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b) u t2)))) 
-(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) t2 t (pc3_refl c0 (THead (Bind 
-b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) H7)) H6))))))))))))) (\lambda 
-(c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
-u0)).(\lambda (_: (((eq T w (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) u0))) (\lambda (_: 
-T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: T).(\lambda (t: T).(\lambda 
-(_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (_: (((eq T v (THead 
-(Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 
-(THead (Bind b) u t2) (THead (Bind Abst) u0 t)))) (\lambda (_: T).(\lambda 
-(t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 
-(Bind b) u) t1 t2))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead 
-(Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-_) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T 
-(\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Flat 
-Appl) w (THead (Bind Abst) u0 t))))) (\lambda (_: T).(\lambda (t0: T).(ty3 g 
-c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t2)))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t3) t2))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: 
-T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t4) t3))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: 
-(eq T (THead (Flat Cast) t2 t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind 
-T (THead (Flat Cast) t2 t0) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t4: T).(\lambda 
-(_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 t2)))) (\lambda 
-(_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 
-g (CHead c0 (Bind b) u) t1 t4)))) H6))))))))))) c y x H0))) H))))))).
-
-lemma ty3_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
-T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: 
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
-T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead 
-(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T 
-(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind 
-Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: 
-T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
-(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u 
-t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u 
-t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
-(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: 
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda 
-(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat 
-Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead 
-(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
-T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u 
-(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u 
-(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq 
-T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda 
-(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda 
-(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) 
-in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead 
-(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w 
-v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
-Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
-T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
-(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
-t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: 
-(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
-(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w 
-(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) 
-(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat 
-Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow False])) I (THead (Flat Appl) w v) H1) in (False_ind (ex3_2 T T 
-(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) (TSort (next g m))))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v 
-(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) 
-H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
-(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
-T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
-(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
-(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
-u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
-(lift (S n) O t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
-(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
-H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
-(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
-T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
-(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
-(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
-u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
-(lift (S n) O u)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
-(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
-H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
-(ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T 
-T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind 
-Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w 
-u0))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 
-g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w 
-v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) 
-u) (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) v (THead (Bind Abst) u0 
-t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w 
-u0))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (THead (Flat Appl) w 
-v))).(let H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T 
-(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u0 t0)) (THead (Bind b) u t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 
-g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g 
-c0 w u0)))) H6))))))))))))) (\lambda (c0: C).(\lambda (w0: T).(\lambda (u: 
-T).(\lambda (H1: (ty3 g c0 w0 u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) 
-w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat 
-Appl) w (THead (Bind Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 
-g c0 v (THead (Bind Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 
-w u0))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 
-(THead (Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) 
-\to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w 
-(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat 
-Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow w0 | (TLRef _) \Rightarrow w0 | 
-(THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) 
-w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow 
-t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: 
-(eq T w0 w)).(let H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead 
-(Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 
-(THead (Flat Appl) w (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) 
-(\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) 
-(\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 
-\def (eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 
-v H7) in (let H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat 
-Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead 
-(Flat Appl) w (THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: 
-T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: 
-T).(ty3 g c0 w u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: 
-T).(ty3 g c0 t0 u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T 
-(\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: 
-T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: 
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
-(THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda 
-(t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda 
-(_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T 
-t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: 
-T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: 
-T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g 
-c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T 
-(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind 
-Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda 
-(H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def 
-(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
-(Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (THead (Flat Cast) 
-t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u 
-t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) c y x 
-H0))) H)))))).
-
-lemma ty3_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall 
-(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: 
-T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
-(\lambda (t0: T).(ty3 g c t2 t0))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T 
-(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 
-T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 
-g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: 
-(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 
-(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda 
-(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) 
-t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) 
-(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 
-t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
-t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
-(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 
-t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 
-t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal 
-T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def 
-(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T 
-(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 
-g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 
-t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 
-(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat 
-Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 
-t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) 
-(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: 
-T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: 
-T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 
-g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: 
-T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) 
-(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) 
-H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: 
-nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def 
-(eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 
-(THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_: T).(ty3 g c0 t1 
-t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n: nat).(\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d 
-(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: 
-(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d 
-(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0: 
-T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2 
-t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T 
-(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t))) 
-(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) 
-H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
-(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 
-T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 
-g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef 
-n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in 
-(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S 
-n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
-t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda 
-(_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to 
-(ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t)) (\lambda (_: 
-T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (b: 
-B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
-u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T 
-(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Cast) t4 t2) t3)) 
-(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t4: T).(ty3 g 
-(CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T (THead (Bind b) u t0) 
-(THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Bind b) u t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t2 
-t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 
-t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: 
-T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: 
-T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (THead 
-(Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 
-t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
-t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead 
-(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 
-T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u 
-t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
-t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 
-t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
-_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
-\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
-False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda 
-(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind 
-Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
-t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 (THead (Flat 
-Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) 
-t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 
-t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: (((eq 
-T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead 
-(Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: 
-T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) (THead 
-(Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) 
-\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in 
-((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
-(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq 
-T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat 
-Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
-t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
-H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 
-t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat 
-Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
-t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
-H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 
-t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 
-(THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g 
-c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 
-(\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: 
-T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) 
-(\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T 
-t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: 
-T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: 
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 
-(THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) 
-H)))))).
-
-lemma tys3_gen_nil:
- \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
-(\lambda (u0: T).(ty3 g c u u0))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil 
-u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: 
-TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda 
-(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq 
-TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: 
-T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList 
-TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda 
-(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: 
-TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to 
-(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t 
-ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: 
-TList).(match ee with [TNil \Rightarrow False | (TCons _ _) \Rightarrow 
-True])) I TNil H4) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u0 u1))) 
-H5))))))))) y u H0))) H)))).
-
-lemma tys3_gen_cons:
- \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
-(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts 
-u)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda 
-(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) 
-(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) 
-(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind 
-g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to 
-(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: 
-T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t 
-ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee with 
-[TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons t ts) H2) 
-in (False_ind (land (ty3 g c t u0) (tys3 g c ts u0)) H3)))))) (\lambda (t0: 
-T).(\lambda (u0: T).(\lambda (H1: (ty3 g c t0 u0)).(\lambda (ts0: 
-TList).(\lambda (H2: (tys3 g c ts0 u0)).(\lambda (H3: (((eq TList ts0 (TCons 
-t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0))))).(\lambda (H4: (eq TList 
-(TCons t0 ts0) (TCons t ts))).(let H5 \def (f_equal TList T (\lambda (e: 
-TList).(match e with [TNil \Rightarrow t0 | (TCons t1 _) \Rightarrow t1])) 
-(TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal TList TList 
-(\lambda (e: TList).(match e with [TNil \Rightarrow ts0 | (TCons _ t1) 
-\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 
-t)).(let H8 \def (eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons 
-t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def 
-(eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let 
-H10 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj 
-(ty3 g c t u0) (tys3 g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma
deleted file mode 100644 (file)
index d98b402..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd_nf2.ma
+++ /dev/null
@@ -1,284 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/arity_props.ma".
-
-include "basic_1/pc3/nf2.ma".
-
-include "basic_1/nf2/fwd.ma".
-
-lemma ty3_gen_appl_nf2:
- \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
-T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: 
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
-T).(nf2 c (THead (Bind Abst) u t))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
-T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda 
-(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
-x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: 
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
-x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g 
-c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in 
-(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 
-x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) 
-w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v 
-(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) 
-(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda 
-(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def 
-(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: 
-T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead 
-(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: 
-T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 
-g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) 
-(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c 
-(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind 
-Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c 
-(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda 
-(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
-x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c 
-(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def 
-(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: 
-T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
-(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
-T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c 
-x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
-b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 
-(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 
-(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda 
-(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: 
-T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: 
-T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c 
-(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead 
-(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) 
-(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w 
-Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead 
-(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind 
-Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 
-x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 
-(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) 
-(ty3_gen_appl g c w v x H))))))).
-
-lemma ty3_inv_lref_nf2_pc3:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c 
-(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to 
-((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda 
-(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t 
-u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c 
-u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda 
-(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: 
-T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: 
-T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift 
-(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 
-c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T 
-(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda 
-(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to 
-((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T 
-(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 
-t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda 
-(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 
-\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 
-\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to 
-(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: 
-T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def 
-(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y 
-\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 
-H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq 
-T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: 
-T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) 
-u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort 
-_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (TLRef i) H1) in (False_ind (ex T (\lambda (u: T).(eq T u2 (lift 
-(S i) O u)))) H5))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
-(TLRef i)) \to ((nf2 d u) \to (\forall (u2: T).((nf2 d u2) \to ((pc3 d t u2) 
-\to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H4: 
-(eq T (TLRef n) (TLRef i))).(\lambda (H5: (nf2 c0 (TLRef n))).(\lambda (u2: 
-T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: (pc3 c0 (lift (S n) O t) u2)).(let 
-H8 \def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n 
-| (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
-i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
-O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 
-(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl 
-n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 
-(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) 
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
-d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: 
-T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S 
-i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: 
-(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: 
-(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow n | (TLRef n0) \Rightarrow n0 | 
-(THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef i) H4) in (let H9 \def 
-(eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) O u) u2)) H7 i H8) in 
-(let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 (TLRef n0))) H5 i H8) 
-in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind 
-Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 (lift (S i) O u) u2 
-H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y d (getl_drop Abst 
-c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 (lift (S i) O t2))) 
-(\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
-u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) O x))).(\lambda 
-(_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: T).(ex T (\lambda 
-(u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda (u0: T).(eq T 
-(lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S i) O x))) u2 
-H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 
-c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T (\lambda 
-(u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: B).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) t1) 
-\to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 
-(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
-u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda 
-(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 
-u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T 
-(THead (Bind b) u t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef i) H5) in (False_ind (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
-u0)))) H9))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef i)) \to ((nf2 
-c0 w) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 u u2) \to (ex T (\lambda 
-(u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (v: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v 
-(TLRef i)) \to ((nf2 c0 v) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 
-(THead (Bind Abst) u t) u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
-u0))))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef 
-i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) w v))).(\lambda (u2: T).(\lambda 
-(_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) 
-u t)) u2)).(let H9 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T 
-(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda 
-(c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: 
-T).((nf2 c0 u2) \to ((pc3 c0 t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift 
-(S i) O u))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda 
-(_: (((eq T t2 (TLRef i)) \to ((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) 
-\to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O 
-u))))))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TLRef 
-i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) t2 t1))).(\lambda (u2: 
-T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead (Flat Cast) t0 t2) 
-u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match 
-ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ 
-_ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: 
-T).(eq T u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
-
-lemma ty3_inv_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c 
-(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: 
-T).(eq T u (lift (S i) O u0))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: 
-(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
-
-lemma ty3_inv_appls_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: 
-T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to 
-((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S 
-i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) 
-u1))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: 
-TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t 
-(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: 
-T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t 
-(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: 
-(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c 
-u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in 
-(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: 
-T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) 
-(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def 
-(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r 
-T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) 
-O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda 
-(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) 
-(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) 
-(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall 
-(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef 
-i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
-(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) 
-u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead 
-(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c 
-(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t 
-(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T 
-T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind 
-Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: 
-T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) 
-u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: 
-T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) 
-u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat 
-Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat 
-Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t 
-x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def 
-(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) 
-x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 
-c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) 
-(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) 
-x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def 
-(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c 
-(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) 
-O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O 
-u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
-(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O 
-x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead 
-(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
-(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S 
-i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c 
-(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c 
-(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t 
-Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
-
-lemma ty3_inv_lref_lref_nf2:
- \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c 
-(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i 
-j)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda 
-(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda 
-(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 
-H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift 
-(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S 
-i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 
-in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x 
-(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x 
-(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt 
-j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda 
-(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S 
-i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) 
-j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) 
-H2))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma
deleted file mode 100644 (file)
index f5a2ebb..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/nf2.ma
+++ /dev/null
@@ -1,455 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/arity.ma".
-
-include "basic_1/pc3/nf2.ma".
-
-include "basic_1/nf2/arity.ma".
-
-definition ty3_nf2_inv_abst_premise:
- C \to (T \to (T \to Prop))
-\def
- \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: 
-T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: 
-TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w 
-u)) \to False)))))))).
-
-lemma ty3_nf2_inv_abst_premise_csort:
- \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise 
-(CSort m) w u)))
-\def
- \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda 
-(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) 
-wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs 
-(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d 
-(Bind Abst) wi) H False))))))))).
-
-lemma ty3_nf2_inv_all:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
-u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
-t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
-(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat 
-(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
-(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) 
-in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda 
-(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
-(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
-T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) 
-u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: 
-(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g 
-c t x H2 H0)))) H1)))))))).
-
-lemma ty3_nf2_inv_sort:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t 
-(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort 
-n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda 
-(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def 
-(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 
-T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) 
-(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: 
-T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t 
-(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t 
-(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) 
-(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat 
-m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
-i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
-(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
-(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) 
-u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind 
-Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: 
-T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda 
-(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 
-x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) 
-x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H 
-(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) 
-(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda 
-(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: 
-T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: 
-T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead 
-(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) 
-(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind 
-Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: 
-(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 
-x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst 
-c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat 
-(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: 
-nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
-(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c 
-x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: 
-nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) 
-(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat 
-m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
-i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def 
-(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in 
-(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T 
-t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) 
-(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort 
-n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef 
-i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda 
-(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) 
-(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
-(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda 
-(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
-i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda 
-(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) 
-(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g 
-c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda 
-(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: 
-nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 
-TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) 
-ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) 
-(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: 
-TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef 
-x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 
-\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat 
-Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) 
-(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda 
-(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda 
-(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: 
-nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
-(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws 
-(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda 
-(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat 
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef 
-x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
-nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) 
-x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) 
-H2)) H1)))))))).
-
-fact ty3_nf2_gen__ty3_nf2_inv_abst_aux:
- \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise 
-c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead 
-(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to 
-(ty3_nf2_inv_abst_premise c w2 u2))))))))
-\def
- \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: 
-C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) 
-\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) 
-(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: 
-T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind 
-Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: 
-T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) 
-wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift 
-(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t 
-(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t 
-(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) 
-vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) 
-w1 u1) H0))))))))))))))).
-
-lemma ty3_nf2_inv_abst:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: 
-T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to 
-((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: 
-T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g 
-c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v 
-u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) 
-v))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c 
-t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w 
-u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in 
-(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq 
-T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c 
-w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) 
-(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
-TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
-TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: 
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: 
-T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 
-c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) 
-u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind 
-Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: 
-T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda 
-(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t 
-(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 
-(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 
-g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r 
-T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
-T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) 
-w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
-x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w 
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
-(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T 
-(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) 
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: 
-T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w 
-x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T 
-(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead 
-(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) 
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) 
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
-(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) 
-(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
-(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) 
-(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g 
-(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: 
-B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda 
-(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w 
-v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: 
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c 
-x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind 
-b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def 
-(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w 
-H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) 
-t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
-T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: 
-T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w 
-x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) 
-H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c 
-x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) 
-(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t 
-H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort 
-n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda 
-(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let 
-H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H 
-(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda 
-(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c 
-(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq 
-T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 
-g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v 
-u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t 
-H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
-nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda 
-(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
-TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
-nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t 
-(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) 
-(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
-(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 
-(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef 
-x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind 
-Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads 
-(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
-T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
-T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
-(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
-(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u 
-(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind 
-Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda 
-(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind 
-Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: 
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w 
-(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) 
-(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T 
-T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) 
-(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) 
-(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c 
-(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to 
-((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda 
-(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: 
-T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda 
-(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind 
-Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c 
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 
-T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x 
-v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C 
-T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O 
-t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind 
-Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c 
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef 
-x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x 
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda 
-(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O 
-x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind 
-Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind 
-Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda 
-(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c 
-(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) 
-(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda 
-(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
-Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
-(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: 
-(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let 
-H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind 
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind 
-Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
-T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) 
-(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: 
-T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: 
-T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to 
-((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda 
-(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: 
-T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
-(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: 
-(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda 
-(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind 
-Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) 
-t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: 
-T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) 
-(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 
-(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let 
-H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let 
-H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T 
-(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) 
-(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 
-w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) 
-(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T 
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat 
-Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
-Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads 
-(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c 
-x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: 
-(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T 
-(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T 
-(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat 
-Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
-(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
-(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
-Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) 
-(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x3 x5) H22) in 
-(False_ind (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat 
-Appl) t0 (THeads (Flat Appl) TNil (TLRef x1))) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v)))) H23))) (\lambda (t2: T).(\lambda (t3: 
-TList).(\lambda (_: (((eq T (THeads (Flat Appl) t3 (TLRef x1)) (THead (Bind 
-Abst) x3 x5)) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead 
-(Flat Appl) t0 (THeads (Flat Appl) t3 (TLRef x1))) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v))))))).(\lambda (H22: (eq T (THeads (Flat 
-Appl) (TCons t2 t3) (TLRef x1)) (THead (Bind Abst) x3 x5))).(let H23 \def 
-(eq_ind T (THead (Flat Appl) t2 (THeads (Flat Appl) t3 (TLRef x1))) (\lambda 
-(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
-False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | 
-(Flat _) \Rightarrow True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind 
-(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 
-(THeads (Flat Appl) (TCons t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) 
-(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
-(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
-T).(nf2 (CHead c (Bind Abst) x) v)))) H23)))))) t1 H18))))))) H17))))))))) 
-(ty3_gen_appl g c t0 (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x 
-x2) H12))))))))) x0)) H10)) H9)) t H5))))))) H4)) H3))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
deleted file mode 100644 (file)
index 33f41eb..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3.ma
+++ /dev/null
@@ -1,676 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/csubt/ty3.ma".
-
-include "basic_1/ty3/subst1.ma".
-
-include "basic_1/ty3/fsubst0.ma".
-
-include "basic_1/pc3/pc1.ma".
-
-include "basic_1/pc3/wcpr0.ma".
-
-include "basic_1/pc1/props.ma".
-
-lemma ty3_sred_wcpr0_pr0:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
-t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) 
-\to (ty3 g c2 t2 t)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
-(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
-(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to 
-(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: 
-T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c 
-c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
-C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 
-t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 
-c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 
-H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 
-H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: 
-C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) 
-t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) 
-(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda 
-(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
-Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: 
-((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g 
-c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: 
-T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: 
-T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
-(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 
-(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
-(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda 
-(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) 
-(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n 
-H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) 
-\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: 
-C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) 
-t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) 
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind 
-Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind 
-Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g 
-c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u 
-(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S 
-n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) 
-O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 
-H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n 
-H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda 
-(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to 
-(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: 
-B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) 
-u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) 
-\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: 
-C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead 
-(Bind b) u t2) t4)).(let H6 \def (match H5 with [(pr0_refl t5) \Rightarrow 
-(\lambda (H6: (eq T t5 (THead (Bind b) u t2))).(\lambda (H7: (eq T t5 
-t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 
-g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: (eq T (THead (Bind b) u t2) 
-t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).(ty3 g c2 t6 (THead 
-(Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t2 t3 (H3 
-(CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 
-(pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead (Bind b) u t2) H6) H7))) | 
-(pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 
-t5) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let 
-H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 
-| (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) 
-(THead (Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
-(THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead (Bind b) u t2) H8) in 
-((let H12 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
-(THead k u1 t5) (THead (Bind b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: 
-K).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 
-u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) 
-(\lambda (H13: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to 
-((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 
-g c2 t4 (THead (Bind b) u t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
-(\lambda (t7: T).((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 
-t7 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) (\lambda (H15: (eq T 
-(THead (Bind b) u2 t6) t4)).(eq_ind T (THead (Bind b) u2 t6) (\lambda (t7: 
-T).((pr0 u u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Bind b) u t3))))) 
-(\lambda (H16: (pr0 u u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda 
-(t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) 
-(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H18: (ty3 g (CHead c2 (Bind 
-b) u) t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) 
-(ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x0: 
-T).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead 
-(Bind b) u t3) (THead (Bind b) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl 
-u)) b t3 x H18) (THead (Bind b) u2 t6) (THead (Bind b) u2 t3) (ty3_bind g c2 
-u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 
-H4 u u2 H16 (Bind b)) t6 H17)) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead 
-(Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) 
-(ty3_correct g (CHead c2 (Bind b) u2) t6 t3 (H3 (CHead c2 (Bind b) u2) 
-(wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))))) (ty3_correct g (CHead c2 
-(Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl 
-u) (Bind b)) t2 (pr0_refl t2)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 
-(sym_eq T u1 u H13))) k (sym_eq K k (Bind b) H12))) H11)) H10)) H9 H6 H7))) | 
-(pr0_beta u0 v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
-Appl) v1 (THead (Bind Abst) u0 t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq 
-T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind Abst) u0 t5)) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) 
-\to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) 
-H10)) H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) 
-\Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 
-t5)) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat 
-Appl) v1 (THead (Bind b0) u1 t5)) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t2) H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 
-v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u 
-t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
-\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
-t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
-_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
-t5) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
-t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
-(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2) H9) in (eq_ind B Abbr 
-(\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead (Bind Abbr) 
-u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 
-g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T 
-u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind Abbr) u2 w) t4) \to 
-((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead 
-(Bind Abbr) u t3)))))))) (\lambda (H15: (eq T t5 t2)).(eq_ind T t2 (\lambda 
-(t7: T).((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) 
-\to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3))))))) 
-(\lambda (H16: (eq T (THead (Bind Abbr) u2 w) t4)).(eq_ind T (THead (Bind 
-Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to ((subst0 O u2 
-t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u t3)))))) (\lambda (H17: (pr0 u 
-u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda (H19: (subst0 O u2 t6 w)).(let H20 
-\def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind 
-b0) u) c3) \to (\forall (t7: T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 
-Abbr H13) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c 
-(Bind b0) u) t2 t3)) H2 Abbr H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead 
-c2 (Bind Abbr) u) t3 t7)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind 
-Abbr) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) 
-t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) 
-(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: 
-T).(\lambda (_: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 
-(THead (Bind Abbr) u t3) (THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 
-H4 u (pr0_refl u)) Abbr t3 x H22) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) 
-u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead 
-c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u 
-u2 H17 (Bind Abbr)) t6 H18) c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x 
-c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 
-(pr2_free c2 u u2 H17) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind 
-Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 
-(Bind Abbr)) t6 H18))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 
-(CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 
-(pr0_refl t2))))))))) t4 H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u 
-H14))) b H13)) H12)) H11)) H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) 
-\Rightarrow (\lambda (H8: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) (THead 
-(Bind b) u t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
-nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
-nat).(plus x (S O))) O t5) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
-b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H11 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
-_) \Rightarrow u0 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 
-(lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
-\Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
-\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) 
-O t5)) (THead (Bind b) u t2) H8) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) 
-\to ((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b1 Abst)) 
-\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: 
-(eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t5) t2) \to 
-((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 
-(THead (Bind b) u t3))))))) (\lambda (H14: (eq T (lift (S O) O t5) 
-t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: T).((eq T t6 t4) \to ((not (eq 
-B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) 
-(\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((not (eq B b 
-Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) (\lambda 
-(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t4)).(let H18 \def 
-(eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) 
-c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 t3)))))) H3 (lift (S 
-O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda (t7: T).(ty3 g (CHead 
-c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in (ex_ind T (\lambda (t7: 
-T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 (THead (Bind b) u t3)) 
-(\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u) t3 x)).(B_ind 
-(\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead c2 (Bind b1) u) t3 
-x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) t3) \to (ty3 g c2 t4 
-(THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda 
-(H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 
-(Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def (ty3_gen_cabbr g (CHead 
-c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O (getl_refl Abbr c2 u) 
-(CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2 
-(drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) (lift (S O) O y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Abbr) u 
-t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: (subst1 O u (lift (S O) 
-O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u t3 (lift (S O) O 
-x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind T x0 (\lambda 
-(t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O (subst1_gen_lift_eq 
-t4 u (lift (S O) O x0) (S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) 
-(\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O 
-(S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) 
-u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) t4 x1 H28 
-(pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift 
-(S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3) 
-(THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3) 
-(THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3 
-(pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u 
-(lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 
-(pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H24))))) (\lambda (H21: 
-(not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 
-x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t4) t3)).(let 
-H24 \def (match (H21 (refl_equal B Abst)) in False with []) in H24)))) 
-(\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind 
-Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O 
-t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) 
-O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 
-(drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
-t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) 
-(ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: 
-(eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def 
-(eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) u) t7 x)) H22 
-(lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) (\lambda (t7: T).(ty3 
-g c2 t4 (THead (Bind Void) u t7))) (let H29 \def (eq_ind_r T x0 (\lambda (t7: 
-T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S O) O H25)) in (ty3_conv g c2 
-(THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u x) (ty3_bind g 
-c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S O) O x1) x H28) t4 x1 H29 
-(pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pc3_pr2_r c2 (THead 
-(Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Void) u (lift 
-(S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u)))))) t3 H26))))))) 
-H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u 
-(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O))))) 
-(ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 (H18 (CHead c2 
-(Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) 
-(pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 H15))) t2 H14)) u0 
-(sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) H9 H6 H7))) | 
-(pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 
-t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def (eq_ind 
-T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
-(THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 t5 t6) \to 
-(ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 (refl_equal T 
-(THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) (\lambda (c: 
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: 
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g 
-c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v 
-(THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to 
-(\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u 
-t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: 
-T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5 
-with [(pr0_refl t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w 
-v))).(\lambda (H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda 
-(t4: T).((eq T t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))))) (\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T 
-(THead (Flat Appl) w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 
-(H3 c2 H4 v (pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) 
-H6) H7))) | (pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T 
-(THead k u1 t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) 
-t2)).((let H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5])) 
-(THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T 
-(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
-\Rightarrow u1 | (THead _ t5 _) \Rightarrow t5])) (THead k u1 t3) (THead 
-(Flat Appl) w v) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e 
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K 
-(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead 
-k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind 
-T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) 
-\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v 
-(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to 
-((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T 
-(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to 
-(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda 
-(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 
-g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 
-g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda 
-(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: 
-T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind 
-Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: 
-(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda 
-(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u 
-x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 
-(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead 
-(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 
-H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead 
-(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 
-c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) 
-(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) 
-u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 
-(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 
-H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T 
-(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w 
-v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
-(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead 
-_ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
-(THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | 
-(THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 
-t3)) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T 
-(THead (Bind Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to 
-((pr0 t5 v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) 
-v)).(eq_ind T (THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind 
-Abbr) v2 t4) t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead 
-(Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead 
-(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: 
-T).((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead 
-(Bind Abst) u t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 
-t4)).(let H16 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c 
-c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u 
-t0))))))) H3 (THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v 
-(\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) 
-u0 t3) H12) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) 
-t5)) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u 
-t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind 
-Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda 
-(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 
-(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u 
-x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind 
-Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 
-(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: 
-T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 
-g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) 
-(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda 
-(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u 
-t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind 
-Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: 
-T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) 
-(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 
-u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) 
-u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w 
-(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 
-H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u 
-(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 
-H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead 
-(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind 
-Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) 
-(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead 
-(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 
-(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x 
-H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) 
-(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 
-t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | 
-(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq 
-T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w 
-v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
-v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead 
-(Bind b) u1 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in ((let H13 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
-_) \Rightarrow v1 | (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 
-(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda 
-(t5: T).((eq T (THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
-v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) 
-v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to 
-((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat 
-Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) 
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b 
-Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 
-(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq 
-B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda 
-(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: 
-C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead 
-(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def 
-(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 
-(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead 
-(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift 
-(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: 
-T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in 
-(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u 
-t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: 
-T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: 
-(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda 
-(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: 
-T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) 
-(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda 
-(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 
-(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind 
-Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead 
-(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 
-x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def 
-(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 
-(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) 
-(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u 
-t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u 
-t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) 
-(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) 
-(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S 
-O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T 
-T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind 
-Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: 
-T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: 
-T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S 
-O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead 
-(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u 
-t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) 
-u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: 
-(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g 
-(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) 
-t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead 
-(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w 
-(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 
-H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead 
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S 
-O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat 
-Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind 
-Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) 
-u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) 
-(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) 
-u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind 
-Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) 
-O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst 
-(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead 
-(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) 
-(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) 
-O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
-(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead 
-(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
-(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl 
-v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O 
-(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) 
-(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w 
-H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind 
-Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u 
-t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) 
-(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat 
-Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) 
-(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) 
-u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) 
-(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind 
-b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g 
-Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind 
-Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 
-(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 
-H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq 
-T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T 
-(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 
-t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) 
-H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to 
-((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 
-H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) 
-(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind 
-T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to 
-((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
-(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) 
-\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) 
-w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) 
-u0 t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
-False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H7) in 
-(False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) 
-w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead 
-(Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: 
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g 
-c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: 
-((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g 
-c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: 
-T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 
-with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat Cast) t3 
-t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3 t2) 
-(\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) 
-(\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat 
-Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3))) 
-(ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl 
-t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp 
-u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead 
-(Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
-_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
-(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match 
-e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
-\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12 
-\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | 
-(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) 
-(THead (Flat Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq 
-T u1 t3) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) 
-\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda 
-(H13: (eq T u1 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T 
-(THead (Flat Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 
-t4 (THead (Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
-(\lambda (t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to 
-((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: 
-(eq T (THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) 
-(\lambda (t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat 
-Cast) t0 t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 
-t6)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat 
-Cast) u2 t6) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g 
-c2 t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
-(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 
-t6) (THead (Flat Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 
-c2 H4 u2 H16) t6 t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 
-H16))) t0 (H3 c2 H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat 
-Cast) t0 t3) (pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 
-u2) (pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct 
-g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) 
-u1 (sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 
-H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda 
-(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead 
-(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
-(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
-(Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to 
-((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) 
-H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow 
-(\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (THead 
-(Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat 
-Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) 
-v1 (THead (Bind b) u1 t5)) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
-with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
-Cast) t3 t2) H10) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
-(lift (S O) O v2) t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
-t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
-\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Flat Cast) 
-t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
-(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
-(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
-(THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) 
-t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 
-t4 (THead (Flat Cast) t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 
-t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) 
-(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def 
-(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
-\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
-False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to 
-((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
-t3))))) H10)) H9 H6 H7))) | (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: 
-(eq T (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq 
-T t6 t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) 
-(THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
-_) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t5) 
-(THead (Flat Cast) t3 t2) H7) in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) 
-\to ((eq T t6 t4) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
-t3)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 
-t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda 
-(H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 
-t4 (THead (Flat Cast) t0 t3)))) (\lambda (H13: (pr0 t2 t4)).(ex_ind T 
-(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 t4 (THead (Flat Cast) t0 t3)) 
-(\lambda (x: T).(\lambda (H14: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead (Flat 
-Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3 
-(pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 H13) (pc3_pr2_x c2 t3 (THead (Flat 
-Cast) t0 t3) (pr2_free c2 (THead (Flat Cast) t0 t3) t3 (pr0_tau t3 t3 
-(pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3))))) 
-t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10))) 
-H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
-t4))))))))))))))) c1 t1 t H))))).
-
-lemma ty3_sred_pr0:
- \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall 
-(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: 
-G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 
-t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
-
-lemma ty3_sred_pr1:
- \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
-(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
-(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: 
-T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: 
-G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: 
-T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: 
-C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: 
-G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t 
-(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
-
-lemma ty3_sred_pr2:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
-G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: 
-G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 
-t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
-G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 
-(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
-H2)))))))))))))) c t1 t2 H)))).
-
-lemma ty3_sred_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
-(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall 
-(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: 
-G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
-(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c 
-t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: 
-(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 
-H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma
deleted file mode 100644 (file)
index 0b897b4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/pr3_props.ma
+++ /dev/null
@@ -1,492 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/pr3.ma".
-
-lemma ty3_cred_pr2:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 
-v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
-(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind 
-b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: 
-C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: 
-B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) 
-t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 
-(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 
-(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda 
-(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 
-t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead 
-c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl 
-c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead 
-c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) 
-(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 
-v2 H))))).
-
-lemma ty3_cred_pr3:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 
-v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
-(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
-(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: 
-B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to 
-(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: 
-B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) 
-t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 
-t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: 
-B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to 
-(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b 
-t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
-
-lemma ty3_gen_lift:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: 
-nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop 
-h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: 
-T).(ty3 g e t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T 
-(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: 
-C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) 
-(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c 
-y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall 
-(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) 
-(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall 
-(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to 
-(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t 
-t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall 
-(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) 
-t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda 
-(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: 
-C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) 
-(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: 
-T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall 
-(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to 
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e 
-x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: 
-(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: 
-T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h 
-x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda 
-(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def 
-(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let 
-H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda 
-(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 
-t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda 
-(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) 
-t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 
-H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: 
-nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift 
-h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort 
-m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort 
-(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e 
-(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: 
-T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 
-(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 
-(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: 
-C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind 
-Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: 
-((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall 
-(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) 
-t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda 
-(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 
-\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 
-h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
-x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: 
-(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda 
-(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 
-(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 
-t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 
-(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
-T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: 
-(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 
-(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 
-\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T 
-t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T 
-(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 
-t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u 
-(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in 
-(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) 
-x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) 
-t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 
-(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) 
-(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) 
-t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S 
-n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift 
-(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda 
-(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) 
-(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift 
-(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) 
-O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus 
-x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) 
-(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g 
-n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) 
-(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) 
-H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n 
-h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
-T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 
-(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
-T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) 
-(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift 
-(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda 
-(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) 
-(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n 
-(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 
-(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus 
-x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n 
-h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O 
-x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n 
-H8) (le_plus_r O (S (minus n h)))) (le_O_n x1))) (ty3_abbr g (minus n h) e d0 
-u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) x0 
-H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
-(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) 
-u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall 
-(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: 
-C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) 
-(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: 
-nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda 
-(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 
-\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 
-h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
-x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: 
-(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda 
-(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 
-(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 
-t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 
-(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C 
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) 
-(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: 
-T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: 
-(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 
-(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 
-\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T 
-t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T 
-(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 
-t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u 
-(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in 
-(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda 
-(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e 
-(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h 
-(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h 
-(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda 
-(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) 
-(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 
-d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 
-x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T 
-(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S 
-n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda 
-(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift 
-h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g 
-e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus 
-x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus 
-(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 
-c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus 
-n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) 
-(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S 
-n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O 
-(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 
-(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst 
-c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land 
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) 
-n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
-t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le 
-(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T 
-(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
-x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T 
-(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: 
-T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T 
-(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O 
-u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 
-O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
-O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) 
-(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h 
-(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) 
-O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus 
-O (S (minus n h))) (le_S_minus x1 h n H8) (le_plus_r O (S (minus n h)))) 
-(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 
-(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) 
-(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u 
-t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
-c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: 
-B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) 
-u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift 
-h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T 
-(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: 
-T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: 
-(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: 
-(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
-(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 
-y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: 
-T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 
-(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: 
-(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda 
-(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u 
-t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda 
-(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to 
-(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda 
-(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 
-g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 
-(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) 
-H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) 
-t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T 
-u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) 
-x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) 
-e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) 
-t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let 
-H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: 
-nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to 
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 
-x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda 
-(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) 
-(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind 
-b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 
-\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda 
-(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) 
-(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: 
-T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 
-x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e 
-(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: 
-T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda 
-(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 
-(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e 
-(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 
-(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead 
-e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) 
-x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) 
-(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) 
-(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 
-T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) 
-t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) 
-x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) 
-(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 
-c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind 
-b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) 
-(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u 
-H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) 
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w 
-u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
-c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: 
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u 
-t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 
-x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
-c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 
-t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead 
-(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 
-e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
-Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) 
-(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) 
-(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 
-x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) 
-x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead 
-(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) 
-(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: 
-nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to 
-(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) 
-(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 
-\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 
-(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall 
-(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: 
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) 
-(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 
-\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in 
-(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 
-(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: 
-T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 
-(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 
-(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
-(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: 
-(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) 
-in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u 
-t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift 
-h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) 
-(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: 
-T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda 
-(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 
-e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u 
-(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda 
-(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
-(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) 
-x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall 
-(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) 
-x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: 
-T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
-Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
-(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 
-e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: 
-T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: 
-T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
-t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda 
-(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda 
-(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: 
-(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 
-x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) 
-(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead 
-(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) 
-(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) 
-x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead 
-(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) 
-(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h 
-(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) 
-(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind 
-Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) 
-(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind 
-Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 
-(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead 
-(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 
-H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) 
-(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind 
-Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 
-H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) 
-(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) 
-(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 
-H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda 
-(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to 
-(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h 
-x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: 
-T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall 
-(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to 
-(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e 
-x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T 
-(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: 
-(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
-(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h 
-x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
-(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda 
-(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq 
-T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 
-x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) 
-x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead 
-(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind 
-T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 
-x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 
-c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h 
-x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 
-(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall 
-(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: 
-C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) 
-(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 
-\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in 
-(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 
-(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead 
-(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g 
-c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 
-(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) 
-\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 
-(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 
-(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) 
-e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) 
-(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 
-t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead 
-(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 
-x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 
-(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 
-(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: 
-T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda 
-(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda 
-(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 
-T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 
-x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat 
-Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) 
-(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 
-c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat 
-Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e 
-x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) 
-H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 
-H5))))))))))))))) c y x H0))))) H))))))).
-
-lemma ty3_tred:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
-t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
-(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T 
-(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: 
-(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g 
-c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
-
-theorem ty3_sconv_pc3:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
-u2) \to (pc3 c t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
-u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
-(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: 
-T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def 
-(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g 
-t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
-
-lemma ty3_sred_back:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c 
-t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 
-t) \to (ty3 g c t1 t)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda 
-(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
-(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t 
-t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g 
-c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t 
-H1)))) (ty3_correct g c t2 t H1)))))))))).
-
-theorem ty3_sconv:
- \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
-u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
-u2) \to (ty3 g c u1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
-(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
-u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
-(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda 
-(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back 
-g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma
deleted file mode 100644 (file)
index d9a82d4..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/props.ma
+++ /dev/null
@@ -1,669 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/fwd.ma".
-
-include "basic_1/pc3/fwd.ma".
-
-lemma ty3_lift:
- \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e 
-t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c 
-e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
-\def
- \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to 
-(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: 
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h 
-d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 
-g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: 
-nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d 
-t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: 
-nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h 
-d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) 
-(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: 
-nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop 
-h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort 
-(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 
-(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort 
-(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: 
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
-(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: 
-nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 
-t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: 
-(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 
-(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le 
-n d0 (le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) 
-d0 H4)))) c0 c h H3 (CHead d (Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda 
-(e0: C).(\lambda (_: C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
-C).(drop h (minus d0 n) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
-(CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift 
-(S n) O t))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 
-x0)).(\lambda (H7: (drop h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 
-(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
-nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let 
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S n)) H9 Abbr d u H8) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d0 (S n)) 
-u)))) (\lambda (c1: C).(drop h (minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 
-(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x: C).(\lambda (H11: 
-(clear x0 (CHead x (Bind Abbr) (lift h (minus d0 (S n)) u)))).(\lambda (H12: 
-(drop h (minus d0 (S n)) x d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g 
-c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat (plus (S n) (minus d0 (S 
-n))) (\lambda (n0: nat).(ty3 g c0 (TLRef n) (lift h n0 (lift (S n) O t)))) 
-(eq_ind_r T (lift (S n) O (lift h (minus d0 (S n)) t)) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) 
-(lift (S n) O (lift h (minus d0 (S n)) t)))) (ty3_abbr g n c0 x (lift h 
-(minus d0 (S n)) u) (getl_intro n c0 (CHead x (Bind Abbr) (lift h (minus d0 
-(S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) t) (H2 x (minus d0 (S n)) h 
-H12)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H4)) (lift h 
-(plus (S n) (minus d0 (S n))) (lift (S n) O t)) (lift_d t h (S n) (minus d0 
-(S n)) O (le_O_n (minus d0 (S n))))) d0 (le_plus_minus_r (S n) d0 H4)) (lift 
-h d0 (TLRef n)) (lift_lref_lt n h d0 H4))))) H10)))))))) H5))) (\lambda (H4: 
-(le d0 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(ty3 g c0 t0 (lift 
-h d0 (lift (S n) O t)))) (eq_ind nat (S n) (\lambda (_: nat).(ty3 g c0 (TLRef 
-(plus n h)) (lift h d0 (lift (S n) O t)))) (eq_ind_r T (lift (plus h (S n)) O 
-t) (\lambda (t0: T).(ty3 g c0 (TLRef (plus n h)) t0)) (eq_ind_r nat (plus (S 
-n) h) (\lambda (n0: nat).(ty3 g c0 (TLRef (plus n h)) (lift n0 O t))) 
-(ty3_abbr g (plus n h) c0 d u (drop_getl_trans_ge n c0 c d0 h H3 (CHead d 
-(Bind Abbr) u) H0 H4) t H1) (plus h (S n)) (plus_sym h (S n))) (lift h d0 
-(lift (S n) O t)) (lift_free t (S n) h O d0 (le_S_n d0 (S n) (le_S (S d0) (S 
-n) (le_n_S d0 n H4))) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
-n) (\lambda (n0: nat).(eq nat (S n) n0)) (le_antisym (S n) (plus (S O) n) 
-(le_n (plus (S O) n)) (le_n (S n))) (plus n (S O)) (plus_sym n (S O)))) (lift 
-h d0 (TLRef n)) (lift_lref_ge n h d0 H4)))))))))))))))) (\lambda (n: 
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: 
-nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 
-t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: 
-(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 
-(lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le 
-n d0 (le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) 
-d0 H4)))) c0 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda 
-(e0: C).(\lambda (_: C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
-C).(drop h (minus d0 n) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
-(CHead d (Bind Abst) u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift 
-(S n) O u))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 
-x0)).(\lambda (H7: (drop h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 
-(CHead d (Bind Abst) u))).(let H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
-nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let 
-H10 \def (drop_clear_S x1 x0 h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C 
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abst) (lift h (minus d0 (S n)) 
-u)))) (\lambda (c1: C).(drop h (minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 
-(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (x: C).(\lambda (H11: 
-(clear x0 (CHead x (Bind Abst) (lift h (minus d0 (S n)) u)))).(\lambda (H12: 
-(drop h (minus d0 (S n)) x d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g 
-c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S 
-n))) (\lambda (n0: nat).(ty3 g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) 
-(eq_ind_r T (lift (S n) O (lift h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) 
-(lift (S n) O (lift h (minus d0 (S n)) u)))) (ty3_abst g n c0 x (lift h 
-(minus d0 (S n)) u) (getl_intro n c0 (CHead x (Bind Abst) (lift h (minus d0 
-(S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) t) (H2 x (minus d0 (S n)) h 
-H12)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H4)) (lift h 
-(plus (S n) (minus d0 (S n))) (lift (S n) O u)) (lift_d u h (S n) (minus d0 
-(S n)) O (le_O_n (minus d0 (S n))))) d0 (le_plus_minus_r (S n) d0 H4)) (lift 
-h d0 (TLRef n)) (lift_lref_lt n h d0 H4))))) H10)))))))) H5))) (\lambda (H4: 
-(le d0 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(ty3 g c0 t0 (lift 
-h d0 (lift (S n) O u)))) (eq_ind nat (S n) (\lambda (_: nat).(ty3 g c0 (TLRef 
-(plus n h)) (lift h d0 (lift (S n) O u)))) (eq_ind_r T (lift (plus h (S n)) O 
-u) (\lambda (t0: T).(ty3 g c0 (TLRef (plus n h)) t0)) (eq_ind_r nat (plus (S 
-n) h) (\lambda (n0: nat).(ty3 g c0 (TLRef (plus n h)) (lift n0 O u))) 
-(ty3_abst g (plus n h) c0 d u (drop_getl_trans_ge n c0 c d0 h H3 (CHead d 
-(Bind Abst) u) H0 H4) t H1) (plus h (S n)) (plus_sym h (S n))) (lift h d0 
-(lift (S n) O u)) (lift_free u (S n) h O d0 (le_S_n d0 (S n) (le_S (S d0) (S 
-n) (le_n_S d0 n H4))) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
-n) (\lambda (n0: nat).(eq nat (S n) n0)) (le_antisym (S n) (plus (S O) n) 
-(le_n (plus (S O) n)) (le_n (S n))) (plus n (S O)) (plus_sym n (S O)))) (lift 
-h d0 (TLRef n)) (lift_lref_ge n h d0 H4)))))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: 
-((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to 
-(ty3 g c0 (lift h d u) (lift h d t)))))))).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda 
-(H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 
-(CHead c (Bind b) u)) \to (ty3 g c0 (lift h d t0) (lift h d 
-t3)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
-(drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) 
-d) t0)) (\lambda (t4: T).(ty3 g c0 t4 (lift h d (THead (Bind b) u t3)))) 
-(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) t3)) (\lambda 
-(t4: T).(ty3 g c0 (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) 
-t4)) (ty3_bind g c0 (lift h d u) (lift h d t) (H1 c0 d h H4) b (lift h (S d) 
-t0) (lift h (S d) t3) (H3 (CHead c0 (Bind b) (lift h d u)) (S d) h 
-(drop_skip_bind h d c0 c H4 b u))) (lift h d (THead (Bind b) u t3)) 
-(lift_head (Bind b) u t3 h d)) (lift h d (THead (Bind b) u t0)) (lift_head 
-(Bind b) u t0 h d)))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda 
-(u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c0: C).(\forall 
-(d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d w) (lift 
-h d u)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d v) (lift h d (THead (Bind 
-Abst) u t))))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: 
-nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d 
-w) (lift h (s (Flat Appl) d) v)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d 
-(THead (Flat Appl) w (THead (Bind Abst) u t))))) (eq_ind_r T (THead (Flat 
-Appl) (lift h d w) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))) 
-(\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) (lift h (s (Flat 
-Appl) d) v)) t0)) (eq_ind_r T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) 
-(lift h (s (Bind Abst) (s (Flat Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 
-(THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) (THead (Flat 
-Appl) (lift h d w) t0))) (ty3_appl g c0 (lift h d w) (lift h d u) (H1 c0 d h 
-H4) (lift h d v) (lift h (S d) t) (eq_ind T (lift h d (THead (Bind Abst) u 
-t)) (\lambda (t0: T).(ty3 g c0 (lift h d v) t0)) (H3 c0 d h H4) (THead (Bind 
-Abst) (lift h d u) (lift h (S d) t)) (lift_bind Abst u t h d))) (lift h (s 
-(Flat Appl) d) (THead (Bind Abst) u t)) (lift_head (Bind Abst) u t h (s (Flat 
-Appl) d))) (lift h d (THead (Flat Appl) w (THead (Bind Abst) u t))) 
-(lift_head (Flat Appl) w (THead (Bind Abst) u t) h d)) (lift h d (THead (Flat 
-Appl) w v)) (lift_head (Flat Appl) w v h d))))))))))))))) (\lambda (c: 
-C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
-(H1: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) 
-\to (ty3 g c0 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda 
-(_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (d: 
-nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d 
-t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
-(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s 
-(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d (THead (Flat Cast) 
-t4 t3)))) (eq_ind_r T (THead (Flat Cast) (lift h d t4) (lift h (s (Flat Cast) 
-d) t3)) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) (lift h d t3) (lift h (s 
-(Flat Cast) d) t0)) t)) (ty3_cast g c0 (lift h (s (Flat Cast) d) t0) (lift h 
-(s (Flat Cast) d) t3) (H1 c0 (s (Flat Cast) d) h H4) (lift h d t4) (H3 c0 d h 
-H4)) (lift h d (THead (Flat Cast) t4 t3)) (lift_head (Flat Cast) t4 t3 h d)) 
-(lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) t3 t0 h 
-d)))))))))))))) e t1 t2 H))))).
-
-lemma ty3_correct:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
-t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda 
-(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: 
-C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda 
-(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: 
-T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g 
-c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g 
-c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T 
-(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) 
-(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
-u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda 
-(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 
-g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda 
-(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 
-(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) 
-(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind 
-Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T 
-(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 
-(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) 
-(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda 
-(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T 
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in 
-(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T 
-(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: 
-T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda 
-(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g 
-c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: 
-T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda 
-(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
-c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 
-(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: 
-T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w 
-(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u 
-x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind 
-Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead 
-(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind 
-Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 
-(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
-t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t 
-t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 
-(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: 
-(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 
-(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w 
-(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 
-(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 
-H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g 
-c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: 
-(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T 
-(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat 
-Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T 
-(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x 
-t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
-
-theorem ty3_unique:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
-t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
-(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
-(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: 
-T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall 
-(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 
-t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s 
-c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: 
-nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g 
-c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
-T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 
-t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) 
-t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) 
-(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 
-(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
-T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda 
-(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
-t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g 
-e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda 
-(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 
-x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n 
-c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n 
-H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
-(CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead 
-_ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) 
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
-(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: 
-T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def 
-(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d 
-H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d 
-H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O 
-(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) 
-H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda 
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) 
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) 
-(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
-T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def 
-(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead 
-x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 
-(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (pc3 c0 
-(lift (S n) O t) t2) H9))))))))) H4)) (ty3_gen_lref g c0 t2 n H3)))))))))))) 
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda 
-(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 
-g d u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t 
-t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
-(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n 
-c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
-(t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda 
-(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) 
-O u0) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
-t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g 
-e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (x2: T).(\lambda (_: (pc3 c0 (lift (S n) O x2) t2)).(\lambda (H6: 
-(getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 
-\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 
-(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
-x0 (Bind Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u0) 
-(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
-\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
-\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
-_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 
-(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) 
-(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 
-t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda 
-(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O 
-x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: 
-(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal 
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) 
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in 
-((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) 
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
-x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 
-H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 
-H10) in (let H14 \def (eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O 
-t0) t2)) H5 u0 H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n 
-c0 (CHead c1 (Bind Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 
-(\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) 
-(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: 
-T).((ty3 g c0 u0 t2) \to (pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t0 
-t2)).(\lambda (H3: ((\forall (t3: T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) 
-\to (pc3 (CHead c0 (Bind b) u0) t2 t3))))).(\lambda (t3: T).(\lambda (H4: 
-(ty3 g c0 (THead (Bind b) u0 t0) t3)).(ex3_2_ind T T (\lambda (t4: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t4) t3))) (\lambda (_: 
-T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 
-g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead (Bind b) u0 t2) t3) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Bind b) u0 x0) 
-t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 g (CHead c0 (Bind b) 
-u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead (Bind b) u0 t2) 
-(pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) (ty3_gen_bind g b c0 
-u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: 
-T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 w 
-t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 
-g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((ty3 g c0 
-v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) t2))))).(\lambda (t2: T).(\lambda 
-(H4: (ty3 g c0 (THead (Flat Appl) w v) t2)).(ex3_2_ind T T (\lambda (u1: 
-T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u1 t0)) 
-t2))) (\lambda (u1: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u1 
-t0)))) (\lambda (u1: T).(\lambda (_: T).(ty3 g c0 w u1))) (pc3 c0 (THead 
-(Flat Appl) w (THead (Bind Abst) u0 t)) t2) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
-t2)).(\lambda (H6: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 
-g c0 w x0)).(pc3_t (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) c0 (THead 
-(Flat Appl) w (THead (Bind Abst) u0 t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 
-t) (THead (Bind Abst) x0 x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 
-H5)))))) (ty3_gen_appl g c0 w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda 
-(t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: 
-((\forall (t3: T).((ty3 g c0 t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: 
-T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda (H3: ((\forall (t4: T).((ty3 g c0 
-t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead 
-(Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda (t5: T).(pc3 c0 (THead (Flat 
-Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 t2)) (\lambda (t5: T).(ty3 g 
-c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: T).(\lambda 
-(H5: (pc3 c0 (THead (Flat Cast) x0 t2) t4)).(\lambda (_: (ty3 g c0 t0 
-t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t (THead (Flat Cast) x0 t2) c0 
-(THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 (H3 x0 H7) (Flat Cast) t2) t4 
-H5))))) (ty3_gen_cast g c0 t0 t2 t4 H4)))))))))))) c u t1 H))))).
-
-lemma ty3_gen_abst_abst:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
-(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 
-T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) 
-u) t1 t2))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u 
-t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T 
-(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) 
-t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) 
-x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) 
-u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: 
-T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda 
-(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u 
-x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind 
-Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c 
-(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda 
-(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind 
-Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 
-g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
-(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: 
-(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y 
-\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g 
-c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 
-(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) 
-(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g 
-Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind 
-Abst) u t2) H))))))).
-
-lemma ty3_typecheck:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t 
-v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: 
-(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: 
-T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g 
-c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) 
-(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v 
-H)))))).
-
-lemma ty3_getl_subst0:
- \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
-u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t 
-t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d 
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
-(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
-T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) 
-\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d 
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda 
-(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
-w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 
-t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: 
-nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
-w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: 
-T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: 
-B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) 
-v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: 
-nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: 
-(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: 
-T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m 
-H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n 
-c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 
-t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: 
-T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v 
-w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
-(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda 
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n 
-i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) 
-(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 
-\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) 
-H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: 
-C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind 
-Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono 
-c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def 
-(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) 
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in 
-((let H11 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abbr) u0) 
-(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 
-(Bind b) v) H7)) in (\lambda (H12: (eq B Abbr b)).(\lambda (H13: (eq C d 
-d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind 
-b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: 
-T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n 
-c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T 
-(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: 
-B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr H12) in (ex_intro T (\lambda 
-(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) 
-(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (n: nat).(\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d 
-(Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda 
-(_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) 
-\to (\forall (b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 
-(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda 
-(v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef 
-n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl 
-i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O 
-v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n 
-i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i 
-(\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 
-\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 
-(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 
-(Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind 
-Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 
-(CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
-(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind 
-Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) 
-\Rightarrow t2])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono 
-c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: 
-(eq B Abst b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v 
-(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T 
-u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def 
-(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d 
-H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) 
-(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) 
-u0))) H15 Abst H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 
-H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n 
-H3)))))))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: 
-T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: ((\forall (v0: T).(\forall 
-(t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall 
-(d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda 
-(w: T).(ty3 g d v w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: 
-((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to 
-(\forall (b0: B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) 
-u0) (CHead d (Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
-w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda 
-(H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind 
-(ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: 
-T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 
-t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: 
-T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) 
-(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) 
-(\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 
-(THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda 
-(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) 
-x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) 
-(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) 
-(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: 
-T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) 
-v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: 
-(eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 
-x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) 
-H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
-T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b) 
-i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
-(THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 
-u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex 
-T (\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(_: (eq T t3 (THead (Bind b) x0 x1))).(\lambda (H8: (subst0 i v0 u0 
-x0)).(\lambda (_: (subst0 (s (Bind b) i) v0 t1 x1)).(H1 v0 x0 i H8 b0 d v 
-H5)))))) H6)) (subst0_gen_head (Bind b) v0 u0 t1 t3 i H4)))))))))))))))))))) 
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
-u0)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: 
-nat).((subst0 i v0 w t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w0: T).(ty3 g d v 
-w0))))))))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 v 
-(THead (Bind Abst) u0 t0))).(\lambda (H3: ((\forall (v0: T).(\forall (t1: 
-T).(\forall (i: nat).((subst0 i v0 v t1) \to (\forall (b: B).(\forall (d: 
-C).(\forall (v1: T).((getl i c0 (CHead d (Bind b) v1)) \to (ex T (\lambda 
-(w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: T).(\lambda (t1: 
-T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat Appl) w v) 
-t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda (H5: (getl i 
-c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t1 (THead 
-(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T (\lambda 
-(t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat 
-Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t1 
-(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w 
-u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)))) 
-(ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda (u2: 
-T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) 
-(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) 
-(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: 
-(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T 
-(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 
-(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead 
-(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex 
-T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 
-(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v 
-x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: 
-T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 
-(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: 
-(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1 
-H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4))))))))))))))))))) 
-(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
-t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: 
-nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
-w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3: 
-((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to 
-(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) 
-v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0: 
-T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat 
-Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda 
-(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T 
-t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T 
-(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: 
-T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat 
-Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: 
-(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: 
-T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat 
-Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: 
-T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x 
-t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) 
-(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) 
-(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda 
-(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s 
-(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: 
-T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s 
-(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) 
-(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 
-t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
-(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) 
-(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T 
-(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
-(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 
-x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v 
-H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) 
-c t u H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma
deleted file mode 100644 (file)
index e590a65..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/sty0.ma
+++ /dev/null
@@ -1,230 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/pr3_props.ma".
-
-include "basic_1/sty0/fwd.ma".
-
-lemma ty3_sty0:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
-t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
-(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: 
-T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: 
-C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
-(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda 
-(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: 
-((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: 
-(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 
-H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda 
-(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in 
-(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in 
-(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) 
-(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
-u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall 
-(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda 
-(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in 
-(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g 
-e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift 
-(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O 
-t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
-(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: 
-(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
-(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda 
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal 
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) 
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
-((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) 
-(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead 
-x0 (Bind Abbr) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 
-H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 
-u0 H12) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 
-(Bind Abbr) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: 
-C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 
-H17)))))))) H11))) t2 H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) 
-(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
-(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
-u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
-(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq 
-T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
-(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda 
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
-(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind 
-C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind 
-Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) 
-x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O x1)) H11))) t2 
-H9)))))))) H5)) H4))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
-u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 t)).(\lambda (_: ((\forall 
-(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda 
-(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in 
-(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g 
-e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift 
-(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
-u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O 
-t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: 
-T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
-(CHead x0 (Bind Abbr) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq 
-T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
-(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (eq_ind 
-C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind 
-Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) 
-x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 
-H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: 
-C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
-u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
-T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
-(CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: 
-(eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
-(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 
-g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
-(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
-(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (f_equal 
-C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
-\Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) 
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in 
-((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) 
-(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
-x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def 
-(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 
-H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 
-u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O 
-t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 
-(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: 
-C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 
-H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda 
-(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall 
-(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda 
-(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 
-t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) 
-\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: 
-(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 
-t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead 
-c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) 
-(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g 
-(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 
-x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) 
-H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead 
-(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) 
-H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda 
-(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to 
-(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v 
-(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) 
-\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead 
-(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 
-\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: 
-T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) 
-(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead 
-(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead 
-(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: 
-T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 
-\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T 
-(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead 
-(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T 
-(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead 
-(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T 
-(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead 
-(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: 
-(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: 
-T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 
-g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead 
-(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 
-(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda 
-(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat 
-Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w 
-u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind 
-Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) 
-w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v 
-t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead 
-(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 
-H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 
-w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) 
-(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 
-t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 
-t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: 
-((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: 
-T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def 
-(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T 
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda 
-(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead 
-(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 
-t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def 
-(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r 
-T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 
-t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 
-\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 
-x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) 
-(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: 
-T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 
-x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead 
-(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g 
-c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat 
-Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 
-(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) 
-(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma
deleted file mode 100644 (file)
index 43a6d48..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/subst1.ma
+++ /dev/null
@@ -1,1095 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/props.ma".
-
-include "basic_1/pc3/subst1.ma".
-
-include "basic_1/getl/getl.ma".
-
-lemma ty3_gen_cabbr:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
-(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to 
-(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
-e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
-T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T 
-T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: 
-T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: 
-C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) 
-\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d 
-a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S 
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: 
-T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) 
-u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: 
-C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) 
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d 
-u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e 
-u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d 
-x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a 
-x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u 
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift 
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 
-H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 
-H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: 
-C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
-C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
-d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort 
-m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort 
-(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: 
-T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort 
-m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: 
-T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) 
-(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g 
-a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: 
-T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: 
-T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
-C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
-(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
-a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
-O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
-nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) 
-(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 
-(le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 
-H6))))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda 
-(e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) u) e2)) (\lambda (e2: 
-C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 
-(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 
-u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 
-(CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let H10 \def 
-(eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind 
-Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 \def 
-(csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T 
-C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind Abbr) u2)))) 
-(\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda 
-(_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift 
-(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind Abbr) 
-x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: 
-(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: 
-C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind 
-nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S 
-n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
-C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: 
-C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
-C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
-C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: 
-(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S 
-n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S 
-n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 
-e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u 
-(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift 
-(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 
-y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S 
-n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S 
-n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 
-(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 
-x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 
-(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r 
-nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S 
-n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) 
-n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
-u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) 
-(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) 
-(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) 
-d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda 
-(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) 
-(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) 
-H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) 
-(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus 
-d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 
-H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt 
-Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) 
-(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda 
-(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S 
-O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: 
-nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 
-(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind 
-nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
-n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d 
-(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) 
-(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
-(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
-\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) 
-(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e 
-(Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T (\lambda (e0: C).(match 
-e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
-(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) 
-n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 
-\def (eq_ind_r T u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) 
-H10 u H12) in (let H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 
-a0)) H8 u H12) in (eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r 
-C e (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in 
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift 
-(S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) 
-(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(lift n O u) (lift n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n 
-O u)) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u 
-(TLRef n) t0)) (subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n 
-(S O) O n (le_plus_r O n) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) 
-(\lambda (t0: T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S 
-n) O t)) (lift (S O) n (lift n O t)) (lift_free t n (S O) O n (le_plus_r O n) 
-(le_O_n n))) (ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n 
-(csubst1_getl_ge n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a 
-H7)))) u0 H12))))) H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat 
-(S (plus O (minus n (S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) 
-(minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) 
-(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
-d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift 
-(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O 
-t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) 
-(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) 
-t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 
-(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus 
-d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 
-u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 
-(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) 
-(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: 
-nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S 
-O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 
-n (le_S_n d0 n (le_S_n (S d0) (S n) (le_S (S (S d0)) (S n) (le_n_S (S d0) n 
-H6)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat 
-(plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S O)) (plus_sym d0 
-(S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H6)))) (plus 
-(S O) (minus n (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n 
-(S O)))) (refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
-(le_lt_trans O d0 n (le_O_n d0) H6))))))))))))))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
-c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
-d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d0 u0 d a0) \to 
-(\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
-(u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Abbr) 
-u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 a0)).(\lambda (a: 
-C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) 
-d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H6: 
-(lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
-(CHead d (Bind Abst) u) (CHead e (Bind Abbr) u0))) (getl_conf_le d0 (CHead e 
-(Bind Abbr) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S_n (S n) 
-(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H6))))) (S (minus d0 (S n))) 
-(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 
-(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift 
-(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) 
-x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) 
-(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 
-(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) 
-d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
-(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: 
-C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
-C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
-C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 
-(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d 
-x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 
-(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
-(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in 
-(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 
-(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
-O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus 
-d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda 
-(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 
-(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 
-(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind 
-Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in 
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u 
-(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
-(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
-x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) 
-x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda 
-(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S 
-n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: 
-nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S 
-n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
-u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) 
-(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
-a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 
-(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S 
-O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S 
-n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) 
-x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) 
-(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) 
-(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 
-H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 
-H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus 
-d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead 
-d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r 
-nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def 
-(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let 
-H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) 
-u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C 
-(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind 
-Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) 
-H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | 
-Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
-False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n 
-H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) (lift (S O) n y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H11))) d0 H6))))) 
-(\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda 
-(n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef 
-n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 
-(lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: 
-nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) 
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S 
-n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (eq_ind_r nat (plus (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift 
-(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus 
-(minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O u) 
-(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(subst1 d0 
-u0 (TLRef (plus (minus n (S O)) (S O))) t0)) (subst1_refl d0 u0 (TLRef (plus 
-(minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
-(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H6))) (eq_ind_r T 
-(lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 u0 (lift (S n) O u) 
-t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 (lift n O u)) 
-(lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H6)) 
-(le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a 
-(TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u 
-(getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 n (le_S_n 
-d0 n (le_S_n (S d0) (S n) (le_S (S (S d0)) (S n) (le_n_S (S d0) n H6)))) c0 
-a0 u0 H4 (CHead d (Bind Abst) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) 
-d0) (\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S O)) (plus_sym d0 (S O)))) t 
-H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H6)))) (plus (S O) (minus 
-n (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
-(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
-(le_lt_trans O d0 n (le_O_n d0) H6))))))))))))))))))))) (\lambda (c0: 
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: 
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
-(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: 
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: ((\forall 
-(e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) 
-(CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 (CHead c0 (Bind 
-b) u) a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
-(u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) 
-u0))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u0 c0 a0)).(\lambda (a: 
-C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H1 e u0 d H4 a0 H5 a H6) 
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S 
-O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 u (lift (S O) d 
-x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d x1))).(\lambda (H10: (ty3 g a 
-x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
-Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u 
-(lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
-a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S 
-d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S 
-d) u0 t4 (lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
-(CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S O) (S d) 
-x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) x3))).(\lambda 
-(H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S 
-O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Bind 
-b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) (lift (S O) d 
-x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u 
-t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 (lift (S O) (S 
-d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) 
-d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) 
-(\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head u0 u 
-(lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S O) d 
-(THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 H10 
-b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: 
-C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: 
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
-(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: 
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
-c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
-C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift 
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
-(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: 
-(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let 
-H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
-t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d 
-x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d 
-x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) 
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
-(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w 
-(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d 
-x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind 
-Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
-(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S 
-O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u 
-x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S 
-O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: 
-T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
-t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) 
-x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 
-(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 
-(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 
-(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 
-\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 
-x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead 
-(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in 
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat 
-Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 
-(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead 
-(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) 
-(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead 
-(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v 
-(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat 
-Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift 
-(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead 
-(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d 
-x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind 
-Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S 
-d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) 
-t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t 
-(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) 
-(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead 
-(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) 
-(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S 
-O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda 
-(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e 
-(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
-C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 
-t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl 
-d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to 
-(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
-nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
-C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S 
-O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda 
-(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let 
-H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
-(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead 
-(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d 
-u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def 
-(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift 
-t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
-T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) 
-x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d 
-x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) 
-t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) 
-H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) 
-(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda 
-(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S 
-O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat 
-Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 
-H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
-
-lemma ty3_gen_cvoid:
- \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
-t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
-(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T 
-T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2))))))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
-e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: 
-C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to 
-(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u 
-t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
-d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 
-t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d 
-c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 
-a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: 
-(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def 
-(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in 
-(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d 
-x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S 
-O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) 
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 
-d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: 
-(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d 
-x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: 
-T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 
-(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) 
-in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) 
-H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: 
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift 
-(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 
-H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u 
-H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda 
-(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e 
-(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 
-a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T 
-(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m 
-(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g 
-m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) 
-(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
-c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
-d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: 
-T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) 
-u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt 
-n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
-(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e 
-(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S_n (S n) 
-(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H5))))) (S (minus d0 (S n))) 
-(minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
-(S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H5)) in 
-(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift (S O) (minus d0 
-(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abbr) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) d e0))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
-T).(\lambda (x1: C).(\lambda (H8: (eq T u (lift (S O) (minus d0 (S n)) 
-x0))).(\lambda (H9: (getl n a (CHead x1 (Bind Abbr) x0))).(\lambda (H10: 
-(drop (S O) (minus d0 (S n)) d x1)).(let H11 \def (eq_ind T u (\lambda (t0: 
-T).(\forall (e0: C).(\forall (u1: T).(\forall (d1: nat).((getl d1 d (CHead e0 
-(Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d1 d a0) \to (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S O) d1 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t (lift (S O) d1 y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift (S O) (minus d0 (S n)) x0) H8) in 
-(let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g d t0 t)) H1 (lift (S O) 
-(minus d0 (S n)) x0) H8) in (let H13 \def (H11 e u0 (minus d0 (S n)) 
-(getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) u (minus d0 (S n)) H6) x1 
-H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) 
-(minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: 
-(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) 
-x2))).(\lambda (H15: (eq T t (lift (S O) (minus d0 (S n)) x3))).(\lambda 
-(H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t (\lambda (t0: T).(ty3 g d 
-(lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S O) (minus d0 (S n)) x3) 
-H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) (\lambda (t0: T).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H18 \def 
-(eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S 
-O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S 
-n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))))) (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O x3)) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x3)) 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T 
-(TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) 
-(lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O 
-x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) 
-(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) 
-(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t 
-H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 
-(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 
-(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r 
-nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in 
-(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
-T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) 
-(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 
-(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def 
-(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
-Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) 
-H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
-n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
-t) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S 
-O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift 
-(S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus 
-(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S 
-O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda 
-(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef 
-(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
-(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T 
-(lift (plus (S O) n) O t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) 
-(refl_equal T (lift (S n) O t)) (lift (S O) d0 (lift n O t)) (lift_free t n 
-(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) 
-(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
-(S O))) (lift n0 O t))) (ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge 
-n (CHead d (Bind Abbr) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
-(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) 
-n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
-(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
-(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
-(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
-c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
-d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: 
-T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) 
-u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt 
-n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
-(CHead d (Bind Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e 
-(Bind Void) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S_n (S n) 
-(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H5))))) (S (minus d0 (S n))) 
-(minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
-(S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H5)) in 
-(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift (S O) (minus d0 
-(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) 
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) d e0))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
-T).(\lambda (x1: C).(\lambda (H8: (eq T u (lift (S O) (minus d0 (S n)) 
-x0))).(\lambda (H9: (getl n a (CHead x1 (Bind Abst) x0))).(\lambda (H10: 
-(drop (S O) (minus d0 (S n)) d x1)).(let H11 \def (eq_ind T u (\lambda (t0: 
-T).(\forall (e0: C).(\forall (u1: T).(\forall (d1: nat).((getl d1 d (CHead e0 
-(Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d1 d a0) \to (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S O) d1 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t (lift (S O) d1 y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift (S O) (minus d0 (S n)) x0) H8) in 
-(let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g d t0 t)) H1 (lift (S O) 
-(minus d0 (S n)) x0) H8) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x0) 
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) 
-(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
-t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (let H13 \def (H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abst) d 
-(CHead e (Bind Void) u0) u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift 
-(S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift 
-(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 
-y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) 
-d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) 
-(minus d0 (S n)) x0)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T 
-(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda 
-(H15: (eq T t (lift (S O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 
-x3)).(let H17 \def (eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus 
-d0 (S n)) x0) t0)) H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def 
-(eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S 
-O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S 
-n))) (lift (S n) O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))))) (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(TLRef n) (lift (S n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T 
-(TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) 
-(lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O 
-x0))) (ty3_abst g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) 
-(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) 
-(lift_d x0 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))))))))) 
-H13)) u H8)))))))) (getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S 
-n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 
-(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r 
-nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in 
-(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
-T (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) 
-(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 
-(CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def 
-(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee with [(CSort _) 
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
-\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
-Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
-Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) 
-H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
-n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
-u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S 
-O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift 
-(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus 
-(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S 
-O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda 
-(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef 
-(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
-(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T 
-(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) 
-(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n 
-(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) 
-(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
-(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge 
-n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
-(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) 
-n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
-(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
-(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
-(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: 
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda 
-(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
-(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda 
-(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: 
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind 
-b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 
-(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind 
-Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def 
-(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S 
-O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda 
-(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
-(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in 
-(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) 
-H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: 
-T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 
-(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 
-(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
-y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda 
-(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in 
-(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 
-(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) 
-(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T 
-T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda 
-(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S 
-O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S 
-O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S 
-O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: 
-(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) 
-(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
-(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) 
-(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
-x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind 
-b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
-x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead 
-(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) 
-(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) 
-(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 
-(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u 
-H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda 
-(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall 
-(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall 
-(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v 
-(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: 
-C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind 
-Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
-(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: 
-(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) 
-d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d 
-x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: 
-T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in 
-(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
-Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
-y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead 
-(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
-x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
-Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u 
-(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 
-\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 
-x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
-x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
-Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda 
-(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 
-(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in 
-(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
-(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in 
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
-(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: 
-T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: 
-(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 
-x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead 
-(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r 
-T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) 
-in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d 
-x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d 
-(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind 
-Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
-a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) 
-x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 
-(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 
-(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 
-x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 
-x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d 
-x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind 
-Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d 
-x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) 
-(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u 
-H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) 
-H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall 
-(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall 
-(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: 
-((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind 
-Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda 
-(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda 
-(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: 
-C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in 
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
-(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
-T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: 
-(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
-(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in 
-(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def 
-(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) 
-d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: 
-C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) 
-u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 
-(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T 
-(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) 
-d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def 
-(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d 
-x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S 
-O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
-(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda 
-(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 
-x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d 
-x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda 
-(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) 
-(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) 
-d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def 
-(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d 
-H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: 
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) 
-(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) 
-(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) 
-d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
-a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S 
-O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
-(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 
-x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a 
-x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) 
-(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S 
-O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 
-H8))))))) H6)))))))))))))))) c t1 t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma
deleted file mode 100644 (file)
index feef3e8..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/defs.ma
+++ /dev/null
@@ -1,26 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pr0/defs.ma".
-
-include "basic_1/C/defs.ma".
-
-inductive wcpr0: C \to (C \to Prop) \def
-| wcpr0_refl: \forall (c: C).(wcpr0 c c)
-| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall 
-(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k 
-u1) (CHead c2 k u2)))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
deleted file mode 100644 (file)
index 8d4ccb5..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/fwd.ma
+++ /dev/null
@@ -1,105 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wcpr0/defs.ma".
-
-implied rec lemma wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P 
-c c))) (f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) 
-\to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P 
-(CHead c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on 
-w: P c c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp 
-c1 c2 w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1 
-u2 p k)].
-
-lemma wcpr0_gen_sort:
- \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
-n))))
-\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) 
-x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C 
-x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: 
-C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: 
-C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
-C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) 
-(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda 
-(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
-c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
-(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C 
-(CHead c1 k u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
-| (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C 
-(CHead c2 k u2) (CHead c1 k u1)) H5))))))))))) y x H0))) H))).
-
-lemma wcpr0_gen_head:
- \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
-(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
-C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
-\def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
-(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: 
-C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: 
-C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda 
-(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: 
-C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) 
-(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal 
-C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) 
-(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
-T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 
-c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C 
-(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
-T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) 
-(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: 
-C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to 
-(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead 
-c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: 
-T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 
-k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 
-u0) (CHead c1 k u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
-k0 u0) (CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
-(CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda 
-(H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C (CHead c2 k1 u2) 
-(CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead 
-c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 \def (eq_ind T u0 
-(\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 (\lambda (t: T).(or 
-(eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda (c3: C).(\lambda 
-(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
-(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let 
-H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or (eq C 
-c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c2 (CHead c3 k 
-u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def (eq_ind C c0 
-(\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 (\lambda (c: C).(or 
-(eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda 
-(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
-(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) 
-(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
-C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: 
-C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
-u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k 
-u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2 
-k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x H0))) 
-H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma
deleted file mode 100644 (file)
index 2b43bb2..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wcpr0/getl.ma
+++ /dev/null
@@ -1,448 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wcpr0/fwd.ma".
-
-include "basic_1/getl/props.ma".
-
-lemma wcpr0_drop:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
-e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
-(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
-(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
-(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead 
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) 
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u1) 
-(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in 
-((let H5 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 
-u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl 
-(CHead c3 k u1) (CHead e1 k0 u0) H3)) in (\lambda (H7: (eq K k k0)).(\lambda 
-(H8: (eq C c3 e1)).(eq_ind K k (\lambda (k1: K).(ex3_2 C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k1 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3))))) (eq_ind T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t 
-u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
-u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead 
-c4 k u2) (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 (drop_refl (CHead c4 k 
-u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) H5)) H4)))))) (K_ind (\lambda (k0: 
-K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: 
-K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(drop n O (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 
-u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop 
-(S n) O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(drop (S n) O (CHead c4 k0 u2) (CHead e2 k1 u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 
-(Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
-u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
-u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(drop n O c4 (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c4 
-(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (drop n O c4 (CHead x0 k0 x1))).(\lambda 
-(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Bind b) n c4 (CHead x0 k0 x1) H6 u2) 
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 
-(Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
-u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S 
-n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in 
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c4 (CHead e2 
-k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop (S n) O c4 
-(CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
-(Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) 
-c1 c2 H))).
-
-lemma wcpr0_drop_back:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
-e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
-(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
-(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
-(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 
-u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
-(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
-e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead 
-e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
-c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) 
-(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
-[(CSort _) \Rightarrow c4 | (CHead c _ _) \Rightarrow c])) (CHead c4 k u2) 
-(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in 
-((let H5 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 
-u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def 
-(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead 
-_ _ t) \Rightarrow t])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl 
-(CHead c4 k u2) (CHead e1 k0 u0) H3)) in (\lambda (H7: (eq K k k0)).(\lambda 
-(H8: (eq C c4 e1)).(eq_ind K k (\lambda (k1: K).(ex3_2 C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k1 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0))))) (eq_ind T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: 
-T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) 
-(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) 
-(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 
-H2) e1 H8) u0 H6) k0 H7)))) H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall 
-(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O 
-(CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u4: T).(drop n O (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
-u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S 
-n) O (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(drop (S n) O (CHead c3 k0 u1) (CHead e2 k1 u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 
-(Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
-u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
-(\lambda (e2: C).(\lambda (u3: T).(drop n O c3 (CHead e2 k0 u3)))) (\lambda 
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 
-(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H6: (drop n O c3 (CHead x0 k0 x1))).(\lambda 
-(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: 
-C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Bind b) n c3 (CHead x0 k0 x1) H6 u1) 
-H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 
-(Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
-T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
-u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
-(drop (S n) O (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S 
-n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in 
-(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 (CHead e2 
-k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop (S n) O c3 
-(CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
-c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
-(Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) 
-c2 c1 H))).
-
-lemma wcpr0_getl:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
-k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
-(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
-u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
-(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
-u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k 
-u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) 
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda 
-(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
-C (\lambda (e: C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) 
-\Rightarrow c])) (CHead e1 k0 u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 
-(CHead e1 k0 u0) u1 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 
-k0 u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) 
-in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c3 
-(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in (\lambda (H8: 
-(eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K (Bind b) (\lambda 
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind 
-b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: 
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind b) 
-u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c3 (\lambda 
-(c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind 
-b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c 
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind b) u2) (CHead e2 
-(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 (getl_refl b c4 u2) H0 H2) e1 H9) u0 
-H7) k0 H8)))) H6)) H5)))) (\lambda (f: F).(\lambda (H4: (clear (CHead c3 
-(Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 
-(CHead e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u1 
-H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c4 (CHead 
-e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c4 (CHead x0 
-k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) u2) (CHead 
-e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f 
-u2) H7 H8)))))) H5)))) k (getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) 
-H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((getl n (CHead c3 k0 u1) (CHead e1 k1 
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k0 
-u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u1) (CHead e1 
-k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead 
-c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
-e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: 
-B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((getl n (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 (Bind b) u2) (CHead e2 k0 
-u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u1) (CHead 
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 
-u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c4 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 
-k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead 
-x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
-nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
-n (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
-(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let 
-H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) 
-in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 
-k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
-u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead 
-x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 
-(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
-e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat 
-f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 
-H))).
-
-lemma wcpr0_getl_back:
- \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
-nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
-k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
-(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
-(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
-u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
-(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
-u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
-k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
-u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
-c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
-T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
-(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
-u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
-u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k 
-u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) 
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda 
-(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
-C (\lambda (e: C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) 
-\Rightarrow c])) (CHead e1 k0 u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 
-(CHead e1 k0 u0) u2 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 
-k0 u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) 
-in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c4 
-(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in (\lambda (H8: 
-(eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K (Bind b) (\lambda 
-(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind 
-b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: 
-T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) 
-u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c4 (\lambda 
-(c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind 
-b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
-c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T 
-(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u1) (CHead e2 
-(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 (getl_refl b c3 u1) H0 H2) e1 H9) u0 
-H7) k0 H8)))) H6)) H5)))) (\lambda (f: F).(\lambda (H4: (clear (CHead c4 
-(Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c4 
-(CHead e1 k0 u0) c4 (drop_refl c4) (clear_gen_flat f c4 (CHead e1 k0 u0) u2 
-H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead 
-e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 
-k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u1) (CHead 
-e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f 
-u1) H7 H8)))))) H5)))) k (getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) 
-H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((getl n (CHead c4 k0 u2) (CHead e1 k1 
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 
-u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
-(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: 
-C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c4 k0 u2) (CHead e1 
-k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead 
-c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
-e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: 
-B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
-(k0: K).((getl n (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
-(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u1) (CHead e2 k0 
-u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
-T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c4 (Bind b) u2) (CHead 
-e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 
-u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
-(u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 
-k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
-C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) 
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead 
-x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
-nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
-n (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
-C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) 
-(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
-(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
-K).(\lambda (H4: (getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let 
-H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) 
-in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 
-k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
-C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
-T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
-C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
-u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead 
-x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
-(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
-e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat 
-f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 
-H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma
deleted file mode 100644 (file)
index 0199af1..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/clear.ma
+++ /dev/null
@@ -1,87 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wf3/fwd.ma".
-
-include "basic_1/clear/fwd.ma".
-
-lemma wf3_clear_conf:
- \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall 
-(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
-\def
- \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind 
-(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 
-c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: 
-T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) 
-c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e 
-c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g 
-c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def 
-(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
-
-lemma clear_wf3_trans:
- \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall 
-(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda 
-(c2: C).(clear c2 d2))))))))
-\def
- \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind 
-(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 
-d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 
-d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: 
-G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x 
-\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T 
-(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda 
-(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g 
-e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) 
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) 
-\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) 
-(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: 
-C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: 
-C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u 
-w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 
-(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e 
-(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e 
-x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda 
-(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: 
-C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) 
-c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) 
-(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda 
-(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) 
-(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) 
-\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) 
-(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: 
-T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind 
-b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: 
-(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e 
-x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C 
-(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 
-g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C 
-(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 
-(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void 
-g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) 
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: 
-((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: 
-C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: 
-F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c 
-d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda 
-(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: 
-C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda 
-(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C 
-(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 
-d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma
deleted file mode 100644 (file)
index df5eedc..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/defs.ma
+++ /dev/null
@@ -1,29 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/ty3/defs.ma".
-
-inductive wf3 (g: G): C \to (C \to Prop) \def
-| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
-| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
-T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind 
-b) u) (CHead c2 (Bind b) u))))))))
-| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
-T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g 
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
-| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
-T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
deleted file mode 100644 (file)
index d4718ec..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/fwd.ma
+++ /dev/null
@@ -1,377 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wf3/defs.ma".
-
-implied rec lemma wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: 
-nat).(P (CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g 
-c1 c2) \to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
-(\forall (b: B).(P (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))))))))))) (f1: 
-(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall 
-(u: T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(P 
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))))) (f2: (\forall 
-(c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall (u: 
-T).(\forall (f2: F).(P (CHead c1 (Flat f2) u) c2)))))))) (c: C) (c0: C) (w: 
-wf3 g c c0) on w: P c c0 \def match w with [(wf3_sort m) \Rightarrow (f m) | 
-(wf3_bind c1 c2 w0 u t t0 b) \Rightarrow (f0 c1 c2 w0 ((wf3_ind g P f f0 f1 
-f2) c1 c2 w0) u t t0 b) | (wf3_void c1 c2 w0 u f3 b) \Rightarrow (f1 c1 c2 w0 
-((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3 b) | (wf3_flat c1 c2 w0 u f3) 
-\Rightarrow (f2 c1 c2 w0 ((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3)].
-
-lemma wf3_gen_sort1:
- \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to 
-(eq C x (CSort m)))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort 
-m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: 
-C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda 
-(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: 
-nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat 
-(\lambda (e: C).(match e with [(CSort n) \Rightarrow n | (CHead _ _ _) 
-\Rightarrow m0])) (CSort m0) (CSort m) H1) in (eq_ind_r nat m (\lambda (n: 
-nat).(eq C (CSort n) (CSort n))) (refl_equal C (CSort m)) m0 H2)))) (\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 
-(CSort m)) \to (eq C c2 c1)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
-(ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) 
-(CSort m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: 
-C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
-True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 
-(Bind b) u)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
-(wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 
-c1)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to 
-False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort 
-m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee 
-with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
-(CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) (CHead c1 
-(Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 
-g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: 
-T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat f) u) (CSort m))).(let 
-H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) 
-H3) in (False_ind (eq C c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
-
-lemma wf3_gen_bind1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: 
-B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: 
-C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda 
-(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 
-C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: 
-C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to 
-False))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: 
-B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind 
-b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda 
-(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: 
-C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
-v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) 
-(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
-w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g 
-(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or 
-(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) 
-(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) 
-(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: 
-T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C 
-(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda 
-(ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
-\Rightarrow False])) I (CHead c1 (Bind b) v) H1) in (False_ind (or (ex3_2 C T 
-(\lambda (c2: C).(\lambda (_: T).(eq C (CSort m) (CHead c2 (Bind b) v)))) 
-(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C (CSort m) (CHead c2 (Bind 
-Void) (TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall 
-(w: T).((ty3 g c1 v w) \to False))))) H2)))) (\lambda (c0: C).(\lambda (c2: 
-C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) 
-v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
-C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
-(t: T).(\lambda (H3: (ty3 g c0 u t)).(\lambda (b0: B).(\lambda (H4: (eq C 
-(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
-\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
-H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
-H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r B b 
-(\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead 
-c2 (Bind b1) u) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: 
-T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C 
-(\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind Void) (TSort 
-O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g 
-c1 v w) \to False)))))) (let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 
-t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: T).(or (ex3_2 C T (\lambda 
-(c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) t0) (CHead c3 (Bind b) v)))) 
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) t0) 
-(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda 
-(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H11 \def (eq_ind 
-C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 H9) in (let H12 \def (eq_ind C c0 
-(\lambda (c: C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda 
-(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
-w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
-C).(wf3 g c c2)) H1 c1 H9) in (or_introl (ex3_2 C T (\lambda (c3: C).(\lambda 
-(_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: 
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
-v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind 
-Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall 
-(w: T).((ty3 g c1 v w) \to False)))) (ex3_2_intro C T (\lambda (c3: 
-C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) 
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w))) c2 t (refl_equal C (CHead c2 (Bind b) v)) H13 H11))))) u 
-H7)) b0 H8)))) H6)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda 
-(H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) v)) \to (or 
-(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) 
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) 
-(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
-T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall 
-(t: T).((ty3 g c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C 
-(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
-\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
-(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
-H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
-(CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
-H4) in (\lambda (_: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def 
-(eq_ind T u (\lambda (t: T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 
-v H7) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c 
-v t) \to False))) H10 c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: 
-C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: 
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
-C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
-v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
-(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
-w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
-C).(wf3 g c c2)) H1 c1 H9) in (or_intror (ex3_2 C T (\lambda (c3: C).(\lambda 
-(_: T).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind b) v)))) 
-(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
-T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind Void) 
-(TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
-(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))) (ex3_intro C 
-(\lambda (c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) 
-(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
-T).((ty3 g c1 v w) \to False))) c2 (refl_equal C (CHead c2 (Bind Void) (TSort 
-O))) H13 H11))))))))) H6)) H5)))))))))) (\lambda (c0: C).(\lambda (c2: 
-C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) 
-\to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind 
-b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
-c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
-C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
-(f: F).(\lambda (H3: (eq C (CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let 
-H4 \def (eq_ind C (CHead c0 (Flat f) u) (\lambda (ee: C).(match ee with 
-[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind 
-_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead c1 (Bind b) v) 
-H3) in (False_ind (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 
-(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) 
-(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq 
-C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
-(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H4))))))))) y 
-x H0))) H)))))).
-
-lemma wf3_gen_flat1:
- \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: 
-F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: 
-F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat 
-f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: 
-C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: 
-C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: 
-nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def 
-(eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
-True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Flat f) v) H1) in 
-(False_ind (wf3 g c1 (CSort m)) H2)))) (\lambda (c0: C).(\lambda (c2: 
-C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) 
-\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
-t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat 
-f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match 
-ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
-with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 
-(Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 (Bind b) u)) H5))))))))))) 
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: 
-(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
-T).(\lambda (_: ((\forall (t: T).((ty3 g c0 u t) \to False)))).(\lambda (b: 
-B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 
-\def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match ee with [(CSort 
-_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 (Flat f) v) 
-H4) in (False_ind (wf3 g c1 (CHead c2 (Bind Void) (TSort O))) H5)))))))))) 
-(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: 
-(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
-T).(\lambda (f0: F).(\lambda (H3: (eq C (CHead c0 (Flat f0) u) (CHead c1 
-(Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort 
-_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) 
-(CHead c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow (match 
-k with [(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 
-(Flat f0) u) (CHead c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda 
-(e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow 
-t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda (_: (eq F 
-f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 (\lambda (c: 
-C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) in (let H10 
-\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in H10))))) H5)) 
-H4))))))))) y x H0))) H)))))).
-
-lemma wf3_gen_head2:
- \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: 
-K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
-\def
- \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: 
-K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda 
-(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind 
-b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: 
-C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
-k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k 
-v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort 
-_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c k v) H1) 
-in (False_ind (ex B (\lambda (b: B).(eq K k (Bind b)))) H2)))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
-(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda 
-(H4: (eq C (CHead c2 (Bind b) u) (CHead c k v))).(let H5 \def (f_equal C C 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def 
-(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in 
-((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead 
-c k v) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 
-c)).(let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in 
-(let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex 
-B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C 
-c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k 
-(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 
-(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B 
-(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K 
-(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) 
-H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 
-c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
-k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u 
-t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) 
-(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
-with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 
-(Bind Void) (TSort O)) (CHead c k v) H4) in ((let H6 \def (f_equal C K 
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind Void) | (CHead _ 
-k0 _) \Rightarrow k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in 
-((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
-\Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind Void) 
-(TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) k)).(\lambda 
-(H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 
-(CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in 
-(let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let 
-H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B 
-(\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in (eq_ind K 
-(Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind b0))))) 
-(let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind Void) t)) 
-\to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 (TSort O) H7) 
-in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) Void 
-(refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
-(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: 
-T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def 
-(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind 
-C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq 
-K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda 
-(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k 
-v))))))))))))) x y H0))) H)))))).
-
-theorem wf3_mono:
- \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall 
-(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c 
-c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 
-c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: 
-(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C 
-(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 
-H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 
-c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 
-c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u 
-t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) 
-u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in 
-(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind 
-b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead 
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
-C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) 
-c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead 
-c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda 
-(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: 
-C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: 
-C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 
-u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 
-x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda 
-(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind 
-b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 
-H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind 
-Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall 
-(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 
-(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda 
-(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) 
-u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort 
-O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u 
-w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: 
-C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def 
-H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort 
-O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: 
-C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) 
-\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g 
-c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g 
-(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in 
-(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C 
-c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) 
-(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq 
-C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) 
-(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 
-(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda 
-(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: 
-T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u 
-w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 
-(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
-C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) 
-c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind 
-b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r 
-C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort 
-O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C 
-(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) 
-H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) 
-(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: 
-T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead 
-c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
-C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) 
-(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) 
-(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 
-g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda 
-(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead 
-c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K 
-(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) 
-(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: 
-((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: 
-T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) 
-u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) 
-c c1 H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma
deleted file mode 100644 (file)
index 9c5fc73..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/getl.ma
+++ /dev/null
@@ -1,199 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wf3/clear.ma".
-
-include "basic_1/ty3/dec.ma".
-
-lemma wf3_getl_conf:
- \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall 
-(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: 
-C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda 
-(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
-d2)))))))))))))
-\def
- \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: 
-C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to 
-(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g 
-d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) 
-(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: 
-C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda 
-(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda 
-(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) 
-(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def 
-(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T 
-(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda 
-(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 
-g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort 
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 
-g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
-b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda 
-(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
-C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 
-v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
-(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: 
-C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda 
-(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind 
-b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) 
-v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O 
-(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) 
-x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: 
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 
-c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to 
-False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort 
-O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 
-g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) 
-v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 
-(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: 
-((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind 
-Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in 
-(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 
-(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
-d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: 
-((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall 
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
-b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind 
-(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 
-(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall 
-(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: 
-nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) 
-(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g 
-(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort 
-n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda 
-(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead 
-d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to 
-(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda 
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl 
-(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: 
-C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 
-g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r 
-k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: 
-B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r 
-(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t 
-b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda 
-(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: 
-T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C 
-(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: 
-C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to 
-False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) 
-(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: 
-C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: 
-C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t 
-w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
-(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: 
-C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: 
-C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda 
-(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind 
-b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 
-(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g 
-x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 
-(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: 
-C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: 
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind 
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S 
-n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
-d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) 
-c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 
-(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: 
-C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: 
-C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) 
-(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda 
-(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
-d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort 
-O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t 
-w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: 
-C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda 
-(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let 
-H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) 
-v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) 
-(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: 
-C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind 
-b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S 
-n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: 
-C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 
-(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda 
-(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c 
-(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 
-v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n 
-H1)))))))))))))) c1)))) i)).
-
-lemma getl_wf3_trans:
- \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to 
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: 
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
-C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to 
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 
-d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 
-d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x 
-\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x 
-in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) 
-(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) 
-(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x 
-d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 
-d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: 
-nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to 
-(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: 
-C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: 
-C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall 
-(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c 
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda 
-(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda 
-(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C 
-(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 
-d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c 
-d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda 
-(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda 
-(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k 
-t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind 
-(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g 
-(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: 
-B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 
-H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda 
-(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) 
-c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 
-g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in 
-(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall 
-(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c 
-(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T 
-(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) 
-(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: 
-C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t 
-x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda 
-(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 
-b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: 
-T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead 
-c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) 
-(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort 
-O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c 
-d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C 
-(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C 
-(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S 
-n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S 
-n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) 
-(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) 
-k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma
deleted file mode 100644 (file)
index aec1f02..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/props.ma
+++ /dev/null
@@ -1,153 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wf3/ty3.ma".
-
-include "basic_1/app/defs.ma".
-
-lemma wf3_total:
- \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: 
-C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g 
-(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C 
-(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def 
-H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g 
-(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind 
-(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda 
-(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex 
-T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to 
-False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda 
-(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 
-g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda 
-(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g 
-(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 
-b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to 
-False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) 
-(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda 
-(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x 
-(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
-
-lemma ty3_shift1:
- \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall 
-(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c 
-t2)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c 
-(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall 
-(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 
-t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: 
-C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 
-g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) 
-(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C 
-c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g 
-(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda 
-(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C 
-(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C 
-C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) 
-\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind b) u) H4) in (let H7 
-\def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: 
-T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 
-t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 \def (eq_ind_r C c2 (\lambda (c0: 
-C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T (\lambda (t0: T).(ty3 g (CHead c1 
-(Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind b) u t1)) 
-(app1 c1 (THead (Bind b) u t2))) (\lambda (x: T).(\lambda (_: (ty3 g (CHead 
-c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) (THead (Bind b) u t1) (THead 
-(Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 H5)))) (ty3_correct g (CHead c1 
-(Bind b) u) t1 t2 H5))))))))))))))))) (\lambda (c1: C).(\lambda (c2: 
-C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall 
-(t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 
-c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: 
-T).((ty3 g c1 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead 
-c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def 
-(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
-c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort 
-O)) H4) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
-_) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
-\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 
-(Bind Void) (TSort O)) H4) in ((let H8 \def (f_equal C T (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
-(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: 
-(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b 
-(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in 
-(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) 
-(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 
-\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 
-(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: 
-T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) 
-(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 
-(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def 
-(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall 
-(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 
-t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g 
-c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) 
-(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort 
-O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda 
-(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) 
-(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g 
-c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) 
-(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b 
-H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
-(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall 
-(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 
-t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 
-(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead 
-c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 
-(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 
-c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort 
-(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in 
-(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 
-(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in 
-(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 
-g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
-t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def 
-(eq_ind K (Flat f) (\lambda (ee: K).(match ee with [(Bind _) \Rightarrow 
-False | (Flat _) \Rightarrow True])) I (Bind x) H9) in (False_ind (ty3 g 
-(CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
-t2))) H10)))) H8)))))))))))))))) y c H0))) H))).
-
-lemma wf3_idem:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g 
-c2 c2))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 
-c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: 
-nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g 
-c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda 
-(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g 
-c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: 
-(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: 
-((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 
-c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda 
-(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 
-c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
-
-lemma wf3_ty3:
- \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t 
-u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t 
-u)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
-(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind 
-C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) 
-(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 
-x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t 
-u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma
deleted file mode 100644 (file)
index 2958902..0000000
--- a/matita/matita/contribs/lambdadelta/basic_1/wf3/ty3.ma
+++ /dev/null
@@ -1,126 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/wf3/getl.ma".
-
-lemma wf3_pr2_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
-u) \to (pr2 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
-T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 
-c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
-(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: 
-T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
-C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 
-u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def 
-(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in 
-(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: 
-T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 
-g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) 
-(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda 
-(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) 
-H5)))))))))))))))))) c1 t1 t2 H))))).
-
-lemma wf3_pr3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
-u) \to (pr3 c2 t1 t2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall 
-(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t 
-t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 
-c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) 
-(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: 
-T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 
-c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda 
-(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 
-t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 
-H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
-
-lemma wf3_pc3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 
-u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda 
-(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g 
-c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) 
-(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: 
-(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf 
-g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) 
-H3)))))))))))).
-
-lemma wf3_ty3_conf:
- \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
-t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
-(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: 
-C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda 
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: 
-T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall 
-(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 
-t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: 
-T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 
-x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 
-c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: 
-C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g 
-c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
-(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g 
-c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def 
-(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind 
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: 
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d 
-x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: 
-nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
-t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u 
-t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def 
-(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind 
-C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
-C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: 
-C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d 
-x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: 
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: 
-((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: 
-B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) 
-t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to 
-(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g 
-c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t 
-H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
-(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g 
-c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
-(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g 
-c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c 
-c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda 
-(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c 
-c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c 
-c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
-

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/A/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/A/defs.ma
new file mode 100644 (file)
index 0000000..7f893aa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/A/defs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/preamble.ma".
+
+inductive A: Type[0] \def
+| ASort: nat \to (nat \to A)
+| AHead: A \to (A \to A).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/A/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/A/fwd.ma
new file mode 100644 (file)
index 0000000..64aa776
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/A/fwd.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/A/defs.ma".
+
+implied rec lemma A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall 
+(n0: nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: 
+A).((P a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with 
+[(ASort n n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (f0 a0 
+((A_rect P f f0) a0) a1 ((A_rect P f f0) a1))].
+
+implied lemma A_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).(\forall (n0: nat).(P (ASort 
+n n0))))) \to (((\forall (a: A).((P a) \to (\forall (a0: A).((P a0) \to (P 
+(AHead a a0))))))) \to (\forall (a: A).(P a))))
+\def
+ \lambda (P: ((A \to Prop))).(A_rect P).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/C/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/C/defs.ma
new file mode 100644 (file)
index 0000000..aa19799
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/C/defs.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+inductive C: Type[0] \def
+| CSort: nat \to C
+| CHead: C \to (K \to (T \to C)).
+
+rec definition cweight (c: C) on c: nat \def match c with [(CSort _) 
+\Rightarrow O | (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))].
+
+definition clt:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
+
+definition cle:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
+
+rec definition CTail (k: K) (t: T) (c: C) on c: C \def match c with [(CSort 
+n) \Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead 
+(CTail k t d) h u)].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/C/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/C/fwd.ma
new file mode 100644 (file)
index 0000000..3979a5b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/C/fwd.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+implied rec lemma C_rect (P: (C \to Type[0])) (f: (\forall (n: nat).(P (CSort 
+n)))) (f0: (\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P 
+(CHead c k t))))))) (c: C) on c: P c \def match c with [(CSort n) \Rightarrow 
+(f n) | (CHead c0 k t) \Rightarrow (f0 c0 ((C_rect P f f0) c0) k t)].
+
+implied lemma C_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CHead c k 
+t))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(C_rect P).
+
+fact clt_wf__q_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to 
+Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0 
+c))))) P n))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: 
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to 
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c) 
+n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight 
+c)))))).
+
+lemma clt_wf_ind:
+ \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) 
+\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c: 
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to 
+Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d) 
+(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind 
+(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0: 
+C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
+\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat 
+(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
+(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P 
+c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt 
+(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight 
+d))))))))))))) c)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/C/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/C/props.ma
new file mode 100644 (file)
index 0000000..2915384
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/C/props.ma
@@ -0,0 +1,115 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/fwd.ma".
+
+include "basic_1A/T/props.ma".
+
+lemma cle_r:
+ \forall (c: C).(cle c c)
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(le (cweight c0) (cweight c0))) 
+(\lambda (_: nat).(le_O_n O)) (\lambda (c0: C).(\lambda (_: (le (cweight c0) 
+(cweight c0))).(\lambda (_: K).(\lambda (t: T).(le_n (plus (cweight c0) 
+(tweight t))))))) c).
+
+lemma cle_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (u1: T).(\forall 
+(u2: T).((tle u1 u2) \to (\forall (k: K).(cle (CHead c1 k u1) (CHead c2 k 
+u2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
+c2))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (le (tweight u1) 
+(tweight u2))).(\lambda (_: K).(le_plus_plus (cweight c1) (cweight c2) 
+(tweight u1) (tweight u2) H H0))))))).
+
+lemma cle_trans_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (k: K).(\forall 
+(u: T).(cle c1 (CHead c2 k u))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
+c2))).(\lambda (_: K).(\lambda (u: T).(le_plus_trans (cweight c1) (cweight 
+c2) (tweight u) H))))).
+
+lemma clt_cong:
+ \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: 
+T).(clt (CHead c k t) (CHead d k t))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight 
+d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d) 
+(tweight t) H))))).
+
+lemma clt_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight 
+c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_reg_l O 
+(tweight u) (cweight c) (tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
+
+lemma chead_ctail:
+ \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: 
+K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3 
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) 
+(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k: 
+K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
+(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead 
+(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall 
+(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
+(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def 
+H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C 
+(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d: 
+C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d)))))) 
+(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead 
+c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1: 
+C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead 
+c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d: 
+C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0 
+(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0 
+k t) H1))))) H0))))))))) c).
+
+lemma clt_thead:
+ \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt 
+c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0: 
+C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t: 
+T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
+
+lemma c_tail_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to 
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t 
+c))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort 
+n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: 
+T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P 
+c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1) 
+\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d: 
+C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_: 
+((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to 
+(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind 
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) 
+(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2 
+x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def 
+(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P 
+d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2)) 
+(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/G/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/G/defs.ma
new file mode 100644 (file)
index 0000000..a794750
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/G/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/preamble.ma".
+
+record G : Type[0] \def {
+ next: (nat \to nat);
+ next_lt: (\forall (n: nat).(lt n (next n)))
+}.
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/T/dec.ma
new file mode 100644 (file)
index 0000000..6760101
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/T/dec.ma
@@ -0,0 +1,411 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/fwd.ma".
+
+fact terms_props__bind_dec:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
+(P: Prop).P))))
+\def
+ \lambda (b1: B).(B_ind (\lambda (b: B).(\forall (b2: B).(or (eq B b b2) ((eq 
+B b b2) \to (\forall (P: Prop).P))))) (\lambda (b2: B).(B_ind (\lambda (b: 
+B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl 
+(eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B 
+Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
+\Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind P 
+H0))))) (or_intror (eq B Abbr Void) ((eq B Abbr Void) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
+\Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind P 
+H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B 
+Abst b) \to (\forall (P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst 
+Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: 
+Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match ee with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False])) I Abbr 
+H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B Abst Abst) \to 
+(\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B Abst Void) ((eq 
+B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst 
+Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match 
+ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow 
+False])) I Void H) in (False_ind P H0))))) b2)) (\lambda (b2: B).(B_ind 
+(\lambda (b: B).(or (eq B Void b) ((eq B Void b) \to (\forall (P: Prop).P)))) 
+(or_intror (eq B Void Abbr) ((eq B Void Abbr) \to (\forall (P: Prop).P)) 
+(\lambda (H: (eq B Void Abbr)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void 
+(\lambda (ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow 
+False | Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror 
+(eq B Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: 
+(eq B Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda 
+(ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | 
+Void \Rightarrow True])) I Abst H) in (False_ind P H0))))) (or_introl (eq B 
+Void Void) ((eq B Void Void) \to (\forall (P: Prop).P)) (refl_equal B Void)) 
+b2)) b1).
+
+lemma bind_dec_not:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
+\def
+ \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
+in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: 
+Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1 
+b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: 
+(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
+b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
+
+fact terms_props__flat_dec:
+ \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
+(P: Prop).P))))
+\def
+ \lambda (f1: F).(F_ind (\lambda (f: F).(\forall (f2: F).(or (eq F f f2) ((eq 
+F f f2) \to (\forall (P: Prop).P))))) (\lambda (f2: F).(F_ind (\lambda (f: 
+F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl 
+(eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F 
+Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind F Appl (\lambda (ee: F).(match ee with [Appl \Rightarrow True | Cast 
+\Rightarrow False])) I Cast H) in (False_ind P H0))))) f2)) (\lambda (f2: 
+F).(F_ind (\lambda (f: F).(or (eq F Cast f) ((eq F Cast f) \to (\forall (P: 
+Prop).P)))) (or_intror (eq F Cast Appl) ((eq F Cast Appl) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind F Cast (\lambda (ee: F).(match ee with [Appl \Rightarrow False | Cast 
+\Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast 
+Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) 
+f1).
+
+fact terms_props__kind_dec:
+ \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall 
+(P: Prop).P))))
+\def
+ \lambda (k1: K).(K_ind (\lambda (k: K).(\forall (k2: K).(or (eq K k k2) ((eq 
+K k k2) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (k2: K).(K_ind 
+(\lambda (k: K).(or (eq K (Bind b) k) ((eq K (Bind b) k) \to (\forall (P: 
+Prop).P)))) (\lambda (b0: B).(let H_x \def (terms_props__bind_dec b b0) in 
+(let H \def H_x in (or_ind (eq B b b0) ((eq B b b0) \to (\forall (P: 
+Prop).P)) (or (eq K (Bind b) (Bind b0)) ((eq K (Bind b) (Bind b0)) \to 
+(\forall (P: Prop).P))) (\lambda (H0: (eq B b b0)).(eq_ind B b (\lambda (b1: 
+B).(or (eq K (Bind b) (Bind b1)) ((eq K (Bind b) (Bind b1)) \to (\forall (P: 
+Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to 
+(\forall (P: Prop).P)) (refl_equal K (Bind b))) b0 H0)) (\lambda (H0: (((eq B 
+b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq 
+K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) 
+(Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: 
+K).(match e with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) (Bind 
+b) (Bind b0) H1) in (let H3 \def (eq_ind_r B b0 (\lambda (b1: B).((eq B b b1) 
+\to (\forall (P0: Prop).P0))) H0 b H2) in (H3 (refl_equal B b) P))))))) H)))) 
+(\lambda (f: F).(or_intror (eq K (Bind b) (Flat f)) ((eq K (Bind b) (Flat f)) 
+\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Bind b) (Flat f))).(\lambda 
+(P: Prop).(let H0 \def (eq_ind K (Bind b) (\lambda (ee: K).(match ee with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in 
+(False_ind P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda 
+(k: K).(or (eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) 
+(\lambda (b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) 
+\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda 
+(P: Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind b) H) in 
+(False_ind P H0)))))) (\lambda (f0: F).(let H_x \def (terms_props__flat_dec f 
+f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F f f0) \to (\forall (P: 
+Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to 
+(\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind F f (\lambda (f1: 
+F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) \to (\forall (P: 
+Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat f) (Flat f)) \to 
+(\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) (\lambda (H0: (((eq F 
+f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Flat f) (Flat f0)) ((eq 
+K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Flat f) 
+(Flat f0))).(\lambda (P: Prop).(let H2 \def (f_equal K F (\lambda (e: 
+K).(match e with [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat 
+f) (Flat f0) H1) in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) 
+\to (\forall (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) 
+k2))) k1).
+
+lemma term_dec:
+ \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall 
+(P: Prop).P))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (t2: T).(or (eq T t t2) ((eq 
+T t t2) \to (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (t2: 
+T).(T_ind (\lambda (t: T).(or (eq T (TSort n) t) ((eq T (TSort n) t) \to 
+(\forall (P: Prop).P)))) (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in 
+(let H \def H_x in (or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: 
+Prop).P)) (or (eq T (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to 
+(\forall (P: Prop).P))) (\lambda (H0: (eq nat n n0)).(eq_ind nat n (\lambda 
+(n1: nat).(or (eq T (TSort n) (TSort n1)) ((eq T (TSort n) (TSort n1)) \to 
+(\forall (P: Prop).P)))) (or_introl (eq T (TSort n) (TSort n)) ((eq T (TSort 
+n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0)) 
+(\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T 
+(TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) 
+(\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
+(TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort n0) 
+H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to 
+(\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) 
+(\lambda (n0: nat).(or_intror (eq T (TSort n) (TLRef n0)) ((eq T (TSort n) 
+(TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TSort n) (TLRef 
+n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) 
+(\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T 
+(TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or 
+(eq T (TSort n) t0) ((eq T (TSort n) t0) \to (\forall (P: 
+Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) 
+(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) 
+(THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead k t t0) H1) in (False_ind 
+P H2)))))))))) t2))) (\lambda (n: nat).(\lambda (t2: T).(T_ind (\lambda (t: 
+T).(or (eq T (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))) 
+(\lambda (n0: nat).(or_intror (eq T (TLRef n) (TSort n0)) ((eq T (TLRef n) 
+(TSort n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TLRef n) (TSort 
+n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) 
+(\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind 
+(eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) 
+(TLRef n0)) ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P))) (\lambda 
+(H0: (eq nat n n0)).(eq_ind nat n (\lambda (n1: nat).(or (eq T (TLRef n) 
+(TLRef n1)) ((eq T (TLRef n) (TLRef n1)) \to (\forall (P: Prop).P)))) 
+(or_introl (eq T (TLRef n) (TLRef n)) ((eq T (TLRef n) (TLRef n)) \to 
+(\forall (P: Prop).P)) (refl_equal T (TLRef n))) n0 H0)) (\lambda (H0: (((eq 
+nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) 
+((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T 
+(TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow n | (TLRef n1) 
+\Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H1) in 
+(let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to (\forall 
+(P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) (\lambda 
+(k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TLRef n) t) ((eq T (TLRef n) 
+t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T 
+(TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: Prop).P)))).(or_intror 
+(eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) (THead k t t0)) \to (\forall 
+(P: Prop).P)) (\lambda (H1: (eq T (TLRef n) (THead k t t0))).(\lambda (P: 
+Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) 
+(\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t 
+t2) ((eq T t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda 
+(H0: ((\forall (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
+Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t 
+t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: 
+nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort 
+n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort 
+n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind P H2)))))) 
+(\lambda (n: nat).(or_intror (eq T (THead k t t0) (TLRef n)) ((eq T (THead k 
+t t0) (TLRef n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t 
+t0) (TLRef n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in 
+(False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq 
+T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: 
+Prop).P)))).(\lambda (t4: T).(\lambda (H2: (or (eq T (THead k t t0) t4) ((eq 
+T (THead k t t0) t4) \to (\forall (P: Prop).P)))).(let H_x \def (H t3) in 
+(let H3 \def H_x in (or_ind (eq T t t3) ((eq T t t3) \to (\forall (P: 
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k t t0) 
+(THead k0 t3 t4)) \to (\forall (P: Prop).P))) (\lambda (H4: (eq T t t3)).(let 
+H5 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H1 t H4) in (eq_ind T t 
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t5 t4)) ((eq T (THead k t 
+t0) (THead k0 t5 t4)) \to (\forall (P: Prop).P)))) (let H_x0 \def (H0 t4) in 
+(let H6 \def H_x0 in (or_ind (eq T t0 t4) ((eq T t0 t4) \to (\forall (P: 
+Prop).P)) (or (eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) 
+(THead k0 t t4)) \to (\forall (P: Prop).P))) (\lambda (H7: (eq T t0 t4)).(let 
+H8 \def (eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
+(THead k t t0) t5) \to (\forall (P: Prop).P)))) H2 t0 H7) in (eq_ind T t0 
+(\lambda (t5: T).(or (eq T (THead k t t0) (THead k0 t t5)) ((eq T (THead k t 
+t0) (THead k0 t t5)) \to (\forall (P: Prop).P)))) (let H_x1 \def 
+(terms_props__kind_dec k k0) in (let H9 \def H_x1 in (or_ind (eq K k k0) ((eq 
+K k k0) \to (\forall (P: Prop).P)) (or (eq T (THead k t t0) (THead k0 t t0)) 
+((eq T (THead k t t0) (THead k0 t t0)) \to (\forall (P: Prop).P))) (\lambda 
+(H10: (eq K k k0)).(eq_ind K k (\lambda (k1: K).(or (eq T (THead k t t0) 
+(THead k1 t t0)) ((eq T (THead k t t0) (THead k1 t t0)) \to (\forall (P: 
+Prop).P)))) (or_introl (eq T (THead k t t0) (THead k t t0)) ((eq T (THead k t 
+t0) (THead k t t0)) \to (\forall (P: Prop).P)) (refl_equal T (THead k t t0))) 
+k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror 
+(eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) 
+\to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t 
+t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
+\Rightarrow k1])) (THead k t t0) (THead k0 t t0) H11) in (let H13 \def 
+(eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to (\forall (P0: Prop).P0))) 
+H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) t4 H7))) (\lambda (H7: 
+(((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror (eq T (THead k t t0) 
+(THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
+Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t t4))).(\lambda (P: 
+Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) 
+(THead k t t0) (THead k0 t t4) H8) in ((let H10 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in 
+(\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq 
+T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r 
+T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
+\to (\forall (P0: Prop).P0)))) H2 t0 H10) in (H12 (refl_equal T t0) P))))) 
+H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P: 
+Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k 
+t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead 
+k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t3 
+t4) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) 
+(THead k t t0) (THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 t4) H5) in 
+(\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def (eq_ind_r 
+T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
+\to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r T t3 
+(\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in (let 
+H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
+(THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 
+(refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
+
+lemma binder_dec:
+ \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
+T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall 
+(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T 
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind 
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) 
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w 
+u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P 
+H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: 
+B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to 
+(\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P: 
+Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) 
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T 
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 
+u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead 
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 
+B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind 
+b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead 
+(Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda 
+(b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) 
+w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 
+t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) (\lambda (b: 
+B).(\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b0: B).(\lambda 
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b0) w u)))))) (\forall (b0: 
+B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b0) w u)) \to 
+(\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T 
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b0) w 
+u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead 
+(Bind b0) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T 
+(\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) 
+(THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: 
+T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: 
+Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: 
+T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal 
+T (THead (Bind b) t0 t1))))))))) (\lambda (f: F).(\lambda (t0: T).(\lambda 
+(_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 
+(THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: 
+T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda 
+(t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda 
+(u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: 
+T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: 
+Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: 
+T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) 
+(THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead 
+(Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) 
+in (False_ind P H2))))))))))))) k)) t).
+
+lemma abst_dec:
+ \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead 
+(Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to 
+(\forall (P: Prop).P)))))
+\def
+ \lambda (u: T).(T_ind (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda 
+(t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead 
+(Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda 
+(v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v 
+t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall 
+(P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind 
+Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in 
+(False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(or_intror (ex T 
+(\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t: 
+T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) 
+(\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v 
+t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind 
+P H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall (v: 
+T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall 
+(t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P: 
+Prop).P))))))).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(or (ex T 
+(\lambda (t1: T).(eq T t0 (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T 
+t0 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))))).(\lambda (v: 
+T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H1 \def H_x in 
+(or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) 
+(or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) 
+(\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall 
+(P: Prop).P)))) (\lambda (H2: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) 
+(\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead 
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind 
+Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in 
+(let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P: 
+Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H4: (eq T t 
+v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P))))) 
+(or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind 
+Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind 
+Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T 
+(THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead 
+(Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) \to (\forall (P: 
+Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) 
+(THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) 
+(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
+T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
+t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) 
+\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) 
+(THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T 
+t v)).(H4 H8 P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) 
+\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k 
+t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) 
+(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
+T).(\lambda (H3: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: 
+Prop).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
+\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead k t t0) (THead (Bind 
+Abst) v t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: 
+(eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) 
+H1))))))))) u).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/T/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/T/defs.ma
new file mode 100644 (file)
index 0000000..554c4de
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/T/defs.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/preamble.ma".
+
+inductive B: Type[0] \def
+| Abbr: B
+| Abst: B
+| Void: B.
+
+inductive F: Type[0] \def
+| Appl: F
+| Cast: F.
+
+inductive K: Type[0] \def
+| Bind: B \to K
+| Flat: F \to K.
+
+inductive T: Type[0] \def
+| TSort: nat \to T
+| TLRef: nat \to T
+| THead: K \to (T \to (T \to T)).
+
+rec definition tweight (t: T) on t: nat \def match t with [(TSort _) 
+\Rightarrow (S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow 
+(S (plus (tweight u) (tweight t0)))].
+
+definition tle:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(le (tweight t1) (tweight t2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/T/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/T/fwd.ma
new file mode 100644 (file)
index 0000000..570b70c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/T/fwd.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+implied rec lemma T_rect (P: (T \to Type[0])) (f: (\forall (n: nat).(P (TSort 
+n)))) (f0: (\forall (n: nat).(P (TLRef n)))) (f1: (\forall (k: K).(\forall 
+(t: T).((P t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) (t: 
+T) on t: P t \def match t with [(TSort n) \Rightarrow (f n) | (TLRef n) 
+\Rightarrow (f0 n) | (THead k t0 t1) \Rightarrow (f1 k t0 ((T_rect P f f0 f1) 
+t0) t1 ((T_rect P f f0 f1) t1))].
+
+implied lemma T_ind:
+ \forall (P: ((T \to Prop))).(((\forall (n: nat).(P (TSort n)))) \to 
+(((\forall (n: nat).(P (TLRef n)))) \to (((\forall (k: K).(\forall (t: T).((P 
+t) \to (\forall (t0: T).((P t0) \to (P (THead k t t0)))))))) \to (\forall (t: 
+T).(P t)))))
+\def
+ \lambda (P: ((T \to Prop))).(T_rect P).
+
+lemma thead_x_y_y:
+ \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to 
+(\forall (P: Prop).P))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq 
+T (THead k v t0) t0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda 
+(H: (eq T (THead k v (TSort n)) (TSort n))).(\lambda (P: Prop).(let H0 \def 
+(eq_ind T (THead k v (TSort n)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H) in (False_ind P H0))))) (\lambda (n: nat).(\lambda (H: 
+(eq T (THead k v (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def 
+(eq_ind T (THead k v (TLRef n)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H) in (False_ind P H0))))) (\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (_: (((eq T (THead k v t0) t0) \to (\forall (P: 
+Prop).P)))).(\lambda (t1: T).(\lambda (H0: (((eq T (THead k v t1) t1) \to 
+(\forall (P: Prop).P)))).(\lambda (H1: (eq T (THead k v (THead k0 t0 t1)) 
+(THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead 
+k1 _ _) \Rightarrow k1])) (THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) 
+in ((let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) 
+(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in ((let H4 \def (f_equal T 
+T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead k0 t0 t1) | 
+(TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t2) \Rightarrow t2])) 
+(THead k v (THead k0 t0 t1)) (THead k0 t0 t1) H1) in (\lambda (H5: (eq T v 
+t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind T v (\lambda (t2: 
+T).((eq T (THead k t2 t1) t1) \to (\forall (P0: Prop).P0))) H0 t0 H5) in (let 
+H8 \def (eq_ind K k (\lambda (k1: K).((eq T (THead k1 t0 t1) t1) \to (\forall 
+(P0: Prop).P0))) H7 k0 H6) in (H8 H4 P)))))) H3)) H2))))))))) t))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/T/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/T/props.ma
new file mode 100644 (file)
index 0000000..1011e40
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/T/props.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/fwd.ma".
+
+lemma not_abbr_abst:
+ not (eq B Abbr Abst)
+\def
+ \lambda (H: (eq B Abbr Abst)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
+B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
+\Rightarrow False])) I Abst H) in (False_ind False H0)).
+
+lemma not_void_abst:
+ not (eq B Void Abst)
+\def
+ \lambda (H: (eq B Void Abst)).(let H0 \def (eq_ind B Void (\lambda (ee: 
+B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | Void 
+\Rightarrow True])) I Abst H) in (False_ind False H0)).
+
+lemma not_abbr_void:
+ not (eq B Abbr Void)
+\def
+ \lambda (H: (eq B Abbr Void)).(let H0 \def (eq_ind B Abbr (\lambda (ee: 
+B).(match ee with [Abbr \Rightarrow True | Abst \Rightarrow False | Void 
+\Rightarrow False])) I Void H) in (False_ind False H0)).
+
+lemma not_abst_void:
+ not (eq B Abst Void)
+\def
+ \lambda (H: (eq B Abst Void)).(let H0 \def (eq_ind B Abst (\lambda (ee: 
+B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void 
+\Rightarrow False])) I Void H) in (False_ind False H0)).
+
+lemma tweight_lt:
+ \forall (t: T).(lt O (tweight t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(lt O (tweight t0))) (\lambda (_: 
+nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: K).(\lambda 
+(t0: T).(\lambda (H: (lt O (tweight t0))).(\lambda (t1: T).(\lambda (_: (lt O 
+(tweight t1))).(le_S (S O) (plus (tweight t0) (tweight t1)) (le_plus_trans (S 
+O) (tweight t0) (tweight t1) H))))))) t).
+
+lemma tle_r:
+ \forall (t: T).(tle t t)
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(le (tweight t0) (tweight t0))) 
+(\lambda (_: nat).(le_n (S O))) (\lambda (_: nat).(le_n (S O))) (\lambda (_: 
+K).(\lambda (t0: T).(\lambda (_: (le (tweight t0) (tweight t0))).(\lambda 
+(t1: T).(\lambda (_: (le (tweight t1) (tweight t1))).(le_n (S (plus (tweight 
+t0) (tweight t1))))))))) t).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/aplus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/aplus/defs.ma
new file mode 100644 (file)
index 0000000..39f1080
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/aplus/defs.ma
@@ -0,0 +1,21 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/asucc/defs.ma".
+
+rec definition aplus (g: G) (a: A) (n: nat) on n: A \def match n with [O 
+\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/aplus/props.ma
new file mode 100644 (file)
index 0000000..d01417a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/aplus/props.ma
@@ -0,0 +1,240 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/aplus/defs.ma".
+
+include "basic_1A/A/fwd.ma".
+
+include "basic_1A/next_plus/props.ma".
+
+lemma aplus_reg_r:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall 
+(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A 
+(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda 
+(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 
+(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n 
+h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n 
+h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))).
+
+lemma aplus_assoc:
+ \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A 
+(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: 
+nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n 
+h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: 
+nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus 
+g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A 
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) 
+(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g 
+(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O 
+n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) 
+n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda 
+(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g 
+(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) 
+n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) 
+(plus_n_Sm n n0)))) h2)))) h1))).
+
+lemma aplus_asucc:
+ \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) 
+h) (asucc g (aplus g a h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a 
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) 
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) 
+(aplus_assoc g a (S O) h)))).
+
+lemma aplus_sort_O_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O 
+n) (S k)) (aplus g (ASort O (next g n)) k))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc 
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) 
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) 
+k)) (aplus_asucc g k (ASort O n))))).
+
+lemma aplus_sort_S_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A 
+(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind 
+A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g 
+(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g 
+(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
+
+lemma aplus_asort_O_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O 
+n) h) (ASort O (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: 
+nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda 
+(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall 
+(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 
+n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) 
+(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat 
+(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next 
+g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) 
+(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n 
+(ASort O n0)))))) h)).
+
+lemma aplus_asort_le_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h 
+k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: 
+nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort 
+(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O 
+k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) 
+(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: 
+nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A 
+(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: 
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A 
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda 
+(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat 
+O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) 
+h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S 
+x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) 
+H1) in (False_ind (eq A (asucc g (aplus g (ASort O n) h0)) (ASort (minus O (S 
+h0)) n)) H3))))) (le_gen_S h0 O H0)))) (\lambda (n: nat).(\lambda (_: 
+((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n n0) 
+h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: (le (S 
+h0) (S n))).(eq_ind A (aplus g (asucc g (ASort (S n) n0)) h0) (\lambda (a: 
+A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n n0 (le_S_n h0 n H1)) (asucc 
+g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g h0 (ASort (S n) n0))))))) 
+k)))) h)).
+
+lemma aplus_asort_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A 
+(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: 
+nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus 
+g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) 
+(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) 
+(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) 
+(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus 
+h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a 
+(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O 
+(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k 
+h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A 
+(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) 
+(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h 
+(le_S_n (S k) (S h) (le_S (S (S k)) (S h) (le_n_S (S k) h H)))))) (minus k k) 
+(minus_n_n k)) (aplus g (ASort k n) k) (aplus_asort_le_simpl g k k n (le_n 
+k))) (aplus g (ASort k n) (plus k (minus h k))) (aplus_assoc g (ASort k n) k 
+(minus h k))) h (le_plus_minus k h (le_S_n k h (le_S_n (S k) (S h) (le_S (S 
+(S k)) (S h) (le_n_S (S k) h H))))))) (\lambda (H: (le h k)).(eq_ind_r A 
+(ASort (minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g 
+n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) 
+n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) 
+(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) 
+(aplus_asort_le_simpl g h k n H))))))).
+
+lemma aplus_ahead_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A 
+(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: 
+A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 
+n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A 
+(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: 
+A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda 
+(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g 
+(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) 
+(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n 
+a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) 
+h)).
+
+lemma aplus_asucc_false:
+ \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) 
+h) a) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: 
+nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A 
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) 
+\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
+(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O 
+(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) 
+(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 
+\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 
+(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) 
+(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat 
+(\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) 
+\Rightarrow (next_plus g (next g n0) (minus h O))])) (ASort (minus O h) 
+(next_plus g (next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def 
+(eq_ind_r nat (minus h O) (\lambda (n1: nat).(eq nat (next_plus g (next g n0) 
+n1) n0)) H2 h (minus_n_O h)) in (le_lt_false n0 n0 (le_n n0) (eq_ind nat 
+(next_plus g (next g n0) h) (\lambda (n1: nat).(lt n0 n1)) (next_plus_lt g h 
+n0) n0 H3) P))))) (\lambda (n1: nat).(\lambda (_: (((eq A (aplus g (match n1 
+with [O \Rightarrow (ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 
+n0)]) h) (ASort n1 n0)) \to P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) 
+h) (ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) 
+(\lambda (a0: A).(eq A a0 (ASort (S n1) n0))) H0 (ASort (minus n1 h) 
+(next_plus g n0 (minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def 
+(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
+(AHead _ _) \Rightarrow (minus n1 h)])) (ASort (minus n1 h) (next_plus g n0 
+(minus h n1))) (ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda 
+(e: A).(match e with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow 
+(next_plus g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h 
+n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S 
+n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) 
+(minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: A).(\lambda 
+(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: 
+Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus 
+g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: 
+nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 
+a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g 
+a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 (AHead a0 (aplus g 
+(asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def 
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (aplus g 
+(asucc g a1) h) | (AHead _ a2) \Rightarrow a2])) (AHead a0 (aplus g (asucc g 
+a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+
+lemma aplus_inj:
+ \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A 
+(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: 
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
+h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A 
+(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: 
+(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: 
+A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: 
+(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g 
+a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g 
+n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq 
+nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: 
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n 
+h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq 
+A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: 
+A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A 
+(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) 
+n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) 
+(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a 
+n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: 
+(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def 
+(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus 
+g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def 
+(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g 
+a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat 
+nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/app/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/app/defs.ma
new file mode 100644 (file)
index 0000000..af8b3b2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/app/defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+rec definition cbk (c: C) on c: nat \def match c with [(CSort m) \Rightarrow 
+m | (CHead c0 _ _) \Rightarrow (cbk c0)].
+
+rec definition app1 (c: C) on c: T \to T \def \lambda (t: T).(match c with 
+[(CSort _) \Rightarrow t | (CHead c0 k u) \Rightarrow (app1 c0 (THead k u 
+t))]).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/aprem/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/aprem/defs.ma
new file mode 100644 (file)
index 0000000..3b979e0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/aprem/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/A/defs.ma".
+
+inductive aprem: nat \to (A \to (A \to Prop)) \def
+| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
+| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 
+a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/aprem/fwd.ma
new file mode 100644 (file)
index 0000000..38eefdf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/aprem/fwd.ma
@@ -0,0 +1,113 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/aprem/defs.ma".
+
+implied rec lemma aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall 
+(a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: 
+A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to 
+(\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) 
+(a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) 
+\Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4 
+((aprem_ind P f f0) i a2 a3 a4) a5)].
+
+lemma aprem_gen_sort:
+ \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem 
+i (ASort h n) x) \to False))))
+\def
+ \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: 
+nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda 
+(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: 
+(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: 
+A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: 
+A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A 
+(AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False 
+| (AHead _ _) \Rightarrow True])) I (ASort h n) H1) in (False_ind False 
+H2))))) (\lambda (a2: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (_: 
+(aprem i0 a2 a)).(\lambda (_: (((eq A a2 (ASort h n)) \to False))).(\lambda 
+(a1: A).(\lambda (H3: (eq A (AHead a1 a2) (ASort h n))).(let H4 \def (eq_ind 
+A (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow 
+False | (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False 
+H4))))))))) i y x H0))) H))))).
+
+lemma aprem_gen_head_O:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) 
+\to (eq A x a1))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O 
+(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) 
+(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y 
+x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq 
+A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem 
+y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq 
+nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: 
+A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 
+a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) 
+(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) 
+(AHead a1 a2) H3) in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: 
+A).(\lambda (a: A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda 
+(H3: (((eq nat i O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda 
+(a3: A).(\lambda (H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) 
+(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0) 
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) 
+(AHead a1 a2) H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 
+(\lambda (a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a 
+a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i a4 
+a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
+(eq A a a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
+
+lemma aprem_gen_head_S:
+ \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem 
+(S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda 
+(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: 
+A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: 
+A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: 
+nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 
+x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: 
+nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead 
+a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda 
+(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let 
+H4 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow 
+a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 
+\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | 
+(AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: 
+(eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def 
+(eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) 
+\Rightarrow False])) I (S i) H2) in (False_ind (aprem i a2 a1) H7)) a0 H6))) 
+H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (H2: 
+(aprem i0 a0 a)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq A a0 (AHead a1 
+a2)) \to (aprem i a2 a))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S i0) (S 
+i))).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def (f_equal 
+A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) 
+\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A 
+A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead _ a4) 
+\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3 
+a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A 
+a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 a2 H7) in (let H10 \def (eq_ind A 
+a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 H7) in (let H11 \def (f_equal nat 
+nat (\lambda (e: nat).(match e with [O \Rightarrow i0 | (S n) \Rightarrow 
+n])) (S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq 
+nat n (S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in 
+(let H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in 
+H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/aprem/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/aprem/props.ma
new file mode 100644 (file)
index 0000000..e43f533
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/aprem/props.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/aprem/fwd.ma".
+
+include "basic_1A/leq/fwd.ma".
+
+lemma aprem_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
+(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g 
+b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall 
+(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
+(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda 
+(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g 
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: 
+A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 
+i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g 
+b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda 
+(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall 
+(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g 
+b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda 
+(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall 
+(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda 
+(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda 
+(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead 
+a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem 
+n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y 
+\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A 
+(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) 
+(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead 
+a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda 
+(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) 
+(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) 
+(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let 
+H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: 
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: 
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda 
+(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 
+a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 
+H)))).
+
+lemma aprem_asucc:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i 
+a1 a2) \to (aprem i (asucc g a1) a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda 
+(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda 
+(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: 
+A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda 
+(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) 
+a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 
+H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/aprem.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/aprem.ma
new file mode 100644 (file)
index 0000000..22be8e8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/aprem.ma
@@ -0,0 +1,257 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/props.ma".
+
+include "basic_1A/arity/cimp.ma".
+
+include "basic_1A/aprem/props.ma".
+
+lemma arity_aprem:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
+b)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: 
+A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
+b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda 
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H_x \def (aprem_gen_sort b 
+i O n H0) in (let H1 \def H_x in (False_ind (ex2_3 C T nat (\lambda (d: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) H1)))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: 
+(arity g d u a0)).(\lambda (H2: ((\forall (i0: nat).(\forall (b: A).((aprem 
+i0 a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: 
+A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def 
+H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop 
+(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 
+\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def 
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda 
+(c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop 
+(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus 
+i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) 
+H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: 
+((\forall (i0: nat).(\forall (b: A).((aprem i0 (asucc g a0) b) \to (ex2_3 C T 
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 
+d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 
+(asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem 
+i0 a0 b)).(let H_y \def (H2 i0 b) in (let H4 \def (H_y (aprem_asucc g a0 b i0 
+H3)) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda 
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop 
+(plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x 
+\def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abst c0 u i H0) in (let H7 \def 
+H_x in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: 
+C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda 
+(_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop 
+(S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus 
+i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) 
+H9))))) H7)))))))) H4)))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
+g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b0: A).((aprem i a1 b0) 
+\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
+nat).(arity g d u0 (asucc g b0))))))))))).(\lambda (t0: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: 
+((\forall (i: nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead 
+c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity 
+g d u0 (asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda 
+(H5: (aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in 
+(ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: 
+T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop 
+(plus i x2) O x0 (CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc 
+g b0))).(let H9 \def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O 
+x0 c0)) (drop_S b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) 
+in (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d u0 (asucc g b0))))) x0 x1 (S x2) H9 H8))))))) 
+H6))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (_: ((\forall (i: 
+nat).(\forall (b: A).((aprem i (asucc g a1) b) \to (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
+b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead 
+c0 (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (i: nat).(\forall (b: 
+A).((aprem i a2 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind Abst) u))))) 
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
+b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i (AHead 
+a1 a2) b)).(nat_ind (\lambda (n: nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C 
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b)))))))) (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H_y \def 
+(aprem_gen_head_O a1 a2 b H5) in (eq_ind_r A a1 (\lambda (a0: A).(ex2_3 C T 
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g a0))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g a1))))) c0 u O (drop_refl 
+c0) H0) b H_y))) (\lambda (i0: nat).(\lambda (_: (((aprem i0 (AHead a1 a2) b) 
+\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i0 j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
+nat).(arity g d u0 (asucc g b))))))))).(\lambda (H5: (aprem (S i0) (AHead a1 
+a2) b)).(let H_y \def (aprem_gen_head_S a1 a2 b i0 H5) in (let H_x \def (H3 
+i0 b H_y) in (let H6 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda 
+(_: T).(\lambda (j: nat).(drop (plus i0 j) O d (CHead c0 (Bind Abst) u))))) 
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
+b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: 
+nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i0 x2) O x0 (CHead c0 (Bind 
+Abst) u))).(\lambda (H8: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus i0 x2) H7) H8)))))) 
+H6))))))) i H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall 
+(b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: 
+((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T 
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 
+(asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem 
+i a2 b)).(let H_y \def (H3 (S i) b) in (let H5 \def (H_y (aprem_succ a2 b i 
+H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda (u0: 
+T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda 
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda 
+(d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (S 
+(plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(C_ind 
+(\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1 x1 (asucc g 
+b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n: nat).(\lambda (H8: 
+(drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity g (CSort n) x1 
+(asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i x2)) O) (eq 
+nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0 (CSort 
+n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O 
+O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind 
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
+i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d 
+u0 (asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) 
+(\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g 
+d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))).(\lambda (k: 
+K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) 
+c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).(K_ind (\lambda 
+(k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i 
+x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))))))) (\lambda (b0: B).(\lambda (H10: 
+(arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
+(Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda 
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) 
+(CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: 
+nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d 
+c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10)))) (\lambda (f: F).(\lambda 
+(H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r 
+(Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g 
+(CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in 
+(ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: 
+nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda 
+(_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) 
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop 
+(plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g 
+b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))) k H9 
+(drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) 
+H5))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda 
+(_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: 
+A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (t0: 
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall 
+(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))))).(\lambda (i: 
+nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) 
+in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g 
+b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: 
+(drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g 
+b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: 
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda 
+(_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
+t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to 
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus 
+i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d 
+u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
+a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x 
+\def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A 
+(\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T 
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d 
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc 
+g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i 
+a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat 
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g 
+x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop 
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 
+x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: 
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g 
+x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) 
+H4))))))))))))) c t a H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/cimp.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/cimp.ma
new file mode 100644 (file)
index 0000000..9cad1a9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/cimp.ma
@@ -0,0 +1,98 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/fwd.ma".
+
+include "basic_1A/cimp/props.ma".
+
+lemma arity_cimp_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g 
+c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d 
+c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c 
+c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i) 
+a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let 
+H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in 
+(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead 
+d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7: 
+(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def 
+(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead 
+x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind 
+Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow x | (CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind 
+Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 
+(CHead x0 (Bind Abbr) u) H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: 
+C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9 x H10) in (let H12 \def (eq_ind_r 
+C x0 (\lambda (c0: C).(cimp d c0)) H7 x H10) in (arity_abbr g c2 x u i H11 a0 
+(H2 x H12))))))))) H6))))) H4))))))))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda 
+(H2: ((\forall (c2: C).((cimp d c2) \to (arity g c2 u (asucc g 
+a0)))))).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(let H_x \def (H3 Abst d 
+u i H0) in (let H4 \def H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead 
+d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: 
+(getl i c2 (CHead x (Bind Abst) u))).(let H_x0 \def (cimp_getl_conf c c2 H3 
+Abst d u i H0) in (let H6 \def H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d 
+d2)) (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2 
+(TLRef i) a0) (\lambda (x0: C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl 
+i c2 (CHead x0 (Bind Abst) u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) 
+u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 
+(CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow x | (CHead 
+c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind Abst) u) 
+(getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) H8)) in 
+(let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind 
+Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d 
+c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) 
+H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
+a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u 
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b) 
+u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c 
+c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) 
+(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
+((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
+a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to 
+(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c 
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) 
+(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
+(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2: 
+A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: 
+C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2: 
+C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2 
+H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: 
+(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to 
+(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 
+a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0 
+a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1 
+c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda 
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2: 
+C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: 
+(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2 
+t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/defs.ma
new file mode 100644 (file)
index 0000000..e6be51c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/defs.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/leq/defs.ma".
+
+include "basic_1A/getl/defs.ma".
+
+inductive arity (g: G): C \to (T \to (A \to Prop)) \def
+| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O 
+n)))
+| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) 
+\to (arity g c (TLRef i) a)))))))
+| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u 
+(asucc g a)) \to (arity g c (TLRef i) a)))))))
+| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: 
+T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c 
+(THead (Bind b) u t) a2)))))))))
+| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u 
+(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind 
+Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
+| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u 
+a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to 
+(arity g c (THead (Flat Appl) u t) a2)))))))
+| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u 
+(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat 
+Cast) u t) a))))))
+| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t 
+a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/fwd.ma
new file mode 100644 (file)
index 0000000..8f6466e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/fwd.ma
@@ -0,0 +1,1291 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/defs.ma".
+
+include "basic_1A/leq/asucc.ma".
+
+include "basic_1A/getl/drop.ma".
+
+implied rec lemma arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f: 
+(\forall (c: C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0: 
+(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a) 
+\to (P c (TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall 
+(u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: 
+A).((arity g d u (asucc g a)) \to ((P d u (asucc g a)) \to (P c (TLRef i) 
+a)))))))))) (f2: (\forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to 
+(\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to ((P 
+(CHead c (Bind b) u) t a2) \to (P c (THead (Bind b) u t) a2))))))))))))) (f3: 
+(\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1)) 
+\to ((P c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g 
+(CHead c (Bind Abst) u) t a2) \to ((P (CHead c (Bind Abst) u) t a2) \to (P c 
+(THead (Bind Abst) u t) (AHead a1 a2)))))))))))) (f4: (\forall (c: 
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to 
+(\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to ((P c t 
+(AHead a1 a2)) \to (P c (THead (Flat Appl) u t) a2))))))))))) (f5: (\forall 
+(c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to ((P c u 
+(asucc g a)) \to (\forall (t: T).((arity g c t a) \to ((P c t a) \to (P c 
+(THead (Flat Cast) u t) a)))))))))) (f6: (\forall (c: C).(\forall (t: 
+T).(\forall (a1: A).((arity g c t a1) \to ((P c t a1) \to (\forall (a2: 
+A).((leq g a1 a2) \to (P c t a2))))))))) (c: C) (t: T) (a: A) (a0: arity g c 
+t a) on a0: P c t a \def match a0 with [(arity_sort c0 n) \Rightarrow (f c0 
+n) | (arity_abbr c0 d u i g0 a1 a2) \Rightarrow (f0 c0 d u i g0 a1 a2 
+((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) d u a1 a2)) | (arity_abst c0 d u i g0 
+a1 a2) \Rightarrow (f1 c0 d u i g0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 
+f6) d u (asucc g a1) a2)) | (arity_bind b n c0 u a1 a2 t0 a3 a4) \Rightarrow 
+(f2 b n c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u a1 a2) t0 a3 
+a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) (CHead c0 (Bind b) u) t0 a3 a4)) | 
+(arity_head c0 u a1 a2 t0 a3 a4) \Rightarrow (f3 c0 u a1 a2 ((arity_ind g P f 
+f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 a4 ((arity_ind g P f f0 f1 
+f2 f3 f4 f5 f6) (CHead c0 (Bind Abst) u) t0 a3 a4)) | (arity_appl c0 u a1 a2 
+t0 a3 a4) \Rightarrow (f4 c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) 
+c0 u a1 a2) t0 a3 a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 (AHead a1 
+a3) a4)) | (arity_cast c0 u a1 a2 t0 a3) \Rightarrow (f5 c0 u a1 a2 
+((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 
+((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 a1 a3)) | (arity_repl c0 t0 a1 
+a2 a3 l) \Rightarrow (f6 c0 t0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) 
+c0 t0 a1 a2) a3 l)].
+
+lemma arity_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c 
+(TSort n) a) \to (leq g a (ASort O n))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda 
+(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g 
+c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0: 
+(arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: 
+A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: 
+C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
+(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
+n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort O n1) (ASort O 
+n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((eq 
+T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H4: (eq T (TLRef i) 
+(TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) 
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) 
+\to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort 
+n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5))))))))))) 
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T 
+u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t 
+(TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H6: (eq T (THead (Bind b) 
+u t) (TSort n))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind (leq g a2 
+(ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TSort 
+n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T 
+t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H5: (eq T (THead (Bind 
+Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
+(False_ind (leq g (AHead a1 a2) (ASort O n)) H6)))))))))))) (\lambda (c0: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda 
+(_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: 
+(((eq T t (TSort n)) \to (leq g (AHead a1 a2) (ASort O n))))).(\lambda (H5: 
+(eq T (THead (Flat Appl) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0: 
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g 
+a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O 
+n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t 
+(TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat 
+Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
+(False_ind (leq g a0 (ASort O n)) H6))))))))))) (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t a1)).(\lambda (H2: (((eq T t 
+(TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (a2: A).(\lambda (H3: (leq 
+g a1 a2)).(\lambda (H4: (eq T t (TSort n))).(let H5 \def (f_equal T T 
+(\lambda (e: T).e) t (TSort n) H4) in (let H6 \def (eq_ind T t (\lambda (t0: 
+T).((eq T t0 (TSort n)) \to (leq g a1 (ASort O n)))) H2 (TSort n) H5) in (let 
+H7 \def (eq_ind T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TSort n) H5) in 
+(leq_trans g a2 a1 (leq_sym g a1 a2 H3) (ASort O n) (H6 (refl_equal T (TSort 
+n))))))))))))))) c y a H0))) H))))).
+
+lemma arity_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c 
+(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c 
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) 
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda 
+(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g 
+c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl 
+i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind 
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))) 
+(\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T 
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
+(\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
+C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: 
+C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def 
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d 
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u a0)).(\lambda 
+(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
+T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: 
+T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i 
+d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 
+u0 (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 | 
+(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef 
+i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d 
+(Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda 
+(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
+T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: 
+T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda 
+(u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d 
+(Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g 
+a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda 
+(H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead 
+_ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat 
+i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in 
+(or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) 
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) 
+u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) 
+(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
+Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) 
+d u H6 H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
+g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: 
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t 
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead 
+c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i 
+(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda 
+(u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind 
+b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2 
+C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) 
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda 
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g 
+a1))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t 
+a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abbr) 
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T 
+(\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d 
+(Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g 
+a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) u t) (TLRef i))).(let H6 
+\def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6)))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
+a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2: 
+A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef 
+i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d 
+(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1 
+a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind 
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead 
+a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let 
+H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda 
+(u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 (asucc g a2)))))) H6)))))))))))) (\lambda (c0: C).(\lambda 
+(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda 
+(_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: 
+T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 (asucc g a0))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: 
+T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: 
+T).(arity g d u0 (asucc g (asucc g a0)))))))))).(\lambda (t: T).(\lambda (_: 
+(arity g c0 t a0)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T 
+(\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) 
+(\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: 
+C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))))).(\lambda (H5: (eq T 
+(THead (Flat Cast) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Flat Cast) 
+u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in 
+(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead 
+d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) 
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0)))))) 
+H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1: 
+(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T 
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
+(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
+C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2: 
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5 
+\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind 
+T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: 
+C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda 
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind 
+T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6 
+(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind 
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2)))))) 
+(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d 
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d 
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) 
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda 
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11: 
+(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))) 
+x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C 
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) 
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T 
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) 
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T 
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) 
+(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: 
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11: 
+(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda 
+(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: 
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0 
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u 
+(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2) 
+(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
+
+lemma arity_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: 
+C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind 
+b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: 
+A).(arity g (CHead c (Bind b) u) t a2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda 
+(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity 
+g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
+T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u 
+a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y: 
+T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda 
+(t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda 
+(a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
+a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) 
+(THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H2) in (False_ind 
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 
+(Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq 
+T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (_: A).(arity g (CHead d (Bind b) u) t a)))))).(\lambda (H5: (eq T 
+(TLRef i) (THead (Bind b) u t))).(let H6 \def (eq_ind T (TLRef i) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in 
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity 
+g (CHead c0 (Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda 
+(_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u 
+a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g 
+a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def 
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead (Bind b) u t) H5) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u 
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))) H6))))))))))) 
+(\lambda (b0: B).(\lambda (H2: (not (eq B b0 Abst))).(\lambda (c0: 
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: (arity g c0 u0 
+a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
+a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g (CHead c0 
+(Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u t)) \to (ex2 
+A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) (\lambda (_: 
+A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t a0)))))).(\lambda 
+(H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u t))).(let H8 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef 
+_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead 
+(Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
+\Rightarrow t1])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in ((let 
+H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind 
+b0) u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda 
+(H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind 
+b0) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind 
+b) u) t a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: 
+T).(arity g (CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def 
+(eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A 
+(\lambda (a3: A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_: 
+A).(arity g (CHead (CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11) 
+in (let H16 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0) 
+t1) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T 
+t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) 
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let 
+H18 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let 
+H19 \def (eq_ind B b0 (\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to 
+(ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_: 
+A).(arity g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in 
+(let H20 \def (eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t 
+a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 
+Abst))) H2 b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) 
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20))))))))))))) 
+H9)) H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
+A).(\lambda (H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0 
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
+(_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0: 
+T).(\lambda (a0: A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0 
+a0)).(\lambda (H5: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
+A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead 
+(CHead c0 (Bind Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead 
+(Bind Abst) u0 t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst 
+| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abst])])) (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) 
+in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1])) 
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) 
+\Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0) 
+(THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda (H11: (eq B 
+Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
+b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u 
+a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind b) u) t 
+a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g 
+(CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let H14 \def (eq_ind T u0 
+(\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
+A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda (_: A).(arity g (CHead 
+(CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u H10) in (let H15 \def 
+(eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a0)) H13 u 
+H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b) 
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g 
+(CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u H10) in (let H17 \def (eq_ind 
+T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H2 u H10) in (let H18 
+\def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u t)) \to (ex2 
+A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) u a3)) (\lambda (_: 
+A).(arity g (CHead (CHead c0 (Bind Abst) u) (Bind b0) u) t a0))))) H14 Abst 
+H11) in (let H19 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind 
+b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: 
+A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) H16 Abst H11) in (let 
+H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H11) in 
+(eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: A).(arity g c0 u a3)) 
+(\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (AHead a1 a0))))) (let H21 
+\def (match (H20 (refl_equal B Abst)) in False with []) in H21) b 
+H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda (u0: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
+(_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: T).(\lambda 
+(a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: (((eq T t0 
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
+(_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 a0))))))).(\lambda (H6: (eq 
+T (THead (Flat Appl) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T 
+(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u 
+a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) H7)))))))))))) 
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0 
+(asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind 
+b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 
+a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
+a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) u 
+t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind 
+b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: 
+A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) 
+u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g 
+(CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 
+a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T 
+(\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0 
+(\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: 
+A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t 
+a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda 
+(t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7 
+(refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g 
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A 
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind 
+b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: 
+(arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g 
+c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 
+(arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y 
+a2 H1))) H0)))))))).
+
+lemma arity_gen_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
+A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: 
+A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
+(CHead c (Bind Abst) u) t a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: 
+A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead 
+(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A 
+A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: 
+A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: 
+A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0: 
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
+A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: 
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
+(CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n: 
+nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def 
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1: 
+A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
+(CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 
+a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda 
+(a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda 
+(_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
+(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead 
+(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ 
+_ _) \Rightarrow False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2 
+A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl 
+i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 
+(asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A 
+A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) 
+(\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_: 
+A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda 
+(H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef 
+i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u 
+t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g 
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t 
+a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2: 
+(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to 
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) 
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0 
+a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
+(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
+A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda 
+(_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) 
+t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u 
+t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k 
+with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) 
+u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
+(THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0) (THead (Bind Abst) u 
+t) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
+(THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0 
+u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1: 
+T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
+A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda 
+(a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t 
+H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind 
+b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T 
+t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead 
+c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let 
+H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2)) 
+H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead 
+(Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g 
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t 
+a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 
+t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t 
+(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq 
+A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 
+(Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let 
+H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2)) 
+H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in 
+False with []) in H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0: 
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g 
+a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
+(\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 a3)))) 
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0 
+a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
+(\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
+A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3)))) 
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0) 
+(Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
+(THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
+\Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
+(THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T 
+u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
+Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead 
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) 
+u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 
+(Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T 
+t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in 
+(let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t)) 
+\to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) 
+(\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc 
+g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind 
+Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0 
+(\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let 
+H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to 
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3 
+a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) 
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) 
+H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc 
+g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A 
+(AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u 
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind 
+Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) 
+H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to 
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) 
+(\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: 
+A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda 
+(_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda 
+(a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T 
+(THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T 
+(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead c0 (Bind Abst) u) t a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda 
+(u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda 
+(_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: 
+A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (t0: 
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind 
+Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead 
+a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) 
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t 
+a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u 
+t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A 
+(\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda 
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda 
+(c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 
+a1)).(\lambda (H2: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A 
+(\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2: 
+A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda 
+(a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (a2: 
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u 
+t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind Abst) u t) 
+H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind 
+Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead 
+a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) 
+(\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))))) 
+H2 (THead (Bind Abst) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: 
+T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) u t) H5) in (let H8 \def (H6 
+(refl_equal T (THead (Bind Abst) u t))) in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
+A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u 
+(asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind 
+Abst) u) t a4)))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H9: (eq A a1 
+(AHead x0 x1))).(\lambda (H10: (arity g c0 u (asucc g x0))).(\lambda (H11: 
+(arity g (CHead c0 (Bind Abst) u) t x1)).(let H12 \def (eq_ind A a1 (\lambda 
+(a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in (let H13 \def (eq_ind A a1 
+(\lambda (a0: A).(arity g c0 (THead (Bind Abst) u t) a0)) H7 (AHead x0 x1) 
+H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) in (let H14 \def H_x in 
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g x0 a3))) (\lambda (_: 
+A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
+a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g 
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t 
+a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda (H15: (leq g x0 
+x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 (AHead x2 
+x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 x3) H17) in 
+(eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: A).(\lambda 
+(a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: 
+A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g 
+(CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead x2 x3)) 
+(arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 H15)) 
+(arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18))))))) 
+H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
+
+lemma arity_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: 
+A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity 
+g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: 
+A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead 
+(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A 
+(\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 
+a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda 
+(c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) 
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t 
+(AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T 
+(TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H1) in 
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
+g c0 t (AHead a1 (ASort O n))))) H2))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq 
+T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (a1: A).(arity g d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef 
+i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in 
+(False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
+g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda 
+(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g 
+d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda 
+(H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef 
+i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u 
+t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: 
+A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
+A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0: 
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a0)).(\lambda (_: (((eq T 
+t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 
+(Bind b) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) t (AHead 
+a3 a0))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Flat Appl) u 
+t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) u t) H6) in (False_ind (ex2 A 
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
+a0)))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead 
+(Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda 
+(a3: A).(arity g c0 t (AHead a3 (asucc g a1)))))))).(\lambda (t0: T).(\lambda 
+(a0: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (_: 
+(((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g 
+(CHead c0 (Bind Abst) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind 
+Abst) u0) t (AHead a3 a0))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
+(THead (Flat Appl) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) 
+H5) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
+A).(arity g c0 t (AHead a3 (AHead a1 a0))))) H6)))))))))))) (\lambda (c0: 
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 
+a1)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda 
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
+a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 
+(AHead a1 a0))).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A 
+(\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 
+(AHead a1 a0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead 
+(Flat Appl) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) 
+\Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) 
+(THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in (\lambda (H8: (eq T 
+u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat 
+Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
+A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let H10 \def 
+(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t H7) in (let 
+H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to 
+(ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t 
+(AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: 
+T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: A).(arity g c0 
+u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 H10))))))) 
+H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: 
+(arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) 
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t 
+(AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 
+a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 
+a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Appl) u 
+t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
+True])])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a)))) 
+H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
+(H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t)) 
+\to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t 
+(AHead a3 a1))))))).(\lambda (a0: A).(\lambda (H3: (leq g a1 a0)).(\lambda 
+(H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 \def (f_equal T T (\lambda 
+(e: T).e) t0 (THead (Flat Appl) u t) H4) in (let H6 \def (eq_ind T t0 
+(\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: 
+A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1)))))) H2 
+(THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1: 
+T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in (let H8 \def (H6 
+(refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A (\lambda (a3: A).(arity 
+g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1))) (ex2 A (\lambda 
+(a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) 
+(\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda (H10: (arity g c0 t 
+(AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: 
+A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10 
+(AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c 
+y a2 H0))) H)))))).
+
+lemma arity_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: 
+A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) 
+(arity g c t a)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a: 
+A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead 
+(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land 
+(arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0: 
+(arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: 
+A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) 
+(arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T 
+(TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u t) H1) in 
+(False_ind (land (arity g c0 u (asucc g (ASort O n))) (arity g c0 t (ASort O 
+n))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast) 
+u t)) \to (land (arity g d u (asucc g a0)) (arity g d t a0))))).(\lambda (H4: 
+(eq T (TLRef i) (THead (Flat Cast) u t))).(let H5 \def (eq_ind T (TLRef i) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u 
+t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)) 
+H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead 
+(Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t 
+(asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u 
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Cast) u t) H4) in (False_ind (land (arity g c0 u 
+(asucc g a0)) (arity g c0 t a0)) H5))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t 
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 
+(Bind b) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to 
+(land (arity g (CHead c0 (Bind b) u0) u (asucc g a2)) (arity g (CHead c0 
+(Bind b) u0) t a2))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead 
+(Flat Cast) u t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H6) in (False_ind 
+(land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) H7)))))))))))))) 
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
+u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land 
+(arity g c0 u (asucc g (asucc g a1))) (arity g c0 t (asucc g 
+a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 
+(Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to 
+(land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g (CHead c0 
+(Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0) 
+(THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) 
+H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t 
+(AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda 
+(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat 
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t 
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead 
+a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g 
+c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5: 
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def 
+(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u t) H5) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g 
+c0 t a2)) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: 
+A).(\lambda (H1: (arity g c0 u0 (asucc g a0))).(\lambda (H2: (((eq T u0 
+(THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) 
+(arity g c0 t (asucc g a0)))))).(\lambda (t0: T).(\lambda (H3: (arity g c0 t0 
+a0)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 
+u (asucc g a0)) (arity g c0 t a0))))).(\lambda (H5: (eq T (THead (Flat Cast) 
+u0 t0) (THead (Flat Cast) u t))).(let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
+(THead _ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) 
+u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow 
+t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in (\lambda (H8: 
+(eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead 
+(Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)))) 
+H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a0)) 
+H3 t H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead 
+(Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) (arity g c0 
+t (asucc g a0))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1: 
+T).(arity g c0 t1 (asucc g a0))) H1 u H8) in (conj (arity g c0 u (asucc g 
+a0)) (arity g c0 t a0) H12 H10))))))) H6))))))))))) (\lambda (c0: C).(\lambda 
+(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: 
+(((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1)) 
+(arity g c0 t a1))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda 
+(H4: (eq T t0 (THead (Flat Cast) u t))).(let H5 \def (f_equal T T (\lambda 
+(e: T).e) t0 (THead (Flat Cast) u t) H4) in (let H6 \def (eq_ind T t0 
+(\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u 
+(asucc g a1)) (arity g c0 t a1)))) H2 (THead (Flat Cast) u t) H5) in (let H7 
+\def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) 
+u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in 
+(land_ind (arity g c0 u (asucc g a1)) (arity g c0 t a1) (land (arity g c0 u 
+(asucc g a2)) (arity g c0 t a2)) (\lambda (H9: (arity g c0 u (asucc g 
+a1))).(\lambda (H10: (arity g c0 t a1)).(conj (arity g c0 u (asucc g a2)) 
+(arity g c0 t a2) (arity_repl g c0 u (asucc g a1) H9 (asucc g a2) (asucc_repl 
+g a1 a2 H3)) (arity_repl g c0 t a1 H10 a2 H3)))) H8))))))))))))) c y a H0))) 
+H)))))).
+
+lemma arity_gen_appls:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall 
+(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: 
+A).(arity g c t a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads 
+(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda 
+(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c 
+t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall 
+(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a: 
+A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g 
+c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g 
+c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 
+a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_: 
+(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x 
+a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A 
+(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a))) 
+(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a: 
+A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
+
+lemma arity_gen_lift:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: 
+nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: 
+C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T 
+(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall 
+(c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda 
+(H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) 
+\to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat 
+d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: 
+C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq 
+T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
+a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: 
+T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda 
+(_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 
+(ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) 
+(\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: 
+(arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u 
+(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0 
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) 
+(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def 
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq 
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) 
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef 
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: 
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda 
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: 
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) 
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S 
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) 
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: 
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 
+(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 
+a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u 
+(\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in 
+(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h 
+(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h 
+(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i) 
+(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef 
+(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda 
+(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda 
+(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h) 
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0 
+H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst) 
+u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda 
+(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall 
+(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda 
+(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x 
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def 
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq 
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h)))) 
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef 
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8: 
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda 
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n: 
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8)) 
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S 
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0))) 
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11: 
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2 
+(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14 
+\def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
+t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 
+(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def 
+(eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus 
+x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 
+(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt 
+Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: 
+(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x 
+h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le 
+(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T 
+(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0 
+u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5 
+H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1: 
+(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall 
+(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to 
+(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4: 
+(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x: 
+nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h 
+x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: 
+nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x 
+x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda 
+(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: 
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u 
+(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T 
+(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def 
+(eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to 
+(arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T 
+t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x) 
+x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c 
+(Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def 
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift 
+h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind 
+b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15 
+\def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T 
+t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 
+a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1: 
+T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 
+(H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal 
+T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b 
+x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) 
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u 
+(asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u 
+(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
+(asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g 
+(CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall 
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c 
+(Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda 
+(x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda 
+(c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0: 
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1 
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S 
+x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2 
+t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: 
+nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h 
+x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x) 
+x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c 
+(Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u 
+(\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11 
+(lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: 
+nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall 
+(c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2))))))) 
+H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall 
+(x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: 
+C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1) 
+H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g 
+a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T 
+(lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2)) 
+(CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0 
+H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda 
+(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall 
+(c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4: 
+((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall 
+(c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda 
+(x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift 
+h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T 
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) 
+(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1 
+x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x 
+x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1 
+a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall 
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
+(arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def 
+(eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2) 
+H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall 
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
+(arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u 
+(\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2 
+x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2 
+(refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0 
+x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: 
+A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x: 
+nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x 
+c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3: 
+(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T 
+t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead 
+(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c 
+c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
+Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0) 
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast) 
+x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h 
+x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1 
+a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall 
+(x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to 
+(arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0 
+(\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def 
+(eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1 
+(lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4 
+(asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u 
+(\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in 
+(arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 
+x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast 
+u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall 
+(x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to 
+(arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 
+a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x 
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 
+(H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
+
+theorem arity_mono:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c 
+t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H: 
+(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a: 
+A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0: 
+C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n) 
+a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u 
+a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a 
+a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 
+\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda 
+(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
+Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) 
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead 
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind 
+Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind 
+Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 
+(CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d 
+x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind 
+Abbr) t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity 
+g x0 t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl 
+i c0 (CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 
+(\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9))))))) 
+H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
+(asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 
+(CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
+(asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: 
+(getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g 
+a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i 
+c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i 
+H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind 
+Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 
+(CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind 
+(leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u (asucc g a))).(\lambda 
+(H2: ((\forall (a2: A).((arity g d u a2) \to (leq g (asucc g a) 
+a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 
+\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: 
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda 
+(H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind 
+Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 
+a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 
+(Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) 
+(leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def 
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead 
+x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind 
+Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (leq g a a2) 
+H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: 
+T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: 
+T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda 
+(u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda 
+(u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: 
+(arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C (CHead d (Bind Abst) u) 
+(\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 
+(CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
+c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) 
+(getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in 
+((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u) 
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead 
+x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u 
+H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g 
+a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0 
+(CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 
+(\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2 
+(H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall 
+(a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda 
+(a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4: 
+((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3 
+a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0) 
+a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A 
+(\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind 
+b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u 
+x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8)))) 
+H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda 
+(_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0 
+u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3: 
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3: 
+((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3 
+a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0) 
+a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda 
+(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda 
+(_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity 
+g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0: 
+A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7: 
+(arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u) 
+t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a)) 
+(leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8)) 
+a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: 
+A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g 
+c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: 
+(arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0 
+t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity 
+g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 
+H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity 
+g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g 
+c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x 
+a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: 
+((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda 
+(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: 
+A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: 
+(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u 
+t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g 
+a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0 
+a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3: 
+A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2: 
+(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans 
+g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/lift1.ma
new file mode 100644 (file)
index 0000000..8d52c97
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/lift1.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/props.ma".
+
+include "basic_1A/drop1/fwd.ma".
+
+lemma arity_lift1:
+ \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: 
+PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t 
+a) \to (arity g c1 (lift1 hds t) a))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: 
+PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: 
+T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) 
+(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda 
+(H0: (arity g c2 t a)).(let H_y \def (drop1_gen_pnil c1 c2 H) in (eq_ind_r C 
+c2 (\lambda (c: C).(arity g c t a)) H0 c1 H_y)))))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: 
+C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 
+(lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 
+(PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H_x \def 
+(drop1_gen_pcons c1 c2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c3: C).(drop n n0 c1 c3)) (\lambda (c3: C).(drop1 p c3 c2)) (arity g c1 
+(lift n n0 (lift1 p t)) a) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 
+x)).(\lambda (H4: (drop1 p x c2)).(arity_lift g x (lift1 p t) a (H x t H4 H1) 
+c1 n n0 H3)))) H2))))))))))) hds)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/pr3.ma
new file mode 100644 (file)
index 0000000..fb730be
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/pr3.ma
@@ -0,0 +1,579 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/arity.ma".
+
+include "basic_1A/pr3/fwd.ma".
+
+include "basic_1A/pr1/fwd.ma".
+
+include "basic_1A/wcpr0/getl.ma".
+
+include "basic_1A/pr0/props.ma".
+
+include "basic_1A/arity/subst0.ma".
+
+lemma arity_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
+c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 
+t2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n) 
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
+(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda 
+(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to 
+(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i) 
+(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
+(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2 
+(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u 
+x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 
+H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g 
+a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: 
+T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2: 
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) 
+t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
+T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 
+d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 
+H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i 
+H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
+a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 
+u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda 
+(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq 
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g 
+c2 t2 a2)) (\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: 
+T).(\lambda (t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) 
+(\lambda (t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def 
+(f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T 
+(THead (Bind b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 
+c2 u a1 (H2 c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp 
+c c2 H5 u u (pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 
+(THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) 
+u t)) \to (arity g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k 
+u1 t3) (THead (Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead 
+k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind b) u t) H12) in ((let 
+H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
+| (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Bind b) u t) H12) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead (Bind b) u t) H12) in 
+(\lambda (H16: (eq T u1 u)).(\lambda (H17: (eq K k (Bind b))).(eq_ind_r K 
+(Bind b) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H18 \def 
+(eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 
+t4 a2))) H11 t H15) in (let H19 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 
+t4)) H10 t H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 
+(THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 u H16) in (let H21 \def 
+(eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H16) in (arity_bind g b H0 c2 
+u2 a1 (H2 c2 H5 u2 H21) t4 a2 (H4 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H5 
+u u2 H21 (Bind b)) t4 H19)))))) k H17)))) H14)) H13)))))))))))) (\lambda (u0: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
+(Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (H12: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Bind b) u t))).(let H13 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u t) H12) in (False_ind (arity g 
+c2 (THead (Bind Abbr) v2 t4) a2) H13)))))))))))) (\lambda (b0: B).(\lambda 
+(_: (not (eq B b0 Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t)) \to (arity g c2 v2 
+a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (THead (Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead 
+(Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (H15: (eq T (THead (Flat 
+Appl) v1 (THead (Bind b0) u1 t3)) (THead (Bind b) u t))).(let H16 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t3)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u t) H15) in (False_ind (arity g 
+c2 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) a2) 
+H16))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H8: (pr0 u1 
+u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u t)) \to (arity g c2 u2 
+a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H10: (pr0 t3 t4)).(\lambda 
+(H11: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (w: 
+T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda (H13: (eq T (THead (Bind Abbr) 
+u1 t3) (THead (Bind b) u t))).(let H14 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | 
+(THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in 
+((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in ((let H16 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) u1 
+t3) (THead (Bind b) u t) H13) in (\lambda (H17: (eq T u1 u)).(\lambda (H18: 
+(eq B Abbr b)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead 
+(Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H16) in (let H20 \def (eq_ind T 
+t3 (\lambda (t0: T).(pr0 t0 t4)) H10 t H16) in (let H21 \def (eq_ind T u1 
+(\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 u2 a2))) H9 
+u H17) in (let H22 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H8 u H17) 
+in (let H23 \def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u 
+t)) \to (arity g c2 t4 a2))) H19 Abbr H18) in (let H24 \def (eq_ind_r B b 
+(\lambda (b0: B).((eq T u (THead (Bind b0) u t)) \to (arity g c2 u2 a2))) H21 
+Abbr H18) in (let H25 \def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: 
+C).((wcpr0 (CHead c (Bind b0) u) c3) \to (\forall (t5: T).((pr0 t t5) \to 
+(arity g c3 t5 a2)))))) H4 Abbr H18) in (let H26 \def (eq_ind_r B b (\lambda 
+(b0: B).(arity g (CHead c (Bind b0) u) t a2)) H3 Abbr H18) in (let H27 \def 
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H0 Abbr H18) in 
+(arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w a2 (arity_subst0 g (CHead 
+c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H5 u 
+u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr c2 u2) w 
+H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda (H8: (not 
+(eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 
+t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S O) 
+O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | 
+(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u 
+t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) \Rightarrow 
+t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in 
+((let H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) 
+u t) H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B b0 b)).(let H17 
+\def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1 Abst))) H8 b H16) in (let 
+H18 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Bind b) u t0)) \to 
+(arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def (eq_ind_r T 
+t (\lambda (t0: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) c3) \to 
+(\forall (t5: T).((pr0 t0 t5) \to (arity g c3 t5 a2)))))) H4 (lift (S O) O 
+t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g (CHead c 
+(Bind b) u) t0 a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2 
+(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u 
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2 
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12)))))))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: 
+(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: 
+T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let 
+H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) 
+H11)))))))) y t2 H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g 
+c2 t2 (asucc g a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: 
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: 
+C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) 
+t2)).(insert_eq T (THead (Bind Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) 
+(\lambda (_: T).(arity g c2 t2 (AHead a1 a2))) (\lambda (y: T).(\lambda (H6: 
+(pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Bind 
+Abst) u t)) \to (arity g c2 t3 (AHead a1 a2))))) (\lambda (t0: T).(\lambda 
+(H7: (eq T t0 (THead (Bind Abst) u t))).(let H8 \def (f_equal T T (\lambda 
+(e: T).e) t0 (THead (Bind Abst) u t) H7) in (eq_ind_r T (THead (Bind Abst) u 
+t) (\lambda (t3: T).(arity g c2 t3 (AHead a1 a2))) (arity_head g c2 u a1 (H1 
+c2 H4 u (pr0_refl u)) t a2 (H3 (CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u 
+u (pr0_refl u) (Bind Abst)) t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 
+(THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (k: 
+K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Bind Abst) u t))).(let H12 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | 
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) 
+(THead (Bind Abst) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) (THead (Bind Abst) u t) H11) 
+in ((let H14 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) 
+(THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda (H15: (eq T u1 
+u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) (\lambda 
+(k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17 \def (eq_ind T 
+t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: 
+T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq 
+T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2)))) H8 u H15) 
+in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H7 u H15) in 
+(arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead c2 (Bind Abst) u2) 
+(wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k H16)))) H13)) 
+H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity 
+g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead 
+(Bind Abbr) v2 t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 
+(AHead a1 a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 
+(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) 
+u1 t3)) (THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) 
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: 
+(((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 
+a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
+(_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 
+a2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T 
+(THead (Bind Abbr) u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T 
+(THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g c2 (THead 
+(Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: B).(\lambda 
+(H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 
+(AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0 
+(lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 \def (f_equal T B 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef _) 
+\Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O 
+t3)) (THead (Bind Abst) u t) H10) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
+(THead _ t0 _) \Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead 
+(Bind Abst) u t) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) 
+| (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind b) u0 (lift (S O) O t3)) (THead 
+(Bind Abst) u t) H10) in (\lambda (_: (eq T u0 u)).(\lambda (H15: (eq B b 
+Abst)).(let H16 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
+Abst H15) in (let H17 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead 
+(Bind Abst) u t0)) \to (arity g c2 t4 (AHead a1 a2)))) H9 (lift (S O) O t3) 
+H13) in (let H18 \def (eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 
+(CHead c (Bind Abst) u) c3) \to (\forall (t5: T).((pr0 t0 t5) \to (arity g c3 
+t5 a2)))))) H3 (lift (S O) O t3) H13) in (let H19 \def (eq_ind_r T t (\lambda 
+(t0: T).(arity g (CHead c (Bind Abst) u) t0 a2)) H2 (lift (S O) O t3) H13) in 
+(let H20 \def (match (H16 (refl_equal B Abst)) in False with []) in 
+H20)))))))) H12)) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity 
+g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat 
+Cast) u0 t3) (THead (Bind Abst) u t))).(let H10 \def (eq_ind T (THead (Flat 
+Cast) u0 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u 
+t) H9) in (False_ind (arity g c2 t4 (AHead a1 a2)) H10)))))))) y t2 H6))) 
+H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c u a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to 
+(\forall (t2: T).((pr0 u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: 
+T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g 
+c2 t2 (AHead a1 a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c 
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) u t) 
+t2)).(insert_eq T (THead (Flat Appl) u t) (\lambda (t0: T).(pr0 t0 t2)) 
+(\lambda (_: T).(arity g c2 t2 a2)) (\lambda (y: T).(\lambda (H6: (pr0 y 
+t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Flat Appl) 
+u t)) \to (arity g c2 t3 a2)))) (\lambda (t0: T).(\lambda (H7: (eq T t0 
+(THead (Flat Appl) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 
+(THead (Flat Appl) u t) H7) in (eq_ind_r T (THead (Flat Appl) u t) (\lambda 
+(t3: T).(arity g c2 t3 a2)) (arity_appl g c2 u a1 (H1 c2 H4 u (pr0_refl u)) t 
+a2 (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 (THead (Flat Appl) u 
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: 
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g 
+c2 t4 a2)))).(\lambda (k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead 
+(Flat Appl) u t))).(let H12 \def (f_equal T K (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) (THead k u1 t3) 
+(THead (Flat Appl) u t) H11) in ((let H14 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead (Flat Appl) u t) H11) 
+in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r 
+K (Flat Appl) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 
+\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to 
+(arity g c2 t4 a2))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t0: 
+T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq 
+T t0 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 
+\def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) H7 u H15) in (arity_appl g c2 
+u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) 
+H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H7: (pr0 v1 v2)).(\lambda (H8: (((eq T v1 (THead (Flat Appl) u t)) \to 
+(arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 
+t4)).(\lambda (H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 
+a2)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Flat Appl) u t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead 
+(Flat Appl) u t) H11) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow 
+(THead (Bind Abst) u0 t3) | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) u t) H11) in (\lambda 
+(H14: (eq T v1 u)).(let H15 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 
+(THead (Flat Appl) u t)) \to (arity g c2 v2 a2))) H8 u H14) in (let H16 \def 
+(eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2)) H7 u H14) in (let H17 \def 
+(eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead (Flat Appl) u t0)) \to (arity 
+g c2 t4 a2))) H10 (THead (Bind Abst) u0 t3) H13) in (let H18 \def (eq_ind_r T 
+t (\lambda (t0: T).((eq T u (THead (Flat Appl) u t0)) \to (arity g c2 v2 
+a2))) H15 (THead (Bind Abst) u0 t3) H13) in (let H19 \def (eq_ind_r T t 
+(\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall (t5: T).((pr0 t0 
+t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3 (THead (Bind Abst) u0 t3) H13) 
+in (let H20 \def (eq_ind_r T t (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) 
+H2 (THead (Bind Abst) u0 t3) H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let 
+H22 \def (H19 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) 
+t3 t4 H9 (Bind Abst))) in (let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 
+a2) H22) in (ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 
+a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g 
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 
+a4))) (arity g c2 (THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda 
+(x1: A).(\lambda (H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: 
+(arity g c2 u0 (asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) 
+u0) t4 x1)).(let H27 \def (f_equal A A (\lambda (e: A).(match e with [(ASort 
+_ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0])) (AHead a1 a2) (AHead x0 
+x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e with [(ASort 
+_ _) \Rightarrow a2 | (AHead _ a0) \Rightarrow a0])) (AHead a1 a2) (AHead x0 
+x1) H24) in (\lambda (H29: (eq A a1 x0)).(let H30 \def (eq_ind_r A x1 
+(\lambda (a0: A).(arity g (CHead c2 (Bind Abst) u0) t4 a0)) H26 a2 H28) in 
+(let H31 \def (eq_ind_r A x0 (\lambda (a0: A).(arity g c2 u0 (asucc g a0))) 
+H25 a1 H29) in (arity_bind g Abbr not_abbr_abst c2 v2 a1 H21 t4 a2 
+(csuba_arity g (CHead c2 (Bind Abst) u0) t4 a2 H30 (CHead c2 (Bind Abbr) v2) 
+(csuba_abst g c2 c2 (csuba_refl g c2) u0 a1 H31 v2 H21))))))) H27))))))) 
+H23)))))))))))) H12)))))))))))) (\lambda (b: B).(\lambda (H7: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H8: (pr0 v1 v2)).(\lambda 
+(H9: (((eq T v1 (THead (Flat Appl) u t)) \to (arity g c2 v2 a2)))).(\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (((eq T 
+u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H12: (pr0 t3 t4)).(\lambda (H13: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (H14: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) u t))).(let 
+H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 
+| (TLRef _) \Rightarrow v1 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat 
+Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ 
+t0) \Rightarrow t0])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
+(Flat Appl) u t) H14) in (\lambda (H17: (eq T v1 u)).(let H18 \def (eq_ind T 
+v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g c2 v2 
+a2))) H9 u H17) in (let H19 \def (eq_ind T v1 (\lambda (t0: T).(pr0 t0 v2)) 
+H8 u H17) in (let H20 \def (eq_ind_r T t (\lambda (t0: T).((eq T t3 (THead 
+(Flat Appl) u t0)) \to (arity g c2 t4 a2))) H13 (THead (Bind b) u1 t3) H16) 
+in (let H21 \def (eq_ind_r T t (\lambda (t0: T).((eq T u1 (THead (Flat Appl) 
+u t0)) \to (arity g c2 u2 a2))) H11 (THead (Bind b) u1 t3) H16) in (let H22 
+\def (eq_ind_r T t (\lambda (t0: T).((eq T u (THead (Flat Appl) u t0)) \to 
+(arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) H16) in (let H23 \def 
+(eq_ind_r T t (\lambda (t0: T).(\forall (c3: C).((wcpr0 c c3) \to (\forall 
+(t5: T).((pr0 t0 t5) \to (arity g c3 t5 (AHead a1 a2))))))) H3 (THead (Bind 
+b) u1 t3) H16) in (let H24 \def (eq_ind_r T t (\lambda (t0: T).(arity g c t0 
+(AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) in (let H25 \def (H1 c2 H4 v2 
+H19) in (let H26 \def (H23 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H10 
+t3 t4 H12 (Bind b))) in (let H27 \def (arity_gen_bind b H7 g c2 u2 t4 (AHead 
+a1 a2) H26) in (ex2_ind A (\lambda (a3: A).(arity g c2 u2 a3)) (\lambda (_: 
+A).(arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))) (arity g c2 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) a2) (\lambda (x: 
+A).(\lambda (H28: (arity g c2 u2 x)).(\lambda (H29: (arity g (CHead c2 (Bind 
+b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 c2 u2 x H28 (THead (Flat Appl) 
+(lift (S O) O v2) t4) a2 (arity_appl g (CHead c2 (Bind b) u2) (lift (S O) O 
+v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 (Bind b) u2) (S O) O (drop_drop 
+(Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 H29))))) H27))))))))))))) 
+H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2 
+a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda 
+(_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda 
+(w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind 
+Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T (THead (Bind 
+Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u 
+t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) 
+H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 
+(THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda 
+(H10: (eq T (THead (Bind b) u0 (lift (S O) O t3)) (THead (Flat Appl) u 
+t))).(let H11 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H10) in (False_ind 
+(arity g c2 t4 a2) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity 
+g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) 
+(THead (Flat Appl) u t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u t) H9) in 
+(False_ind (arity g c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) 
+(\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u 
+(asucc g a0))).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall 
+(t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (t: 
+T).(\lambda (_: (arity g c t a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a0))))))).(\lambda 
+(c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 
+(THead (Flat Cast) u t) t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda 
+(t0: T).(pr0 t0 t2)) (\lambda (_: T).(arity g c2 t2 a0)) (\lambda (y: 
+T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda (t3: T).((eq 
+T t0 (THead (Flat Cast) u t)) \to (arity g c2 t3 a0)))) (\lambda (t0: 
+T).(\lambda (H7: (eq T t0 (THead (Flat Cast) u t))).(let H8 \def (f_equal T T 
+(\lambda (e: T).e) t0 (THead (Flat Cast) u t) H7) in (eq_ind_r T (THead (Flat 
+Cast) u t) (\lambda (t3: T).(arity g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 
+H4 u (pr0_refl u)) t (H3 c2 H4 t (pr0_refl t))) t0 H8)))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (((eq T u1 
+(THead (Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Flat 
+Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (k: K).(\lambda (H11: (eq T 
+(THead k u1 t3) (THead (Flat Cast) u t))).(let H12 \def (f_equal T K (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | 
+(THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat Cast) u t) H11) 
+in ((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H14 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t0) \Rightarrow t0])) (THead k u1 t3) (THead 
+(Flat Cast) u t) H11) in (\lambda (H15: (eq T u1 u)).(\lambda (H16: (eq K k 
+(Flat Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(arity g c2 (THead k0 
+u2 t4) a0)) (let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead 
+(Flat Cast) u t)) \to (arity g c2 t4 a0))) H10 t H14) in (let H18 \def 
+(eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H9 t H14) in (let H19 \def (eq_ind 
+T u1 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 u2 
+a0))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t0: T).(pr0 t0 u2)) 
+H7 u H15) in (arity_cast g c2 u2 a0 (H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 
+H18)))))) k H16)))) H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead 
+(Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
+\to (arity g c2 t4 a0)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u0 t3)) (THead (Flat Cast) u t))).(let H12 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u t) H11) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a0) 
+H12)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Flat Cast) u t)) \to (arity g c2 v2 a0)))).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) 
+u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity 
+g c2 t4 a0)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t3)) (THead (Flat Cast) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead 
+(Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) 
+\to (arity g c2 t4 a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4 
+w)).(\lambda (H12: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Cast) u 
+t))).(let H13 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (THead (Flat Cast) u t) H12) in (False_ind (arity 
+g c2 (THead (Bind Abbr) u2 w) a0) H13))))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 
+a0)))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0 (lift (S O) O 
+t3)) (THead (Flat Cast) u t))).(let H11 \def (eq_ind T (THead (Bind b) u0 
+(lift (S O) O t3)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Cast) u t) H10) in (False_ind (arity g c2 t4 a0) H11)))))))))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 
+(THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda 
+(H9: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Cast) u t))).(let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
+_) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Cast) u0 
+t3) (THead (Flat Cast) u t) H9) in ((let H11 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Flat Cast) u0 t3) (THead (Flat Cast) 
+u t) H9) in (\lambda (_: (eq T u0 u)).(let H13 \def (eq_ind T t3 (\lambda 
+(t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) 
+in (let H14 \def (eq_ind T t3 (\lambda (t0: T).(pr0 t0 t4)) H7 t H11) in (H3 
+c2 H4 t4 H14))))) H10)))))))) y t2 H6))) H5))))))))))))) (\lambda (c: 
+C).(\lambda (t: T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda 
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to 
+(arity g c2 t2 a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
+a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda 
+(H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2 H3 t2 H4) a2 H2)))))))))))) c1 
+t1 a H))))).
+
+lemma arity_sred_wcpr0_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
+(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 
+c2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0 
+a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: 
+A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1 
+c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a: 
+A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5 
+a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3: 
+(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a 
+(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl 
+c2)))))))))))))) t1 t2 H))).
+
+lemma arity_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
+G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda 
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda 
+(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0 
+g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
+G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a 
+(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
+H2)))))))))))))) c t1 t2 H)))).
+
+lemma arity_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a: 
+A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c 
+t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3: 
+(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2 
+H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/props.ma
new file mode 100644 (file)
index 0000000..712c5fa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/props.ma
@@ -0,0 +1,266 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/fwd.ma".
+
+lemma node_inh:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: 
+C).(\lambda (t: T).(arity g c t (ASort k n)))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: 
+nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n)))))) 
+(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n)))) 
+(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda 
+(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 
+n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t: 
+T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t: 
+T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c: 
+C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1) 
+(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 
+x1) (ASort (S n0) n) H1))))) H0)))) k))).
+
+lemma arity_lift:
+ \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 
+t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 
+c2) \to (arity g c1 (lift h d t) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to 
+(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n: 
+nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop 
+h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O 
+n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d)))))))) 
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: 
+(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall 
+(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda 
+(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1 
+c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i 
+d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def 
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S d0) (le_S (S (S i)) (S 
+d0) (le_n_S (S i) d0 H4)))) c1 c h H3 (CHead d (Bind Abbr) u) H0) in 
+(ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda 
+(e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: 
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity g c1 (TLRef 
+i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O c1 
+x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
+(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C 
+(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i)) 
+u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) 
+a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h 
+(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x 
+d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead 
+x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S 
+i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
+H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: 
+T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i 
+c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) 
+(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g 
+a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0: 
+nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g 
+a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda 
+(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) 
+(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 
+t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S 
+d0) (le_S (S (S i)) (S d0) (le_n_S (S i) d0 H4)))) c1 c h H3 (CHead d (Bind 
+Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 
+e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) 
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) u)))) (arity 
+g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O 
+c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
+(CHead d (Bind Abst) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abst d u H8) in (ex2_ind C 
+(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abst) (lift h (minus d0 (S i)) 
+u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i) 
+a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift h 
+(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x 
+d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead 
+x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S 
+i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
+H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0: 
+T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h) (drop_getl_trans_ge i 
+c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1) (lift h d0 (TLRef i)) 
+(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b: B).(\lambda (H0: (not 
+(eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: 
+(arity g c u a1)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall 
+(d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) a1))))))).(\lambda 
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 
+a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1 (lift h d t0) 
+a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H5: 
+(drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) 
+d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind g b H0 c1 (lift h d 
+u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4 (CHead c1 (Bind b) 
+(lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5 b u))) (lift h d 
+(THead (Bind b) u t0)) (lift_head (Bind b) u t0 h d))))))))))))))))) (\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g 
+a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g 
+a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1 
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d 
+u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1 
+a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst) 
+d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d) 
+(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0)) 
+(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity 
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead 
+a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s 
+(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1 
+(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s 
+(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat 
+Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: 
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1: 
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift 
+h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 
+a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1: 
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1 
+c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) 
+t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0 
+(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4)) 
+(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h 
+d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) 
+a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1: 
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1 
+c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a 
+H))))).
+
+lemma arity_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (a1: 
+A).((arity g (CHead c (Bind Abst) w) t a1) \to (\forall (a2: A).((arity g c 
+(THead (Bind Abst) w t) a2) \to ((leq g a1 a2) \to (\forall (P: 
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (a1: 
+A).(\lambda (H: (arity g (CHead c (Bind Abst) w) t a1)).(\lambda (a2: 
+A).(\lambda (H0: (arity g c (THead (Bind Abst) w t) a2)).(\lambda (H1: (leq g 
+a1 a2)).(\lambda (P: Prop).(let H_y \def (arity_repl g (CHead c (Bind Abst) 
+w) t a1 H a2 H1) in (let H2 \def (arity_gen_abst g c w t a2 H0) in (ex3_2_ind 
+A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: 
+A).(\lambda (_: A).(arity g c w (asucc g a3)))) (\lambda (_: A).(\lambda (a4: 
+A).(arity g (CHead c (Bind Abst) w) t a4))) P (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H3: (eq A a2 (AHead x0 x1))).(\lambda (_: (arity g c w (asucc g 
+x0))).(\lambda (H5: (arity g (CHead c (Bind Abst) w) t x1)).(let H6 \def 
+(eq_ind A a2 (\lambda (a: A).(arity g (CHead c (Bind Abst) w) t a)) H_y 
+(AHead x0 x1) H3) in (leq_ahead_false_2 g x1 x0 (arity_mono g (CHead c (Bind 
+Abst) w) t (AHead x0 x1) H6 x1 H5) P))))))) H2)))))))))))).
+
+theorem arity_appls_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: 
+TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to 
+((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) 
+vs (THead (Flat Cast) u t)) a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads 
+(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a) 
+\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a))))) 
+(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity 
+g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1: 
+TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u) 
+(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a: 
+A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u)) 
+(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat 
+Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1 
+t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1: 
+A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda 
+(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat 
+Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat 
+Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) 
+(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g 
+a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat 
+Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda 
+(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g 
+a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t)) 
+a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g 
+a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7 
+(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc 
+g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g 
+(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
+
+lemma arity_appls_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall 
+(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c 
+(THeads (Flat Appl) vs (TLRef i)) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs: 
+TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads 
+(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t 
+(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v) 
+a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop 
+Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: 
+((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to 
+(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda 
+(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
+v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S 
+i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: 
+A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity 
+g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x: 
+A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat 
+Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat 
+Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
+
+theorem arity_appls_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: 
+C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: 
+T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) 
+(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat 
+Appl) vs (THead (Bind b) v t)) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda 
+(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v 
+a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
+TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind 
+b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v) 
+t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1: 
+TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads 
+(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1 
+(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead 
+c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts 
+(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v) 
+(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in 
+(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0) 
+a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda 
+(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity 
+g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x 
+a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O) 
+O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1 
+(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/arity/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1A/arity/subst0.ma
new file mode 100644 (file)
index 0000000..bdfb5e5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/arity/subst0.ma
@@ -0,0 +1,1115 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/props.ma".
+
+include "basic_1A/fsubst0/fwd.ma".
+
+include "basic_1A/csubst0/getl.ma".
+
+include "basic_1A/subst0/dec.ma".
+
+include "basic_1A/subst0/fwd.ma".
+
+include "basic_1A/getl/getl.ma".
+
+lemma arity_gen_cvoid_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to 
+(\forall (P: Prop).P))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
+A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d 
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to 
+(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d 
+(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w 
+(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P))))))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: 
+(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall 
+(i0: nat).((getl i0 d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall 
+(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 
+(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w 
+(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) 
+O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O 
+w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0 
+(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) 
+(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0 
+(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind 
+Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) 
+u0) H7)) in (False_ind P H9)))))) (subst0_gen_lref w v i0 i 
+H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: 
+((\forall (d0: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d0 
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i0 w u v) \to 
+(\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda 
+(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w: 
+T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P: 
+Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq 
+nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat 
+i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let 
+H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 
+(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead 
+d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d 
+(Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (False_ind P H9)))))) 
+(subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: ((\forall (d: C).(\forall 
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
+(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: 
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g 
+(CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: C).(\forall (u0: 
+T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d (Bind Void) u0)) 
+\to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
+T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) v)).(\lambda (P: 
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) 
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
+(Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2))) (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
+(t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T (\lambda 
+(u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i w u 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda 
+(u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead 
+(Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 P)))) 
+H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u t2))) 
+(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda (t2: 
+T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w 
+t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u 
+x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i) 
+(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d 
+(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u 
+x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9 
+P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6))))))))))))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u 
+(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: 
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 
+a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl 
+i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w: 
+T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
+T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P: 
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) 
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
+(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6: 
+(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2: 
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind 
+Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda 
+(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d 
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v 
+(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2))) 
+(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x: 
+T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s 
+(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind 
+Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8 
+P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v 
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u 
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda 
+(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda 
+(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1 
+d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i 
+H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall 
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
+(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P: 
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 
+t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall 
+(i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall 
+(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind 
+Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead 
+(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq 
+T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T 
+(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 
+(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
+T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w 
+u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 
+t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2 
+t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T 
+v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda 
+(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0 
+i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: 
+T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) 
+i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) 
+(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x: 
+T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s 
+(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
+(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w 
+u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 
+P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5))))))))))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u 
+(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v: 
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0: 
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall 
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall 
+(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P: 
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v: 
+T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P: 
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) 
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead 
+(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6: 
+(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2: 
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat 
+Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda 
+(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d 
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v 
+(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2))) 
+(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x: 
+T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s 
+(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda 
+(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w 
+u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8 
+P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5)))))))))))))))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 
+t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: 
+T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2: 
+A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w: 
+T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d 
+u i H3 w v H4 P)))))))))))))))) c t a H))))).
+
+lemma arity_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
+(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in 
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i 
+v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i 
+v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T 
+t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift 
+(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3: 
+(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u 
+(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))) 
+(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda 
+(t0: T).(arity g c t0 a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i 
+x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v))))) 
+(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x 
+(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
+
+lemma arity_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g 
+c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u 
+c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda 
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead 
+d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 
+t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n: 
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i 
+c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: 
+(fsubst0 i u c (TSort n) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TSort 
+n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u 
+(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i 
+u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: 
+(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i 
+u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c 
+c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: 
+C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 
+(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0 
+i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 
+(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c 
+c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n))) 
+(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n) 
+t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c 
+c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n) 
+t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g 
+c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: 
+((\forall (d1: C).(\forall (u0: T).(\forall (i0: nat).((getl i0 d (CHead d1 
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i0 u0 d u c2 
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
+T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) 
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef 
+i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in 
+(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) 
+(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) 
+t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) 
+(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) 
+t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 
+(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq 
+nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat 
+i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O 
+u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda 
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind 
+Abbr) u0) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
+Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 
+(CHead d1 (Bind Abbr) u0) H11)) in ((let H14 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda (H15: (eq C d 
+d1)).(let H16 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind 
+Abbr) t))) H12 u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O 
+t) a0)) (let H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 
+(Bind Abbr) u))) H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop 
+Abbr c d u i H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i 
+H8)) c2 H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c 
+c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) 
+(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c 
+c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 
+(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def 
+(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl i c2 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0) 
+(\lambda (H11: (getl i c2 (CHead d (Bind Abbr) u))).(let H12 \def (eq_ind nat 
+(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 
+(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d 
+(Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) 
+(le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in 
+(arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11: (ex3_4 B C T T (\lambda 
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind 
+x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def 
+(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) 
+(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 
+(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S 
+i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i 
+H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in 
+(\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def 
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) 
+in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind 
+x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i 
+c2 (CHead d (Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 
+(H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead 
+d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) 
+(minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda 
+(H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda 
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq 
+C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
+d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
+Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
+(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
+(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
+(Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H12) in (\lambda (H19: (eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let 
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) 
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus 
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abbr H19) in (arity_abbr g c2 x2 u 
+i H23 a0 (H2 d1 u0 (r (Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d 
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind 
+Abbr) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) 
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind 
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) 
+(\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) 
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 
+(le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0 
+H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal 
+C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in 
+((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abbr 
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: 
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C 
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let 
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) 
+H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) 
+(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u 
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S i))) u0 
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: 
+(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead 
+d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) 
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) 
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: 
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda 
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind 
+Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind 
+Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 
+(CHead d1 (Bind Abbr) u0) H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d 
+d1)).(let H17 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind 
+Abbr) t))) H13 u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: 
+T).(csubst0 i t c c2)) H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 
+(lift (S i) O t) a0)) (let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c 
+(CHead c0 (Bind Abbr) u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O 
+(getl_drop Abbr c2 d u i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d 
+(Bind Abbr) u) H19)))) u0 H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 
+i0 i H7)))) H6)) H5)))))))))))))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g 
+a0))).(\lambda (H2: ((\forall (d1: C).(\forall (u0: T).(\forall (i0: 
+nat).((getl i0 d (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall 
+(t2: T).((fsubst0 i0 u0 d u c2 t2) \to (arity g c2 t2 (asucc g 
+a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda 
+(H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: 
+T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x \def 
+(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (let H5 \def H_x in (or3_ind 
+(land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) 
+(csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) 
+(arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) 
+t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) 
+(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind 
+C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2 
+(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
+T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n 
+c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d 
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in 
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
+I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
+d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c (lift (S i) O u0) a0) 
+H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda 
+(H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (eq T (TLRef 
+i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) 
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: 
+T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: 
+(lt i i0)).(let H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind 
+Abst) u) H0) in (or4_ind (getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
+(arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d (Bind Abst) 
+u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
+(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
+(minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0 H1))) (\lambda (H11: 
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda 
+(b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C 
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2 
+(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2 
+x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
+(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
+(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
+(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let 
+H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) 
+H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(getl i c2 (CHead 
+c0 (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in (arity_abst g c2 d x3 
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d 
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind 
+Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) 
+(\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda 
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq 
+C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 
+(CHead x2 (Bind x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 
+x2)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead 
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind 
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S_n (S i) (S i0) 
+(le_S (S (S i)) (S i0) (le_n_S (S i) i0 H9))))) (S (minus i0 (S i))) 
+(minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
+(Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def (f_equal C B 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C d x1)).(let 
+H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2 (Bind x0) t))) 
+H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus 
+i0 (S i)) u0 c0 x2)) H14 d H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in (arity_abst g c2 x2 u 
+i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d 
+(CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind 
+Abst) (minus i0 (S i))) u0 d u x2 H22))))))))) H17)) H16)))))))))) H11)) 
+(\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq C (CHead d (Bind 
+Abst) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind 
+x0) x4))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) 
+(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) 
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 
+(le_S_n i i0 (le_S_n (S i) (S i0) (le_S (S (S i)) (S i0) (le_n_S (S i) i0 
+H9))))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal 
+C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in 
+((let H18 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead 
+x1 (Bind x0) x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20: (eq B Abst 
+x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3 (\lambda (t: 
+T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23 \def (eq_ind_r C 
+x1 (\lambda (c0: C).(csubst0 (minus i0 (S i)) u0 c0 x2)) H15 d H21) in (let 
+H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) x4))) 
+H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) 
+(minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u 
+(minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S i))) u0 
+d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9: 
+(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead 
+d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0 
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2) 
+(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i) 
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2 
+(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda 
+(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: 
+T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: 
+nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r nat i0 (\lambda 
+(n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in (let H13 \def 
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead 
+d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind 
+Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d 
+(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in (False_ind (arity g c2 
+(lift (S i) O u0) a0) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) 
+H6)) H5)))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity 
+g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 a1)))))))))).(\lambda (t: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t 
+a2)).(\lambda (H4: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr) u0)) \to (\forall 
+(c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) u) t c2 t2) \to 
+(arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i: 
+nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: 
+C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead (Bind b) u t) c2 
+t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind b) u t) t2 u0 i H6) in 
+(let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 (THead (Bind b) u 
+t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (land 
+(subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 a2) 
+(\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind b) u t) 
+t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) (arity g c2 
+t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 (THead (Bind b) 
+u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T 
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
+u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) 
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x: 
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2)) 
+(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t 
+a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b 
+c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S 
+i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u 
+x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2 
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
+(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x: 
+T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s 
+(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity 
+g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i) 
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
+(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c 
+(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda 
+(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t 
+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2)) 
+(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13)) 
+x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind 
+b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both 
+(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0) 
+(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head 
+(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead 
+(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t) 
+t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind 
+b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u 
+t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 
+i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) 
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c 
+(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) 
+t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) 
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2) 
+(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead 
+(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i 
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda 
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2 
+T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) 
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x: 
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0 
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2)) 
+(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13 
+c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u 
+(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t 
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x) 
+(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda 
+(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: 
+T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)) 
+(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u 
+x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind 
+b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 
+d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i) 
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c 
+(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 
+u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) 
+i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))) 
+(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 
+(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14: 
+(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda 
+(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 
+(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind 
+b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) 
+(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 
+H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 
+H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) 
+H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1: 
+C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0)) 
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g 
+c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: 
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: 
+C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead 
+d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 
+(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda 
+(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 
+(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i 
+u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 
+(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq 
+C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind 
+Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u 
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+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
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+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) 
+u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i 
+u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq 
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+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S 
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+u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind 
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+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S 
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+(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
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+(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 
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+u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i) 
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) 
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S 
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+(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10)) 
+(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) 
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u 
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
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+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 
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+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
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+(Flat Appl) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
+Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) 
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+(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
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+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
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+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
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+(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0 
+(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c 
+t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9)) 
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+i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 
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+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 
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+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i 
+u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T 
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+(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x) 
+(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 
+u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c 
+t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat 
+Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0: 
+T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0 
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1 
+(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head 
+(Flat Appl) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))) (\lambda (c: 
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+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i: 
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g 
+a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3: 
+((\forall (d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 
+(Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c t c2 
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0: 
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr) 
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead 
+(Flat Cast) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Flat 
+Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) 
+t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) 
+(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) 
+(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 
+(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c 
+c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c 
+(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T 
+t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T 
+(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 
+(s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3)))) (arity g c t2 a0) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 
+(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 
+i u0 u u2)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead 
+(Flat Cast) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat 
+Cast) x t) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 
+i H4 c x (fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t3: 
+T).(subst0 (s (Flat Cast) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3)) (arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat 
+Cast) u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T 
+(THead (Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u 
+a0 H0 x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) 
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity 
+g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead 
+(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: 
+(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) 
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0 
+(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t 
+x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2 
+H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i 
+u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2) 
+(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda 
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0: 
+T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst 
+i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2 
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2) 
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2) 
+(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead 
+(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10: 
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: 
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat 
+Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0) 
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda 
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0: 
+T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x 
+(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 
+c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 
+(THead (Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0) 
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda 
+(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x) 
+(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 
+u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t 
+x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Cast) i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0 
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat 
+Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: 
+T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0 
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i 
+u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t 
+t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c: C).(\lambda (t: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall 
+(d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) 
+u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity 
+g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda 
+(d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 
+(Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u 
+c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def 
+H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2) 
+(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 
+t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c 
+c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda 
+(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) 
+(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) 
+c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind 
+(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t 
+t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g 
+c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 
+H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u 
+c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) 
+(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c 
+c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7 
+c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
+
+lemma arity_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c 
+t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead 
+d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 
+a)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: 
+(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u 
+c t1 t2 H1)))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/asucc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/asucc/defs.ma
new file mode 100644 (file)
index 0000000..1829672
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/asucc/defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/A/defs.ma".
+
+include "basic_1A/G/defs.ma".
+
+rec definition asucc (g: G) (l: A) on l: A \def match l with [(ASort n0 n) 
+\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) 
+\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g 
+a2))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/asucc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/asucc/fwd.ma
new file mode 100644 (file)
index 0000000..98dd03a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/asucc/fwd.ma
@@ -0,0 +1,92 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/asucc/defs.ma".
+
+include "basic_1A/A/fwd.ma".
+
+lemma asucc_gen_sort:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A 
+(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: 
+nat).(eq A a (ASort h0 n0)))))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind 
+(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda 
+(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: 
+nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 
+n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 
+with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 
+n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A 
+(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) 
+(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat 
+nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 
+n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) 
+\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 
+n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 
+\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (asucc g (AHead a0 a1)) 
+H1) in (False_ind (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A 
+(AHead a0 a1) (ASort h0 n0))))) H2))))))) a)))).
+
+lemma asucc_gen_head:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A 
+(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind 
+(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: 
+A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc 
+g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g 
+(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead 
+a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda 
+(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
+\Rightarrow True])) I (ASort O (next g n0)) H0) in (False_ind (ex2 A (\lambda 
+(a0: A).(eq A (ASort O n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g 
+a0)))) H1))) (\lambda (n1: nat).(\lambda (_: (((eq A (AHead a1 a2) (asucc g 
+(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H0: (eq A (AHead 
+a1 a2) (asucc g (ASort (S n1) n0)))).(let H1 \def (eq_ind A (AHead a1 a2) 
+(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
+\Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: 
+A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g 
+a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) 
+(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda 
+(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A 
+(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 
+a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead 
+a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: 
+A).(match e with [(ASort _ _) \Rightarrow a1 | (AHead a4 _) \Rightarrow a4])) 
+(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in ((let H3 \def (f_equal A A 
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) 
+\Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: 
+(eq A a1 a0)).(let H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 
+a2) (asucc g a4)) \to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) 
+(\lambda (a5: A).(eq A a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda 
+(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda 
+(a5: A).(eq A a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: 
+A).((eq A (AHead a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 
+(AHead a1 a5))) (\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) 
+H3) in (let H7 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc 
+g a1)) \to (ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: 
+A).(eq A a4 (asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) 
+(\lambda (a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) 
+(\lambda (a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq 
+A (AHead a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g 
+a4))) a3 (refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) 
+a0 H4)))) H2))))))) a)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/cimp/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/cimp/defs.ma
new file mode 100644 (file)
index 0000000..d39a85b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/cimp/defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/defs.ma".
+
+definition cimp:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall 
+(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C 
+(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/cimp/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/cimp/props.ma
new file mode 100644 (file)
index 0000000..231609f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/cimp/props.ma
@@ -0,0 +1,125 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/cimp/defs.ma".
+
+include "basic_1A/getl/getl.ma".
+
+lemma cimp_flat_sx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) 
+c)))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: 
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) 
+v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat 
+f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 
+(Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind 
+b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 
+(getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c 
+(CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) 
+w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v) 
+(CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind 
+b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind 
+b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w))) 
+d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
+
+lemma cimp_flat_dx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) 
+v))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: 
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind 
+b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2 
+(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
+
+lemma cimp_bind:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
+(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: 
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to 
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda 
+(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: 
+T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 
+(Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) 
+(CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) 
+v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v) 
+(CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 
+(Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) 
+w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let 
+H3 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 
+| (CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat 
+_) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) 
+(clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) 
+v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) 
+(CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 
+(Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) 
+in (\lambda (H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v 
+(\lambda (t: T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead 
+d2 (Bind b0) t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: 
+C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C 
+(\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 
+(getl_refl b c2 v)) b0 H5) w H4)))) H3)) H2))) (\lambda (h0: nat).(\lambda 
+(_: (((getl h0 (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w)) \to (ex C 
+(\lambda (d2: C).(getl h0 (CHead c2 (Bind b) v) (CHead d2 (Bind b0) 
+w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind b) v) (CHead d1 (Bind b0) 
+w))).(let H_x \def (H b0 d1 w (r (Bind b) h0) (getl_gen_S (Bind b) c1 (CHead 
+d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x in (ex_ind C (\lambda (d2: 
+C).(getl h0 c2 (CHead d2 (Bind b0) w))) (ex C (\lambda (d2: C).(getl (S h0) 
+(CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: 
+(getl h0 c2 (CHead x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) 
+(CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) h0 c2 
+(CHead x (Bind b0) w) H3 v)))) H2)))))) h H0)))))))))).
+
+lemma cimp_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall 
+(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) 
+\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead 
+d2 (Bind b) w)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: 
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to 
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda 
+(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def 
+H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C 
+(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall 
+(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4: 
+C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x 
+(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3: 
+C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) 
+\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0: 
+B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h 
+d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1 
+(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0 
+(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in 
+(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (S (plus h i)) c2 
+(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind 
+b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (S (plus h i)) c2 (CHead x0 
+(Bind b0) w0))).(let H_y0 \def (getl_conf_le (S (plus h i)) (CHead x0 (Bind 
+b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (refl_equal nat 
+(plus (S h) i)) in (let H7 \def (eq_ind nat (S (plus h i)) (\lambda (n: 
+nat).(getl (minus n i) (CHead x (Bind b) w) (CHead x0 (Bind b0) w0))) (H_y0 
+(le_S i (plus h i) (le_plus_r h i))) (plus (S h) i) H6) in (let H8 \def 
+(eq_ind nat (minus (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind 
+b) w) (CHead x0 (Bind b0) w0))) H7 (S h) (minus_plus_r (S h) i)) in (ex_intro 
+C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0 (getl_gen_S (Bind 
+b) x (CHead x0 (Bind b0) w0) w h H8)))))))) H4))))))))) H2))) H1)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clear/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/clear/defs.ma
new file mode 100644 (file)
index 0000000..ce009d9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clear/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+inductive clear: C \to (C \to Prop) \def
+| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e 
+(Bind b) u) (CHead e (Bind b) u))))
+| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: 
+F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clear/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/clear/drop.ma
new file mode 100644 (file)
index 0000000..4ecf977
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clear/drop.ma
@@ -0,0 +1,168 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/clear/fwd.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+lemma drop_clear:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to 
+(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead 
+e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
+c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: 
+nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda 
+(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda 
+(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind 
+(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda 
+(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat 
+O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (ex2_3 B 
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e 
+(Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
+c2))))) H3))))) (drop_gen_sort n (S i) O c2 H)))))) (\lambda (c: C).(\lambda 
+(H: ((\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T 
+(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) 
+v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
+c2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (i: 
+nat).(\lambda (H0: (drop (S i) O (CHead c k t) c2)).(K_ind (\lambda (k0: 
+K).((drop (r k0 i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear (CHead c k0 t) (CHead e (Bind b) v))))) (\lambda 
+(_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))) (\lambda (b: 
+B).(\lambda (H1: (drop (r (Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda 
+(b0: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e 
+(Bind b0) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e 
+c2)))) b c t (clear_bind b c t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r 
+(Flat f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda 
+(b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C 
+T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) 
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: 
+T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O 
+x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: 
+T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c 
+(CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) k (drop_gen_drop k c c2 t i 
+H0))))))))) c1).
+
+lemma drop_clear_O:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c 
+(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 
+e2) \to (drop (S i) O c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: 
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: 
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) 
+(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort 
+n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: 
+(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O 
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: 
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: 
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 
+e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: 
+T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0: 
+K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 
+k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) 
+(CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) (CHead e1 
+(Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) 
+u) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
+_) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) u) (CHead c0 
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H5 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) 
+(clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b 
+b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: 
+C).(drop i O c1 e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O 
+(CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) 
+H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 
+(Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead 
+e1 (Bind b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
+
+lemma drop_clear_S:
+ \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop 
+h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear 
+x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 
+(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
+\def
+ \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: 
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: 
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
+c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda 
+(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) 
+u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: 
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
+c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: 
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: 
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
+c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k 
+t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear 
+(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 
+(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C 
+(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: 
+C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift 
+h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k 
+(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead 
+c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind 
+(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h 
+(r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) 
+t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) 
+(\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind 
+b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind 
+b0 c (CHead c2 (Bind b) u) t H4)) in ((let H7 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match 
+k0 with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 
+(Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) 
+t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead 
+c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda 
+(H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: 
+T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) 
+t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) 
+(eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind 
+b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda 
+(c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda 
+(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind 
+b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda 
+(c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind 
+b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x 
+(lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f: 
+F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda 
+(H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u 
+(clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: 
+C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 
+c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) 
+t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) 
+(\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d 
+u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear 
+(CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d 
+u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) 
+(lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1 
+H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clear/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/clear/fwd.ma
new file mode 100644 (file)
index 0000000..1b12b5f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clear/fwd.ma
@@ -0,0 +1,192 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/clear/defs.ma".
+
+include "basic_1A/C/fwd.ma".
+
+implied rec lemma clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: 
+B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) 
+u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to 
+(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) 
+(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) 
+\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 
+((clear_ind P f f0) e c2 c3) f1 u)].
+
+lemma clear_gen_sort:
+ \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: 
+Prop).P)))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda 
+(P: Prop).(insert_eq C (CSort n) (\lambda (c: C).(clear c x)) (\lambda (_: 
+C).P) (\lambda (y: C).(\lambda (H0: (clear y x)).(clear_ind (\lambda (c: 
+C).(\lambda (_: C).((eq C c (CSort n)) \to P))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u: T).(\lambda (H1: (eq C (CHead e (Bind b) u) (CSort n))).(let 
+H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
+H1) in (False_ind P H2)))))) (\lambda (e: C).(\lambda (c: C).(\lambda (_: 
+(clear e c)).(\lambda (_: (((eq C e (CSort n)) \to P))).(\lambda (f: 
+F).(\lambda (u: T).(\lambda (H3: (eq C (CHead e (Flat f) u) (CSort n))).(let 
+H4 \def (eq_ind C (CHead e (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) 
+H3) in (False_ind P H4))))))))) y x H0))) H)))).
+
+lemma clear_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
+(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
+(clear (CHead e (Bind b) u) x)).(insert_eq C (CHead e (Bind b) u) (\lambda 
+(c: C).(clear c x)) (\lambda (c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: 
+(clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e 
+(Bind b) u)) \to (eq C c0 c)))) (\lambda (b0: B).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) 
+u))).(let H2 \def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) 
+\Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0) 
+(CHead e (Bind b) u) H1) in ((let H3 \def (f_equal C B (\lambda (e1: 
+C).(match e1 with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead 
+e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in ((let H4 \def (f_equal C T 
+(\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H1) in (\lambda 
+(H5: (eq B b0 b)).(\lambda (H6: (eq C e0 e)).(eq_ind_r T u (\lambda (t: 
+T).(eq C (CHead e0 (Bind b0) t) (CHead e0 (Bind b0) t))) (eq_ind_r C e 
+(\lambda (c: C).(eq C (CHead c (Bind b0) u) (CHead c (Bind b0) u))) (eq_ind_r 
+B b (\lambda (b1: B).(eq C (CHead e (Bind b1) u) (CHead e (Bind b1) u))) 
+(refl_equal C (CHead e (Bind b) u)) b0 H5) e0 H6) u0 H4)))) H3)) H2)))))) 
+(\lambda (e0: C).(\lambda (c: C).(\lambda (_: (clear e0 c)).(\lambda (_: 
+(((eq C e0 (CHead e (Bind b) u)) \to (eq C c e0)))).(\lambda (f: F).(\lambda 
+(u0: T).(\lambda (H3: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) 
+u))).(let H4 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind 
+b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0))) 
+H))))).
+
+lemma clear_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear 
+(CHead e (Flat f) u) x) \to (clear e x)))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: 
+(clear (CHead e (Flat f) u) x)).(insert_eq C (CHead e (Flat f) u) (\lambda 
+(c: C).(clear c x)) (\lambda (_: C).(clear e x)) (\lambda (y: C).(\lambda 
+(H0: (clear y x)).(clear_ind (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead 
+e (Flat f) u)) \to (clear e c0)))) (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) 
+u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat 
+f) u) H1) in (False_ind (clear e (CHead e0 (Bind b) u0)) H2)))))) (\lambda 
+(e0: C).(\lambda (c: C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C e0 
+(CHead e (Flat f) u)) \to (clear e c)))).(\lambda (f0: F).(\lambda (u0: 
+T).(\lambda (H3: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(let H4 
+\def (f_equal C C (\lambda (e1: C).(match e1 with [(CSort _) \Rightarrow e0 | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) 
+H3) in ((let H5 \def (f_equal C F (\lambda (e1: C).(match e1 with [(CSort _) 
+\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead e0 (Flat f0) u0) (CHead 
+e (Flat f) u) H3) in ((let H6 \def (f_equal C T (\lambda (e1: C).(match e1 
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 
+(Flat f0) u0) (CHead e (Flat f) u) H3) in (\lambda (_: (eq F f0 f)).(\lambda 
+(H8: (eq C e0 e)).(let H9 \def (eq_ind C e0 (\lambda (c0: C).((eq C c0 (CHead 
+e (Flat f) u)) \to (clear e c))) H2 e H8) in (let H10 \def (eq_ind C e0 
+(\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x 
+H0))) H))))).
+
+lemma clear_gen_flat_r:
+ \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x 
+(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H: 
+(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e 
+(Flat f) u) (\lambda (c: C).(clear x c)) (\lambda (_: C).P) (\lambda (y: 
+C).(\lambda (H0: (clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: 
+C).((eq C c0 (CHead e (Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat 
+f) u))).(let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e 
+(Flat f) u) H1) in (False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: 
+C).(\lambda (H1: (clear e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) 
+\to P))).(\lambda (_: F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat 
+f) u))).(let H4 \def (eq_ind C c (\lambda (c0: C).((eq C c0 (CHead e (Flat f) 
+u)) \to P)) H2 (CHead e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda 
+(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C 
+(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
+
+lemma clear_gen_all:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind 
+(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0: 
+B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0 
+(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e: 
+C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(eq C c (CHead e0 (Bind b) 
+u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b) 
+u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c 
+(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c 
+(\lambda (c0: C).(clear e c0)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C 
+(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B 
+C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind 
+x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) 
+x2))) c H3)))))) H2)))))))) c1 c2 H))).
+
+theorem clear_mono:
+ \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c 
+c2) \to (eq C c1 c2)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to 
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: 
+nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: 
+C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 
+c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to 
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) 
+c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) 
+\to (eq C c1 c2)))) (\lambda (b: B).(\lambda (H2: (clear (CHead c0 (Bind b) 
+t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 
+(Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) 
+(\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 
+(Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t 
+H3))))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) 
+c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f 
+c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
+
+lemma clear_cle:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
+(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda 
+(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight 
+c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
+C).(\lambda (H0: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear 
+(CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) 
+(\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C 
+(CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) 
+(tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c 
+c2 t H1)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) 
+c2)).(le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 
+(clear_gen_flat f c c2 t H1))))) k H0))))))) c1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clear/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/clear/props.ma
new file mode 100644 (file)
index 0000000..e556d59
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clear/props.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/clear/fwd.ma".
+
+lemma clear_clear:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to 
+(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear 
+(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: 
+C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 
+c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear 
+(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to 
+(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t) 
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) 
+(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f: 
+F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c 
+c2 t H1)))) k H0))))))) c1).
+
+theorem clear_trans:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c 
+c2) \to (clear c1 c2)))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to 
+(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: 
+nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: 
+C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) 
+c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to 
+(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) 
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0: 
+K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b: 
+B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 
+(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c 
+c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead 
+c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3))))) 
+(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c 
+c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
+
+lemma clear_ctail:
+ \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 
+(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k 
+u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: 
+K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) 
+u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: 
+(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: 
+T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 
+c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n 
+H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind 
+b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear 
+(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))) 
+k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: 
+T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: 
+T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda 
+(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear 
+(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: 
+T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to 
+(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) 
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind 
+b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) 
+(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in 
+((let H3 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c 
+(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) 
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b 
+b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead 
+(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r 
+C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail 
+k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 
+u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail 
+k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: 
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 
+c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead 
+c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clen/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/clen/defs.ma
new file mode 100644 (file)
index 0000000..ce6bcbf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clen/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+include "basic_1A/s/defs.ma".
+
+rec definition clen (c: C) on c: nat \def match c with [(CSort _) \Rightarrow 
+O | (CHead c0 k _) \Rightarrow (s k (clen c0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/clen/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/clen/getl.ma
new file mode 100644 (file)
index 0000000..665581c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/clen/getl.ma
@@ -0,0 +1,351 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/clen/defs.ma".
+
+include "basic_1A/getl/props.ma".
+
+lemma getl_ctail_clen:
+ \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: 
+nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
+\def
+ \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex 
+nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) 
+(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O 
+(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b 
+(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl 
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: 
+K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl 
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat 
+(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) 
+(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen 
+c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(K_ind (\lambda (k0: 
+K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t 
+c0) k0 t0) (CHead (CSort n) (Bind b) t))))) (\lambda (b0: B).(ex_intro nat 
+(\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) 
+t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail 
+(Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0))) (\lambda (f: 
+F).(ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t 
+c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) 
+t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))) k))) H0)))))) c))).
+
+lemma getl_gen_tail:
+ \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall 
+(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 
+(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
+(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: 
+nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
+\def
+ \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i 
+(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C 
+c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 
+nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind 
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort 
+n)))))))) (\lambda (n: nat).(\lambda (i: nat).(nat_ind (\lambda (n0: 
+nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C 
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort 
+n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) 
+(\lambda (n1: nat).(eq C c2 (CSort n1))))))) (\lambda (H: (getl O (CHead 
+(CSort n) k u1) (CHead c2 (Bind b) u2))).(K_ind (\lambda (k0: K).((clear 
+(CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
+C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e 
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: 
+nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
+nat).(eq C c2 (CSort n0))))))) (\lambda (b0: B).(\lambda (H0: (clear (CHead 
+(CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c _ _) 
+\Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
+(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
+_ k0 _) \Rightarrow (match k0 with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
+(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) 
+(clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: 
+(eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C (CSort n) 
+(\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) 
+(\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda 
+(_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda 
+(_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T 
+u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind 
+b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 
+nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind 
+b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort 
+n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C 
+(CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e 
+(Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: 
+nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda 
+(n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: 
+C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) 
+(CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda 
+(_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda 
+(n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq 
+nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq 
+T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat 
+O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b 
+H4) u2 H3) c2 H5)))) H2)) H1)))) (\lambda (f: F).(\lambda (H0: (clear (CHead 
+(CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 
+(Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or 
+(ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl 
+O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) 
+(\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) 
+(\lambda (n0: nat).(eq C c2 (CSort n0)))))))) k (getl_gen_O (CHead (CSort n) 
+k u1) (CHead c2 (Bind b) u2) H))) (\lambda (n0: nat).(\lambda (_: (((getl n0 
+(CHead (CSort n) k u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
+C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e 
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 O)) (\lambda (_: 
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: 
+nat).(eq C c2 (CSort n1)))))))).(\lambda (H0: (getl (S n0) (CHead (CSort n) k 
+u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) 
+(getl_gen_S k (CSort n) (CHead c2 (Bind b) u2) u1 n0 H0) (or (ex2 C (\lambda 
+(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n1: nat).(eq C c2 (CSort n1))))))))) i))) (\lambda (c: C).(\lambda (H: 
+((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or 
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda 
+(i: nat).(nat_ind (\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) 
+(CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
+e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat 
+(\lambda (_: nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k 
+(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort 
+n0))))))) (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind 
+b) u2))).(K_ind (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead 
+c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
+(\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat 
+(\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind 
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort 
+n))))))) (\lambda (b0: B).(\lambda (H1: (clear (CHead (CTail k u1 c) (Bind 
+b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 
+(CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead _ k1 _) 
+\Rightarrow (match k1 with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
+b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) 
+(clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) 
+(Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) 
+in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind 
+T u2 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) 
+(\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 
+nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_: 
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
+C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e: 
+C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1) 
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 
+(\lambda (c0: C).(\forall (i0: nat).((getl i0 (CTail k u1 c) (CHead c0 (Bind 
+b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: 
+C).(getl i0 c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i0 
+(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in 
+(eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C 
+c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e 
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C 
+(CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) 
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C 
+(\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O 
+(CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c)) 
+(getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2)))) (\lambda (f: 
+F).(\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) 
+u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) 
+(CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) 
+(CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 
+(CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat 
+(\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) 
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or 
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O 
+(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq 
+nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda 
+(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: 
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c 
+(CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 
+e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda 
+(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) 
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4: 
+(eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b) 
+u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: 
+C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) 
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) 
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: 
+C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c 
+(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
+(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n)))) 
+(ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda 
+(e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C 
+(CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3)) 
+(\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: 
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
+C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c))) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
+e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) 
+(ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: 
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq 
+C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen 
+c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda 
+(H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C 
+(\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c 
+(Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
+(Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 
+(\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) 
+(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 
+nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq 
+K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort 
+x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda 
+(e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c 
+(Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s 
+(Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: 
+nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) 
+(or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) 
+(\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 
+nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq 
+K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C 
+(CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat 
+f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: 
+nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 
+(refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) 
+u2 H6) c2 H7)))))) H3)) H2)))) k0 (getl_gen_O (CHead (CTail k u1 c) k0 t) 
+(CHead c2 (Bind b) u2) H0))) (\lambda (n: nat).(\lambda (H0: (((getl n (CHead 
+(CTail k u1 c) k0 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: 
+C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e 
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 (clen c)))) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n0: nat).(eq C c2 (CSort n0)))))))).(\lambda (H1: (getl (S n) (CHead (CTail 
+k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S 
+k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H1)) in (let H2 \def H_x in 
+(or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: 
+C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq 
+nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: 
+nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or (ex2 C 
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
+c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
+k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
+u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H3: (ex2 C 
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c 
+(CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 
+e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C 
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
+c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
+k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
+u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda 
+(H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl (r k0 n) c (CHead x (Bind 
+b) u2))).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k 
+u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind 
+b) u2) t n H1) (CTail k u1 x) H4) in (let H7 \def (eq_ind C c2 (\lambda (c0: 
+C).((getl n (CHead (CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) \to (or (ex2 
+C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c 
+k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (s k0 
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 
+u2)) (\lambda (n0: nat).(eq C c0 (CSort n0))))))) H0 (CTail k u1 x) H4) in 
+(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C 
+c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind 
+b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda 
+(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
+nat).(eq C c0 (CSort n0)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail 
+k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e 
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) 
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda 
+(n0: nat).(eq C (CTail k u1 x) (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq 
+C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) 
+(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_head k0 n c 
+(CHead x (Bind b) u2) H5 t))) c2 H4)))))) H3)) (\lambda (H3: (ex4 nat 
+(\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind 
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort 
+n0))))).(ex4_ind nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda 
+(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: 
+nat).(eq C c2 (CSort n0))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 
+e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 
+nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K 
+k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 
+(CSort n0))))) (\lambda (x0: nat).(\lambda (H4: (eq nat (r k0 n) (clen 
+c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda 
+(H7: (eq C c2 (CSort x0))).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(getl 
+(r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 
+c) (CHead c2 (Bind b) u2) t n H1) (CSort x0) H7) in (let H9 \def (eq_ind C c2 
+(\lambda (c0: C).((getl n (CHead (CTail k u1 c) k0 t) (CHead c0 (Bind b) u2)) 
+\to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: 
+C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: 
+nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) 
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0))))))) 
+H0 (CSort x0) H7) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C 
+(\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead 
+c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s 
+k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T 
+u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H10 \def (eq_ind_r T 
+u2 (\lambda (t0: T).((getl n (CHead (CTail k u1 c) k0 t) (CHead (CSort x0) 
+(Bind b) t0)) \to (or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 
+e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat 
+(\lambda (_: nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind 
+b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) 
+(CSort n0))))))) H9 u1 H6) in (let H11 \def (eq_ind_r T u2 (\lambda (t0: 
+T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t0))) H8 u1 H6) 
+in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) 
+(CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
+t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda 
+(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: 
+nat).(eq C (CSort x0) (CSort n0)))))) (let H12 \def (eq_ind K k (\lambda (k1: 
+K).((getl n (CHead (CTail k1 u1 c) k0 t) (CHead (CSort x0) (Bind b) u1)) \to 
+(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: 
+C).(getl n (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: 
+nat).(eq nat n (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) 
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort 
+n0))))))) H10 (Bind b) H5) in (let H13 \def (eq_ind K k (\lambda (k1: 
+K).(getl (r k0 n) (CTail k1 u1 c) (CHead (CSort x0) (Bind b) u1))) H11 (Bind 
+b) H5) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: 
+C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 
+t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 
+(clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 
+u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n) 
+(\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind 
+b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
+u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_: 
+nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: 
+nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0: 
+nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) 
+(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat 
+(\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind 
+b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) 
+(CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail 
+(Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) 
+u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq 
+K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq 
+C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S 
+n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 
+u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S 
+n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s 
+k0 (r k0 n)) (s_r k0 n)) (clen c) H4) k H5))) u2 H6))) c2 H7)))))))) H3)) 
+H2)))))) i)))))) c1)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/cnt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/cnt/defs.ma
new file mode 100644 (file)
index 0000000..f6a0dc5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/cnt/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+inductive cnt: T \to Prop \def
+| cnt_sort: \forall (n: nat).(cnt (TSort n))
+| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt 
+(THead k v t))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/cnt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/cnt/fwd.ma
new file mode 100644 (file)
index 0000000..aadf1d7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/cnt/fwd.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/cnt/defs.ma".
+
+implied rec lemma cnt_ind (P: (T \to Prop)) (f: (\forall (n: nat).(P (TSort 
+n)))) (f0: (\forall (t: T).((cnt t) \to ((P t) \to (\forall (k: K).(\forall 
+(v: T).(P (THead k v t)))))))) (t: T) (c: cnt t) on c: P t \def match c with 
+[(cnt_sort n) \Rightarrow (f n) | (cnt_head t0 c0 k v) \Rightarrow (f0 t0 c0 
+((cnt_ind P f f0) t0 c0) k v)].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/cnt/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/cnt/props.ma
new file mode 100644 (file)
index 0000000..531fe72
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/cnt/props.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/cnt/fwd.ma".
+
+include "basic_1A/lift/props.ma".
+
+lemma cnt_lift:
+ \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i 
+d t)))))
+\def
+ \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i: 
+nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i: 
+nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0)) 
+(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0: 
+T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d: 
+nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0)) 
+(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k 
+(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/arity.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/arity.ma
new file mode 100644 (file)
index 0000000..088ac63
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/arity.ma
@@ -0,0 +1,319 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/getl.ma".
+
+include "basic_1A/csuba/props.ma".
+
+include "basic_1A/arity/fwd.ma".
+
+include "basic_1A/csubv/getl.ma".
+
+lemma csuba_arity:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c 
+c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall 
+(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda 
+(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in 
+(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda 
+(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d 
+x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc 
+g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u 
+(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def 
+(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc 
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: 
+(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst 
+g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc g a1))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(arity g d u (asucc g a1))))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a1: A).(arity g d2 u2 a1)))) (arity g c2 (TLRef i) a0) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H6: (getl i c2 (CHead x0 
+(Bind Abbr) x1))).(\lambda (_: (csuba g d x0)).(\lambda (H8: (arity g d u 
+(asucc g x2))).(\lambda (H9: (arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) 
+x2 (arity_abbr g c2 x0 x1 i H6 x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u 
+(asucc g x2) H8 (asucc g a0) H1)))))))))) H5)) H4)))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall 
+(c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: 
+((\forall (c2: C).((csuba g (CHead c (Bind b) u) c2) \to (arity g c2 t0 
+a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g c c2)).(arity_bind g b H0 c2 
+u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u) (csuba_head g c c2 H5 (Bind 
+b) u)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: 
+C).((csuba g c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
+a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c (Bind Abst) u) c2) 
+\to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (csuba g c 
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u) 
+(csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1 
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1 
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g 
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2: 
+C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2 
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: 
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity 
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: 
+C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2 
+H2)))))))))) c1 t a H))))).
+
+lemma csuba_arity_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (c2: C).((csuba g c2 c1) \to ((csubv c2 c1) \to (arity g c2 
+t a))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 
+a0))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: 
+(csuba g c2 c)).(\lambda (_: (csubv c2 c)).(arity_sort g c2 n)))))) (\lambda 
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u 
+a0)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to 
+(arity g c2 u a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 
+c)).(\lambda (H4: (csubv c2 c)).(let H_x \def (csuba_getl_abbr_rev g c d u i 
+H0 c2 H3) in (let H5 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 
+(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d 
+u a1))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d)))) (arity 
+g c2 (TLRef i) a0) (\lambda (H6: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)) 
+(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x 
+(Bind Abbr) u))).(\lambda (H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf 
+c2 c H4 Abbr x u i H7) in (let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda 
+(d2: C).(\lambda (v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 
+(TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(H10: (csubv x x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let 
+H12 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x1 
+(Bind x0) x2) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
+(Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) i 
+H0 (CHead x1 (Bind x0) x2) H11)) in ((let H14 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abbr) u) 
+i H0 (CHead x1 (Bind x0) x2) H11)) in ((let H15 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d 
+(Bind Abbr) u) i H0 (CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B 
+Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda 
+(t0: T).(getl i c (CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def 
+(eq_ind_r C x1 (\lambda (c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) 
+in (let H20 \def (eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in 
+(let H21 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) 
+H19 Abbr H16) in (arity_abbr g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) 
+H13)))))))) H9)))))) H6)) (\lambda (H6: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u 
+a1)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a1: A).(arity g d u a1)))) (arity g c2 (TLRef i) a0) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H7: (getl i c2 
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (csuba g x0 d)).(\lambda (H9: (arity 
+g x0 x1 (asucc g x2))).(\lambda (H10: (arity g d u x2)).(arity_repl g c2 
+(TLRef i) x2 (arity_abst g c2 x0 x1 i H7 x2 H9) a0 (arity_mono g d u x2 H10 
+a0 H1))))))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
+i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let 
+H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in 
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: 
+C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef 
+i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda 
+(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d 
+(Bind Abbr) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) 
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind Void) x3) H11)) in 
+(let H13 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow 
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead 
+x2 (Bind Void) x3) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead x2 (Bind 
+Void) x3) H11)) in (False_ind (arity g c2 (TLRef i) a0) H13))))))) H9))))))) 
+H6)) H5)))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: 
+((\forall (c2: C).((csuba g c2 d) \to ((csubv c2 d) \to (arity g c2 u (asucc 
+g a0))))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(\lambda (H4: 
+(csubv c2 c)).(let H_x \def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (let 
+H5 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d)))) (arity g c2 (TLRef i) a0) (\lambda (H6: 
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d)) (arity g c2 (TLRef i) a0) 
+(\lambda (x: C).(\lambda (H7: (getl i c2 (CHead x (Bind Abst) u))).(\lambda 
+(H8: (csuba g x d)).(let H_x0 \def (csubv_getl_conf c2 c H4 Abst x u i H7) in 
+(let H9 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csubv x d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda 
+(v2: T).(getl i c (CHead d2 (Bind b2) v2))))) (arity g c2 (TLRef i) a0) 
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H10: (csubv x 
+x1)).(\lambda (H11: (getl i c (CHead x1 (Bind x0) x2))).(let H12 \def (eq_ind 
+C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x1 (Bind 
+x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 (Bind x0) x2) 
+H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) 
+(CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x1 
+(Bind x0) x2) H11)) in ((let H14 \def (f_equal C B (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) 
+u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
+x1 (Bind x0) x2) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
+(Bind Abst) u) (CHead x1 (Bind x0) x2) (getl_mono c (CHead d (Bind Abst) u) i 
+H0 (CHead x1 (Bind x0) x2) H11)) in (\lambda (H16: (eq B Abst x0)).(\lambda 
+(H17: (eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t0: T).(getl i c 
+(CHead x1 (Bind x0) t0))) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda 
+(c0: C).(getl i c (CHead c0 (Bind x0) u))) H18 d H17) in (let H20 \def 
+(eq_ind_r C x1 (\lambda (c0: C).(csubv x c0)) H10 d H17) in (let H21 \def 
+(eq_ind_r B x0 (\lambda (b: B).(getl i c (CHead d (Bind b) u))) H19 Abst H16) 
+in (arity_abst g c2 x u i H7 a0 (H2 x H8 H20))))))))) H14)) H13)))))))) 
+H9)))))) H6)) (\lambda (H6: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
+i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H7: (getl i c2 (CHead x0 (Bind Void) x1))).(\lambda (_: (csuba g x0 d)).(let 
+H_x0 \def (csubv_getl_conf_void c2 c H4 x0 x1 i H7) in (let H9 \def H_x0 in 
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubv x0 d2))) (\lambda (d2: 
+C).(\lambda (v2: T).(getl i c (CHead d2 (Bind Void) v2)))) (arity g c2 (TLRef 
+i) a0) (\lambda (x2: C).(\lambda (x3: T).(\lambda (_: (csubv x0 x2)).(\lambda 
+(H11: (getl i c (CHead x2 (Bind Void) x3))).(let H12 \def (eq_ind C (CHead d 
+(Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H0 (CHead x2 (Bind Void) x3) 
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead x2 (Bind Void) x3) H11)) in 
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with 
+[(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
+I (CHead x2 (Bind Void) x3) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead 
+x2 (Bind Void) x3) H11)) in (False_ind (arity g c2 (TLRef i) a0) H13))))))) 
+H9))))))) H6)) H5)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
+g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) 
+\to (arity g c2 u a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: 
+(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba 
+g c2 (CHead c (Bind b) u)) \to ((csubv c2 (CHead c (Bind b) u)) \to (arity g 
+c2 t0 a2)))))).(\lambda (c2: C).(\lambda (H5: (csuba g c2 c)).(\lambda (H6: 
+(csubv c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5 H6) t0 a2 (H4 (CHead c2 
+(Bind b) u) (csuba_head g c2 c H5 (Bind b) u) (csubv_bind_same c2 c H6 b u 
+u))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2: C).((csuba g c2 
+c) \to ((csubv c2 c) \to (arity g c2 u (asucc g a1))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0 
+a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c (Bind Abst) u)) 
+\to ((csubv c2 (CHead c (Bind Abst) u)) \to (arity g c2 t0 a2)))))).(\lambda 
+(c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_head 
+g c2 u a1 (H1 c2 H4 H5) t0 a2 (H3 (CHead c2 (Bind Abst) u) (csuba_head g c2 c 
+H4 (Bind Abst) u) (csubv_bind_same c2 c H5 Abst u u))))))))))))))) (\lambda 
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u 
+a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to 
+(arity g c2 u a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity 
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2: C).((csuba g c2 c) \to 
+((csubv c2 c) \to (arity g c2 t0 (AHead a1 a2))))))).(\lambda (c2: 
+C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv c2 c)).(arity_appl g c2 
+u a1 (H1 c2 H4 H5) t0 a2 (H3 c2 H4 H5)))))))))))))) (\lambda (c: C).(\lambda 
+(u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda 
+(H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 u 
+(asucc g a0))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda 
+(H3: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to (arity g c2 t0 
+a0)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(\lambda (H5: (csubv 
+c2 c)).(arity_cast g c2 u a0 (H1 c2 H4 H5) t0 (H3 c2 H4 H5))))))))))))) 
+(\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c t0 
+a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to ((csubv c2 c) \to 
+(arity g c2 t0 a1)))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
+a2)).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(\lambda (H4: (csubv c2 
+c)).(arity_repl g c2 t0 a1 (H1 c2 H3 H4) a2 H2))))))))))) c1 t a H))))).
+
+theorem arity_appls_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c 
+v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: 
+T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead 
+(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H: 
+(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g 
+a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
+TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr) 
+v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead 
+(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead 
+(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr not_abbr_abst g c v 
+t a2 H1) in (let H2 \def H_x in (ex2_ind A (\lambda (a3: A).(arity g c v a3)) 
+(\lambda (_: A).(arity g (CHead c (Bind Abbr) v) t a2)) (arity g c (THead 
+(Flat Appl) v (THead (Bind Abst) u t)) a2) (\lambda (x: A).(\lambda (_: 
+(arity g c v x)).(\lambda (H4: (arity g (CHead c (Bind Abbr) v) t 
+a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t) a2 (arity_head g c u a1 
+H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t a2 H4 (CHead c (Bind 
+Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v H) (csubv_bind c c 
+(csubv_refl c) Abst not_abst_void Abbr u v))))))) H2))))) (\lambda (t0: 
+T).(\lambda (t1: TList).(\lambda (H1: ((\forall (a2: A).((arity g c (THeads 
+(Flat Appl) t1 (THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat 
+Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u t))) a2))))).(\lambda (a2: 
+A).(\lambda (H2: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind Abbr) v t))) a2)).(let H3 \def (arity_gen_appl g c t0 (THeads 
+(Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2) in (ex2_ind A (\lambda (a3: 
+A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c (THeads (Flat Appl) t1 
+(THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u t)))) a2) 
+(\lambda (x: A).(\lambda (H4: (arity g c t0 x)).(\lambda (H5: (arity g c 
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead x a2))).(arity_appl g 
+c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind Abst) u 
+t))) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/clear.ma
new file mode 100644 (file)
index 0000000..69f9d2f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/clear.ma
@@ -0,0 +1,122 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma csuba_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear 
+(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
+b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda 
+(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g 
+c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
+(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
+(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
+e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
+C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
+(\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
+u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
+(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda 
+(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
+(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba 
+g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
+u2) e2)) (CHead c4 (Bind b) u2) (csuba_void g c3 c4 H0 b H2 u1 u2) 
+(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: 
+((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) 
+(\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda 
+(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
+a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) 
+e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) 
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) 
+(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) 
+u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 
+(clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
+
+lemma csuba_clear_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2 
+c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear 
+c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c4 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u) 
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear 
+(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind 
+b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda 
+(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g 
+c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3)))) 
+(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def 
+(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g 
+e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2 
+e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x: 
+C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C 
+(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) 
+u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall 
+(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda 
+(e2: C).(clear c3 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
+(clear (CHead c4 (Bind b) u2) e1)).(eq_ind_r C (CHead c4 (Bind b) u2) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g e2 c)) (\lambda (e2: 
+C).(clear (CHead c3 (Bind Void) u1) e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csuba g e2 (CHead c4 (Bind b) u2))) (\lambda (e2: C).(clear (CHead c3 
+(Bind Void) u1) e2)) (CHead c3 (Bind Void) u1) (csuba_void g c3 c4 H0 b H2 u1 
+u2) (clear_bind Void c3 u1)) e1 (clear_gen_bind b c4 e1 u2 H3)))))))))))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: 
+((\forall (e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) 
+(\lambda (e2: C).(clear c3 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda 
+(H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u 
+a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c4 (Bind Abbr) u) 
+e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) 
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u))) 
+(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst) 
+t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1 
+(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/defs.ma
new file mode 100644 (file)
index 0000000..59c2c71
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/defs.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/arity/defs.ma".
+
+inductive csuba (g: G): C \to (C \to Prop) \def
+| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
+| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
+(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
+| csuba_void: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csuba g 
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall 
+(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: 
+T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind 
+Abbr) u))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/drop.ma
new file mode 100644 (file)
index 0000000..74b30ce
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/drop.ma
@@ -0,0 +1,2453 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/fwd.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+lemma csuba_drop_abbr:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
+O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) 
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 
+(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: 
+(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in 
+(let H_x \def (csuba_gen_abbr g d1 c2 u H1) in (let H2 \def H_x in (ex2_ind C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba 
+g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 
+(CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x 
+(Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda 
+(d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) 
+H2))))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S 
+n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) 
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
+O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind 
+Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (H3: 
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) 
+(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H3) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead 
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))) (drop_gen_sort 
+n0 (S n) O (CHead d1 (Bind Abbr) u) H0))))))))) (\lambda (c: C).(\lambda (H0: 
+((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) 
+\to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: 
+T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abbr) 
+u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) 
+c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) 
+O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (b: 
+B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r 
+(Bind b) n) O c (CHead d1 (Bind Abbr) u))).(B_ind (\lambda (b0: B).((csuba g 
+(CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
+u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H5: (csuba g (CHead c 
+(Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind 
+Abbr) u))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 \def H_x in 
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: 
+C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: 
+(eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C 
+(CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) 
+O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 
+\def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead 
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead 
+x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal 
+nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) 
+n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) 
+(\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r 
+(Bind Abst) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_abst g 
+c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq 
+C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
+g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 
+(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x 
+(Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def 
+(H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead 
+x0 (Bind Abbr) u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal 
+nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x (CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda 
+(H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g 
+c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity 
+g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O 
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 
+(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t 
+(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind 
+Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u 
+H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x 
+(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat 
+(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 
+O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 
+(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))) 
+(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r 
+(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H_x \def (csuba_gen_void g 
+c c2 t H5) in (let H7 \def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda 
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g c d2)))) (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (eq C 
+c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c x1)).(eq_ind_r C (CHead 
+x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def 
+(H c d1 u H6 g x1 H9) in (ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H11: (drop n O x1 (CHead 
+x (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x)).(let H13 \def (refl_equal 
+nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x1 (CHead x (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n 
+x1 (CHead x (Bind Abbr) u) H14 x2) H12)))))) H10)) c2 H8)))))) H7))))) b H3 
+H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) 
+c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let 
+H_x \def (csuba_gen_flat g c c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) 
+O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) 
+x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) 
+(\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 
+H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) 
+u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
+d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) 
+u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g 
+d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) 
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n 
+H1)))))))))))) c1)))) i).
+
+lemma csuba_drop_abst:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
+O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n 
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: 
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: 
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 
+(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl 
+c1 (CHead d1 (Bind Abst) u1) H)) in (let H_x \def (csuba_gen_abst g d1 c2 u1 
+H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: 
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
+O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x 
+(Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind 
+Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) 
+u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind 
+Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
+C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 
+H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind 
+Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 
+(asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind 
+Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) 
+x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
+Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 
+H7)) c2 H4)))))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 
+(Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to 
+(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: 
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind 
+Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) 
+c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) 
+(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (_: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda 
+(H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S 
+n) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H3) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))) H5))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) 
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: 
+T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall 
+(c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda 
+(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) 
+u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) 
+c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) 
+O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O 
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c 
+(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) 
+u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop 
+(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g 
+(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead 
+d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_abbr g c c2 t H5) in (let H7 
+\def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
+(\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) 
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) 
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
+(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead 
+x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def 
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda 
+(n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) 
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) 
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: 
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H12 
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) 
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 
+H8)))) H7))))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda 
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def 
+(csuba_gen_abst g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g 
+c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity 
+g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
+t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: 
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c 
+x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g 
+x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C 
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: 
+(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) 
+u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: 
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 
+x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 
+x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr) x1))) H13 
+(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) 
+n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 
+H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind 
+Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g 
+x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) 
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead 
+x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal 
+nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n0: 
+nat).(drop n0 O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in 
+(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x 
+(Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: 
+(drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 
+x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 
+x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abbr) x4))) 
+H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 
+(drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 
+H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g 
+(CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead 
+d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_void g c c2 t H5) in (let H7 
+\def H_x in (ex2_3_ind B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda (u2: 
+T).(eq C c2 (CHead d2 (Bind b0) u2))))) (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csuba g c d2)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H8: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H9: (csuba g c 
+x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(or (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g 
+x1 H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x1 
+(Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda 
+(d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x1 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O 
+(CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop 
+(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: 
+C).(\lambda (H12: (drop n O x1 (CHead x (Bind Abst) u1))).(\lambda (H13: 
+(csuba g d1 x)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x (Bind Abst) 
+u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind x0) n 
+x1 (CHead x (Bind Abst) u1) H15 x2) H13))))))) H11)) (\lambda (H11: (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x1 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H12: 
+(drop n O x1 (CHead x3 (Bind Abbr) x4))).(\lambda (H13: (csuba g d1 
+x3)).(\lambda (H14: (arity g d1 u1 (asucc g x5))).(\lambda (H15: (arity g x3 
+x4 x5)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def 
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x1 (CHead x3 (Bind Abbr) x4))) 
+H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x1 (Bind x0) x2) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 
+(drop_drop (Bind x0) n x1 (CHead x3 (Bind Abbr) x4) H17 x2) H13 H14 
+H15))))))))))) H11)) H10)) c2 H8)))))) H7))))) b H3 H4)))) (\lambda (f: 
+F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r 
+(Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H_x \def (csuba_gen_flat g c 
+c2 t f H3) in (let H5 \def H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda 
+(u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 
+(Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) 
+x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: 
+C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda 
+(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: 
+(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 
+x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) 
+H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: 
+(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 
+x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 
+x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind 
+Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2 
+(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
+
+lemma csuba_drop_abst_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i 
+O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g 
+c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n 
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))) (\lambda (c1: 
+C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind 
+Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 
+c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 
+(Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in (let H_x 
+\def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in (or_ind (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or 
+(ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C 
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) 
+u))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind Abst) u) 
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O 
+(CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x 
+(Bind Abst) u) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind 
+Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x 
+(drop_refl (CHead x (Bind Abst) u)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop O O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind Void) 
+x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O 
+(CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_refl (CHead x0 (Bind 
+Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 
+(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (or 
+(ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop 
+(S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) 
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) 
+O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind 
+Abst) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (_: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (H3: (eq 
+nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n) 
+(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H3) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) 
+(drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u) H0))))))))) (\lambda (c: 
+C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 
+(Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop 
+(S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: 
+C).(\lambda (u: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind 
+Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g c2 (CHead 
+c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0 t)) \to ((drop (r 
+k0 n) O c (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(drop (S 
+n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: 
+B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
+(Bind b) n) O c (CHead d1 (Bind Abst) u))).(B_ind (\lambda (b0: B).((csuba g 
+c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
+Abst) u)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (H5: (csuba g c2 
+(CHead c (Bind Abbr) t))).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 
+(Bind Abst) u))).(let H_x \def (csuba_gen_abbr_rev g c c2 t H5) in (let H7 
+\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) 
+t))) (\lambda (d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) 
+(\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) 
+t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x (Bind Abbr) t) 
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u H6 g x 
+H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind 
+Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
+g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H14: 
+(csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind 
+Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
+g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) 
+t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
+(Bind Abbr) n x (CHead x0 (Bind Abst) u) H13 t) H14))))) H12)) (\lambda (H12: 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or (ex2 C (\lambda 
+(d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
+x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop 
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
+(drop_drop (Bind Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) 
+H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc 
+g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) 
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g 
+x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t 
+x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1)))))) (let H13 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H16: 
+(csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind 
+Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) 
+x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H15 x1) H16))))) H14)) 
+(\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) 
+x4))).(\lambda (H16: (csuba g x3 d1)).(or_intror (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) 
+H13)) c2 H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: 
+(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r 
+C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: 
+C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
+H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O 
+x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C 
+(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x 
+d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
+(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda 
+(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
+x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) 
+H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) 
+t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) 
+u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in 
+(or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda 
+(d2: C).(csuba g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+c)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
+g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq C c2 
+(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C (CHead x 
+(Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H11 \def (H 
+c d1 u H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) 
+u))).(\lambda (H14: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u) 
+H13 t) H14))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead 
+x0 (Bind Void) x1))).(\lambda (H14: (csuba g x0 d1)).(or_intror (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
+(drop_drop (Bind Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) 
+H11)) c2 H9)))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H9: 
+(eq C c2 (CHead x0 (Bind Void) x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r 
+C (CHead x0 (Bind Void) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: 
+C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
+H11 \def (H c d1 u H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O 
+x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H12: (ex2 C 
+(\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H13: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H14: (csuba g x 
+d1)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
+(Bind Void) n x0 (CHead x (Bind Abst) u) H13 x1) H14))))) H12)) (\lambda 
+(H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x0 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
+x3))).(\lambda (H14: (csuba g x2 d1)).(or_intror (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) 
+H11)) c2 H9))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) 
+t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) 
+u))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in 
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: 
+C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H8: (eq 
+C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x c)).(eq_ind_r C 
+(CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S 
+n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H 
+c d1 u H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) 
+u))).(\lambda (H13: (csuba g x0 d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u) 
+H12 t) H13))))) H11)) (\lambda (H11: (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind 
+Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead 
+x0 (Bind Void) x1))).(\lambda (H13: (csuba g x0 d1)).(or_intror (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 
+(drop_drop (Bind Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) 
+H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 
+(CHead c (Flat f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind 
+Abst) u))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def 
+H_x in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 
+(Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Flat f) 
+x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u H4 g x0 
+H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(drop (S n) 
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g 
+d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat 
+f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C 
+T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) 
+u))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop 
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) 
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H10 
+x1) H11))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 
+(CHead x2 (Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or_intror (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 
+(drop_drop (Flat f) n x0 (CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) 
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n 
+H1)))))))))))) c1)))) i).
+
+lemma csuba_drop_abbr_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i 
+O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba 
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: 
+G).(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n 
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
+(H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: 
+C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl c1 (CHead d1 
+(Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2 u1 H1) in 
+(let H2 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1)) (or3 (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop O O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H4: (eq C c2 (CHead x (Bind Abbr) u1))).(\lambda (H5: (csuba g x 
+d1)).(eq_ind_r C (CHead x (Bind Abbr) u1) (\lambda (c: C).(or3 (ex2 C 
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro0 (ex2 C (\lambda (d2: 
+C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x (Bind Abbr) 
+u1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_refl 
+(CHead x (Bind Abbr) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H5: (csuba g 
+x0 d1)).(\lambda (H6: (arity g x0 x1 (asucc g x2))).(\lambda (H7: (arity g d1 
+u1 x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c: C).(or3 (ex2 C 
+(\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop O O c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop O O c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))) (or3_intro1 (ex2 C (\lambda (d2: 
+C).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Abst) 
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop O O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 
+(drop_refl (CHead x0 (Bind Abst) x1)) H5 H6 H7)) c2 H4)))))))) H3)) (\lambda 
+(H3: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O 
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop O O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H4: (eq C 
+c2 (CHead x0 (Bind Void) x1))).(\lambda (H5: (csuba g x0 d1)).(eq_ind_r C 
+(CHead x0 (Bind Void) x1) (\lambda (c: C).(or3 (ex2 C (\lambda (d2: C).(drop 
+O O c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop O O c (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) 
+(or3_intro2 (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind 
+Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop O O (CHead x0 (Bind Void) x1) (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 
+x1 (drop_refl (CHead x0 (Bind Void) x1)) H5)) c2 H4))))) H3)) H2))))))))))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall 
+(u1: T).((drop n O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall 
+(c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(drop n O c2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: 
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall 
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))))) (\lambda (n0: 
+nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) 
+(CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: 
+(csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) u1) (CSort 
+n0)) (eq nat (S n) O) (eq nat O O) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O 
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u1) 
+(CSort n0))).(\lambda (H3: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let 
+H5 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O \Rightarrow 
+False | (S _) \Rightarrow True])) I O H3) in (False_ind (or3 (ex2 C (\lambda 
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) H5))))) (drop_gen_sort n0 (S n) O 
+(CHead d1 (Bind Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall 
+(d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to 
+(\forall (g: G).(\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda 
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))).(\lambda (k: K).(\lambda 
+(t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c 
+k t) (CHead d1 (Bind Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda 
+(H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead 
+c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (b: 
+B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r 
+(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g 
+c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind 
+Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda 
+(H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6: (drop (r (Bind Abbr) 
+n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abbr_rev g c c2 t 
+H5) in (let H7 \def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead 
+d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: 
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: 
+C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq 
+C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C 
+(CHead x (Bind Abbr) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop 
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind 
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) 
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
+(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
+(Bind Abbr) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda 
+(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 
+(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 
+x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n 
+x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
+x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
+(Bind Abbr) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
+Abbr) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) 
+H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (or3 (ex2 
+C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind 
+Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_: (arity g x0 x1 (asucc 
+g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) 
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) 
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16: (csuba g x 
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) 
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Abst) n x0 
+(CHead x (Bind Abbr) u1) H15 x1) H16))))) H14)) (\lambda (H14: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
+A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16: 
+(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18: 
+(arity g d1 u1 x5)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5 
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H15 x1) H16 H17 
+H18))))))))) H14)) (\lambda (H14: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x3: C).(\lambda (x4: T).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Void) 
+x4))).(\lambda (H16: (csuba g x3 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (drop_drop (Bind 
+Abst) n x0 (CHead x3 (Bind Void) x4) H15 x1) H16)))))) H14)) H13)) c2 
+H9)))))))) H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+c))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) 
+x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x 
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 
+(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: 
+(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: 
+(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 
+H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
+x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind 
+Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) 
+H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abst) t))).(\lambda 
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
+(csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba 
+g d2 c))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c)))) (or3 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H8: 
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: 
+C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind 
+Abst) t))) (\lambda (d2: C).(csuba g d2 c)) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda (H9: (eq 
+C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g x c)).(eq_ind_r C 
+(CHead x (Bind Abst) t) (\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop 
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or3_ind 
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S 
+n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) 
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14: 
+(csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop 
+(Bind Abst) n x (CHead x0 (Bind Abbr) u1) H13 t) H14))))) H12)) (\lambda 
+(H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 
+(Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15: (arity g x0 
+x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n 
+x (CHead x0 (Bind Abst) x1) H13 t) H14 H15 H16))))))))) H12)) (\lambda (H12: 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H13: (drop n O x (CHead x0 (Bind Void) 
+x1))).(\lambda (H14: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
+(Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
+Abst) n x (CHead x0 (Bind Void) x1) H13 t) H14)))))) H12)) H11)) c2 H9)))) 
+H8)) (\lambda (H8: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+c))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H9: (eq C c2 (CHead x0 (Bind Void) 
+x1))).(\lambda (H10: (csuba g x0 c)).(eq_ind_r C (CHead x0 (Bind Void) x1) 
+(\lambda (c0: C).(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c0 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) (let H11 \def (H c d1 u1 H6 g x0 H10) in (or3_ind (ex2 C (\lambda 
+(d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n 
+O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: 
+C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: C).(\lambda 
+(H13: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H14: (csuba g x 
+d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) 
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind Void) n x0 
+(CHead x (Bind Abbr) u1) H13 x1) H14))))) H12)) (\lambda (H12: (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+A).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H14: 
+(csuba g x2 d1)).(\lambda (H15: (arity g x2 x3 (asucc g x4))).(\lambda (H16: 
+(arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S 
+n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
+(drop_drop (Bind Void) n x0 (CHead x2 (Bind Abst) x3) H13 x1) H14 H15 
+H16))))))))) H12)) (\lambda (H12: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop 
+n O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Void) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 
+(Bind Void) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H13: (drop n O x0 (CHead x2 (Bind Void) 
+x3))).(\lambda (H14: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Bind Void) x1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Bind 
+Void) n x0 (CHead x2 (Bind Void) x3) H13 x1) H14)))))) H12)) H11)) c2 H9))))) 
+H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda 
+(H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def 
+(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda 
+(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c)) 
+(or3 (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x: 
+C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g x 
+c)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or3 (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H10 \def (H c d1 u1 H6 g x 
+H9) in (or3_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H11: (ex2 C 
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 
+(Bind Abbr) u1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro0 (ex2 C (\lambda 
+(d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) 
+x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u1) H12 t) H13))))) H11)) 
+(\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 
+(Bind Abst) x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 
+x1 (asucc g x2))).(\lambda (H15: (arity g d1 u1 x2)).(or3_intro1 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n 
+x (CHead x0 (Bind Abst) x1) H12 t) H13 H14 H15))))))))) H11)) (\lambda (H11: 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind 
+C T (\lambda (d2: C).(\lambda (u2: T).(drop n O x (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H12: (drop n O x (CHead x0 (Bind Void) 
+x1))).(\lambda (H13: (csuba g x0 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x 
+(Bind Void) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x0 x1 (drop_drop (Bind 
+Void) n x (CHead x0 (Bind Void) x1) H12 t) H13)))))) H11)) H10)) c2 H8)))) 
+H7))))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat 
+f) t))).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) 
+u1))).(let H_x \def (csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x in 
+(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 c))) (or3 (ex2 C (\lambda 
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 
+c)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or3 (ex2 C (\lambda 
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(drop (S n) O c0 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H8 \def (H0 d1 u1 H4 g x0 
+H7) in (or3_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O x0 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H9: (ex2 C (\lambda 
+(d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr) 
+u1))).(\lambda (H11: (csuba g x d1)).(or3_intro0 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop 
+(Flat f) n x0 (CHead x (Bind Abbr) u1) H10 x1) H11))))) H9)) (\lambda (H9: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abst) 
+x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda (H12: (arity g x2 x3 (asucc g 
+x4))).(\lambda (H13: (arity g d1 u1 x4)).(or3_intro1 (ex2 C (\lambda (d2: 
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) 
+x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x2 x3 x4 
+(drop_drop (Flat f) n x0 (CHead x2 (Bind Abst) x3) H10 x1) H11 H12 
+H13))))))))) H9)) (\lambda (H9: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O x0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 
+(Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O 
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (drop (S n) O x0 (CHead x2 
+(Bind Void) x3))).(\lambda (H11: (csuba g x2 d1)).(or3_intro2 (ex2 C (\lambda 
+(d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop (S n) O (CHead 
+x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x2 x3 (drop_drop (Flat f) n x0 
+(CHead x2 (Bind Void) x3) H10 x1) H11)))))) H9)) H8)) c2 H6))))) H5)))))) k 
+H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/fwd.ma
new file mode 100644 (file)
index 0000000..b99cd59
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/fwd.ma
@@ -0,0 +1,1024 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/defs.ma".
+
+implied rec lemma csuba_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csuba 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
+(CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csuba g c1 
+c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
+T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
+u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to ((P 
+c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to 
+(\forall (u: T).((arity g c2 u a) \to (P (CHead c1 (Bind Abst) t) (CHead c2 
+(Bind Abbr) u)))))))))))) (c: C) (c0: C) (c1: csuba g c c0) on c1: P c c0 
+\def match c1 with [(csuba_sort n) \Rightarrow (f n) | (csuba_head c2 c3 c4 k 
+u) \Rightarrow (f0 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) k u) | 
+(csuba_void c2 c3 c4 b n u1 u2) \Rightarrow (f1 c2 c3 c4 ((csuba_ind g P f f0 
+f1 f2) c2 c3 c4) b n u1 u2) | (csuba_abst c2 c3 c4 t a a0 u a1) \Rightarrow 
+(f2 c2 c3 c4 ((csuba_ind g P f f0 f1 f2) c2 c3 c4) t a a0 u a1)].
+
+lemma csuba_gen_abbr:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g 
+(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
+(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) 
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C 
+c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) 
+u) H1) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (k: 
+K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) 
+u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 
+(Bind Abbr) u) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
+u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in (\lambda (H7: (eq K k (Bind 
+Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u (\lambda (t: T).(ex2 C 
+(\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C 
+(\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C 
+c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) H2 d1 H8) in (let H10 
+\def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex_intro2 C 
+(\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abbr) u)) 
+H10))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (b: B).(\lambda (_: (not (eq B 
+b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 
+(Bind Void) u1) (CHead d1 (Bind Abbr) u))).(let H5 \def (eq_ind C (CHead c1 
+(Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
+| (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 
+with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
+True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H4) in 
+(False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C 
+c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: 
+T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) 
+t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H5) in (False_ind (ex2 C 
+(\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) H6)))))))))))) y c H0))) H))))).
+
+lemma csuba_gen_void:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
+(CHead d1 (Bind Void) u1) c) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(H: (csuba g (CHead d1 (Bind Void) u1) c)).(insert_eq C (CHead d1 (Bind Void) 
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_3 B C T (\lambda 
+(b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind b) u2))))) 
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) 
+(\lambda (y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: 
+C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T 
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind b) 
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind 
+Void) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
+(Bind Void) u1) H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Bind b) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
+(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead d1 (Bind Void) 
+u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 
+(Bind Void) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
+u) (CHead d1 (Bind Void) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 k u) (CHead d1 (Bind Void) u1) H3) in (\lambda (H7: (eq K k (Bind 
+Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_3 B C 
+T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k t) 
+(CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csuba g d1 d2)))))) (eq_ind_r K (Bind Void) (\lambda (k0: K).(ex2_3 B C T 
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) 
+(CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csuba g d1 d2)))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 
+(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H8) in (let 
+H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in 
+(ex2_3_intro B C T (\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C 
+(CHead c2 (Bind Void) u1) (CHead d2 (Bind b) u2))))) (\lambda (_: B).(\lambda 
+(d2: C).(\lambda (_: T).(csuba g d1 d2)))) Void c2 u1 (refl_equal C (CHead c2 
+(Bind Void) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 
+(CHead d1 (Bind Void) u1)) \to (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) u2))))) (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) 
+u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind Void) u0) 
+(CHead d1 (Bind Void) u1) H4) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 (Bind Void) u0) (CHead d1 (Bind Void) u1) H4) in (\lambda (H7: (eq 
+C c1 d1)).(let H8 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
+(Bind Void) u1)) \to (ex2_3 B C T (\lambda (b0: B).(\lambda (d2: C).(\lambda 
+(u3: T).(eq C c2 (CHead d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csuba g d1 d2))))))) H2 d1 H7) in (let H9 \def (eq_ind C 
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H7) in (ex2_3_intro B C T (\lambda 
+(b0: B).(\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c2 (Bind b) u2) (CHead 
+d2 (Bind b0) u3))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba 
+g d1 d2)))) b c2 u2 (refl_equal C (CHead c2 (Bind b) u2)) H9))))) 
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u1)) \to (ex2_3 B C T 
+(\lambda (b: B).(\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Bind b) 
+u2))))) (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2)))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc 
+g a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u1))).(let H6 \def (eq_ind C 
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind 
+Void) u1) H5) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind b) u2))))) 
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) 
+H6)))))))))))) y c H0))) H))))).
+
+lemma csuba_gen_abst:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g 
+(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead 
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) 
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: 
+C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead 
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
+c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind 
+Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) 
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
+(Bind Abst) u1) H1) in (False_ind (or (ex2 C (\lambda (d2: C).(eq C (CSort n) 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CSort n) (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda 
+(H3: (eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C 
+C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H5 
+\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) 
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind 
+Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 
+d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead 
+c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k t) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 
+C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) 
+(let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) 
+u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 d1 H8) 
+in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in 
+(or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind 
+Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 
+(refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) 
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C 
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba 
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b 
+Void))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind 
+Void) u0) (CHead d1 (Bind Abst) u1))).(let H5 \def (eq_ind C (CHead c1 (Bind 
+Void) u0) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 with 
+[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | 
+(Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abst) u1) H4) in (False_ind 
+(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead d2 
+(Bind Abbr) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 
+u3 a)))))) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
+(csuba g c1 c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or 
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda 
+(a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda 
+(H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 
+(Bind Abst) u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind 
+Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in (\lambda (H8: 
+(eq C c1 d1)).(let H9 \def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc 
+g a))) H3 u1 H7) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 
+u1 (asucc g a))) H9 d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: 
+C).((eq C c0 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C 
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc 
+g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
+a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g 
+c0 c2)) H1 d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind 
+Abbr) u) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead 
+c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity 
+g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
+A).(arity g d2 u2 a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 
+H10 H4)))))))) H6)))))))))))) y c H0))) H))))).
+
+lemma csuba_gen_flat:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
+(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d1 d2)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead 
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: 
+(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
+(Flat f) u1) H1) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d1 d2)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
+(csuba g c1 c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead d1 (Flat f) 
+u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 
+(Flat f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) 
+(CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 k u) (CHead d1 (Flat f) u1) H3) in (\lambda (H7: (eq K k (Flat 
+f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 k t) (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) (eq_ind_r K (Flat 
+f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
+c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d1 d2))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 
+(Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 
+(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 
+c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+(CHead c2 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d1 d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H10))) k 
+H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C 
+T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u0) (CHead d1 (Flat f) 
+u1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u0) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
+(CHead d1 (Flat f) u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u3: T).(eq C (CHead c2 (Bind b) u2) (CHead d2 (Flat f) u3)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d1 d2)))) H5))))))))))) (\lambda (c1: C).(\lambda 
+(c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 (CHead d1 (Flat 
+f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 
+d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
+(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C 
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) 
+H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
+c2 (Bind Abbr) u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d1 d2)))) H6)))))))))))) y c H0))) H)))))).
+
+lemma csuba_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: 
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y 
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
+e1 e2))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
+e2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
+(eq K k (Bind b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: 
+T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
+K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 
+H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) 
+in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C 
+(CHead c3 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
+(CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) 
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) 
+in (\lambda (H8: (eq B Void b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
+e2))))))) H2 e1 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csuba g c 
+c3)) H1 e1 H9) in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 
+(CHead e1 (Bind b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H10 Void H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 
+(Bind b) u2)) H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
+C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t: T).(\lambda (a: 
+A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (_: 
+(arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 
+(Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind 
+Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead 
+c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in 
+(\lambda (H9: (eq B Abst b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def 
+(eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let 
+H12 \def (eq_ind C c1 (\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) 
+in (let H13 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) 
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
+(_: T).(csuba g e1 e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 
+(\lambda (c: C).(csuba g c c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 
+(\lambda (b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda 
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) 
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 
+e2))))))) H13 Abst H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) 
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) 
+Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H14))))))))) H7)) 
+H6)))))))))))) y c2 H0))) H)))))).
+
+lemma csuba_gen_abst_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
+(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
+(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) 
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: 
+C).(eq C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (y: 
+C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: 
+C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C 
+c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))) (\lambda (n: 
+nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 \def 
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
+True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) u) H1) in 
+(False_ind (or (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(eq C (CSort n) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))))) H2)))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
+Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq 
+C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H5 
+\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) 
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 
+(Bind Abst) u) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C 
+c2 d1)).(eq_ind_r T u (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C 
+(CHead c1 k t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) 
+(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C 
+(CHead c1 k0 u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let 
+H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to 
+(or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g 
+c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind 
+Abst) u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) u) (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind 
+Abst) u)) H10)))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda 
+(c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 
+(Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: B).(\lambda (H3: (not 
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
+c2 (Bind b) u2) (CHead d1 (Bind Abst) u))).(let H5 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in ((let H6 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
+_ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abst) u) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead 
+d1 (Bind Abst) u) H4) in (\lambda (H8: (eq B b Abst)).(\lambda (H9: (eq C c2 
+d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abst 
+H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind 
+Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: 
+T).(eq C c1 (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 (\lambda 
+(c0: C).(csuba g c1 c0)) H1 d1 H9) in (or_intror (ex2 C (\lambda (d2: C).(eq 
+C (CHead c1 (Bind Void) u1) (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C 
+(CHead c1 (Bind Void) u1) (CHead d2 (Bind Void) u3)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: 
+C).(\lambda (u3: T).(eq C (CHead c1 (Bind Void) u1) (CHead d2 (Bind Void) 
+u3)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C 
+(CHead c1 (Bind Void) u1)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 
+(CHead d1 (Bind Abst) u)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: T).(\lambda (a: 
+A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: T).(\lambda (_: 
+(arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 
+(Bind Abst) u))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind (or 
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) H6)))))))))))) c y H0))) H))))).
+
+lemma csuba_gen_void_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c 
+(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind 
+Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: 
+(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) 
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq 
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda 
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda 
+(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C 
+c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda 
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) 
+u) H1) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind 
+Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (k: 
+K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) 
+u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 
+(Bind Void) u) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k 
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in (\lambda (H7: (eq K k (Bind 
+Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u (\lambda (t: T).(ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 (Bind Void) u))) (\lambda 
+(d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) (\lambda (k0: K).(ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 (Bind Void) u))) (\lambda 
+(d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C 
+c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H8) in (let H10 
+\def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex_intro2 C 
+(\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 (Bind Void) u))) 
+(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Void) u)) 
+H10))) k H7) u0 H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
+Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Void) u))) 
+(\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (b: B).(\lambda (H3: (not (eq 
+B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 
+(Bind b) u2) (CHead d1 (Bind Void) u))).(let H5 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in ((let H6 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead 
+_ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Void) u) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead 
+d1 (Bind Void) u) H4) in (\lambda (H8: (eq B b Void)).(\lambda (H9: (eq C c2 
+d1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Void 
+H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind 
+Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Void) u))) 
+(\lambda (d2: C).(csuba g d2 d1))))) H2 d1 H9) in (let H12 \def (eq_ind C c2 
+(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in (let H13 \def (match (H10 
+(refl_equal B Void)) in False with []) in H13))))))) H6)) H5))))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: 
+(((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c1 
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u0: 
+T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) 
+u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) 
+u0) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k 
+_) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind (ex2 C 
+(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) H6)))))))))))) c y H0))) H))))).
+
+lemma csuba_gen_abbr_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c 
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) 
+u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or3 (ex2 C (\lambda 
+(d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda 
+(c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
+(\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c0 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind 
+Abbr) u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
+(Bind Abbr) u1) H1) in (False_ind (or3 (ex2 C (\lambda (d2: C).(eq C (CSort 
+n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CSort n) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k 
+u) (CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) 
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r 
+T u1 (\lambda (t: T).(or3 (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or3 (ex2 C (\lambda (d2: 
+C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (let H9 \def (eq_ind C 
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C 
+(\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba 
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq 
+C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c1 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba 
+g c1 c0)) H1 d1 H8) in (or3_intro0 (ex2 C (\lambda (d2: C).(eq C (CHead c1 
+(Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C 
+(CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 
+(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (b: 
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) 
+u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2) 
+(CHead d1 (Bind Abbr) u1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match 
+k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c2 
+(Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in ((let H7 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c2 (Bind b) u2) (CHead d1 (Bind Abbr) u1) H4) in 
+(\lambda (H8: (eq B b Abbr)).(\lambda (H9: (eq C c2 d1)).(let H10 \def 
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Void))) H3 Abbr H8) in (let H11 
+\def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to 
+(or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u3: 
+T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u3))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u3: T).(eq C c1 (CHead d2 (Bind Void) u3)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 d1 H9) in (let H12 
+\def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H9) in (or3_intro2 
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u3: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u0) (CHead d2 
+(Bind Abst) u3))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u3: T).(\lambda (a: A).(arity g d2 u3 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C (CHead c1 (Bind 
+Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u3: T).(eq 
+C (CHead c1 (Bind Void) u0) (CHead d2 (Bind Void) u3)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))) c1 u0 (refl_equal C (CHead c1 (Bind 
+Void) u0)) H12)))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind 
+Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: 
+T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C (CHead c2 (Bind Abbr) 
+u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1) 
+H5) in (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u (\lambda (t0: 
+T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 (\lambda (c0: 
+C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 (\lambda (c0: 
+C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or3 (ex2 C (\lambda (d2: C).(eq 
+C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
+(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 
+u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H2 
+d1 H8) in (let H12 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 
+H8) in (or3_intro1 (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) 
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: 
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a0: A).(arity g d1 u1 a0))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 
+(asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 
+u1 a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) 
+H6)))))))))))) c y H0))) H))))).
+
+lemma csuba_gen_flat_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall 
+(f: F).((csuba g c (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda 
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead 
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: 
+(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq 
+C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) 
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 
+(Flat f) u1) H1) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
+(csuba g c1 c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Flat f) u1)) \to (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Flat f) 
+u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 
+(Flat f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) 
+(CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c2 k u) (CHead d1 (Flat f) u1) H3) in (\lambda (H7: (eq K k (Flat 
+f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 (\lambda (t: T).(ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k t) (CHead d2 (Flat f) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (eq_ind_r K (Flat 
+f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
+c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))))) (let H9 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead d1 
+(Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 
+(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 
+c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H10))) k 
+H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat f) u1)) \to (ex2_2 C 
+T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c2 (Bind b) u2) (CHead d1 (Flat f) u1))).(let 
+H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) 
+u1) H4) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u3: T).(eq C 
+(CHead c1 (Bind Void) u0) (CHead d2 (Flat f) u3)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 (CHead d1 (Flat f) 
+u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g 
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C 
+(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C 
+(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) 
+H5) in (False_ind (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead 
+c1 (Bind Abst) t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) H6)))))))))))) c y H0))) H)))))).
+
+lemma csuba_gen_bind_rev:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: 
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 
+y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
+e2 e1))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 
+c3)).(\lambda (H2: (((eq C c3 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 
+e1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c3 k u) 
+(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k 
+u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
+(eq K k (Bind b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: 
+T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c1 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
+K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c1 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 
+H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) 
+in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C 
+(CHead c1 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (b: 
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1))).(let 
+H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c3 
+| (CHead c _ _) \Rightarrow c])) (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) 
+v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
+_) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c3 (Bind b) u2) (CHead e1 
+(Bind b1) v1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 (Bind b) 
+u2) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B b b1)).(\lambda (H9: 
+(eq C c3 e1)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 
+Void))) H3 b1 H8) in (let H11 \def (eq_ind C c3 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H9) in (let 
+H12 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H9) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) Void c1 u1 (refl_equal 
+C (CHead c1 (Bind Void) u1)) H12))))))) H6)) H5))))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t: 
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: 
+T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) 
+u) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 
+(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def (f_equal C B 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+Abbr])])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) 
+v1) H5) in (\lambda (H9: (eq B Abbr b1)).(\lambda (H10: (eq C c3 e1)).(let 
+H11 \def (eq_ind T u (\lambda (t0: T).(arity g c3 t0 a)) H4 v1 H8) in (let 
+H12 \def (eq_ind C c3 (\lambda (c: C).(arity g c v1 a)) H11 e1 H10) in (let 
+H13 \def (eq_ind C c3 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to 
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csuba g e2 e1))))))) H2 e1 H10) in (let H14 \def (eq_ind C c3 (\lambda 
+(c: C).(csuba g c1 c)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda 
+(b: B).((eq C e1 (CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H13 
+Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda 
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t 
+(refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) H6)))))))))))) c2 y 
+H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/getl.ma
new file mode 100644 (file)
index 0000000..a8591bf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/getl.ma
@@ -0,0 +1,1160 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/drop.ma".
+
+include "basic_1A/csuba/clear.ma".
+
+include "basic_1A/getl/clear.ma".
+
+lemma csuba_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
+(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: 
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abbr) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
+(Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda 
+(H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
+(Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 
+d2))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: 
+(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 
+(Bind Abbr) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to 
+((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: 
+C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (b: B).(\lambda 
+(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 
+(Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
+(CHead d1 (Bind Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 
+(Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with 
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind 
+Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
+t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
+(\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
+C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
+T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r 
+B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abbr H10) in 
+(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind 
+Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 g c2 
+H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 (CHead 
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: 
+C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abbr) u))).(\lambda (H18: 
+(csuba g d1 x1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 
+(Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 u)) H18)))) 
+H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead 
+x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind 
+Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c 
+(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n 
+O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) \to (ex2 C 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead 
+x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x1 c2)).(let H10 
+\def (eq_ind C x1 (\lambda (c: C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) 
+(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0 
+(CHead d1 (Bind Abbr) u) (clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) 
+f t) in (let H11 \def (csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead 
+d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 
+(Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+(\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) 
+x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr g d1 x2 u 
+H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) 
+(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abbr) u))).(\lambda 
+(H16: (csuba g d1 x3)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2 
+c)) H13 (CHead x3 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl 
+O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x3 
+(getl_intro O c2 (CHead x3 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) 
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda 
+(d2: C).(csuba g d1 d2))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O 
+x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 
+c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B 
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind 
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead 
+x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: B).(\lambda (x3: 
+C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) 
+x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def 
+(csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C 
+(\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) (\lambda (e2: 
+C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: C).(\lambda (H15: 
+(csuba g (CHead x3 (Bind x2) x4) x5)).(\lambda (H16: (clear c2 x5)).(let H_x 
+\def (csuba_gen_bind g x2 x3 x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B 
+C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g 
+x3 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x6: B).(\lambda (x7: C).(\lambda 
+(x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: 
+(csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 
+(CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (ex2_ind 
+C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x9: C).(\lambda (H22: 
+(getl n x7 (CHead x9 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 
+x9)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) 
+(\lambda (d2: C).(csuba g d1 d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead 
+x9 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17)))))) H14))))))) H11)))))))) 
+i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
+lemma csuba_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) 
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear 
+x (CHead d1 (Bind Abst) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to 
+((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) 
+\to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda 
+(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) 
+(CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc 
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 
+C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (k: K).(\lambda 
+(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear 
+(CHead x0 k t) (CHead d1 (Bind Abst) u1))).(K_ind (\lambda (k0: K).((drop i O 
+c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u1)) 
+\to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i 
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) 
+t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abst) 
+u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u1) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
+in ((let H8 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
+in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abst) u1) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u1) t H6)) 
+in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
+C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
+T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r 
+B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abst H10) in 
+(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind 
+Abst) u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 
+H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
+(H17: (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda 
+(H18: (drop i O c2 (CHead x1 (Bind Abst) u1))).(\lambda (H19: (csuba g d1 
+x1)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) 
+(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)) x1 (getl_intro i c2 (CHead x1 (Bind Abst) u1) (CHead 
+x1 (Bind Abst) u1) H18 (clear_bind Abst x1 u1)) H19))))) H17)) (\lambda (H17: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) 
+x2))).(\lambda (H19: (csuba g d1 x1)).(\lambda (H20: (arity g d1 u1 (asucc g 
+x3))).(\lambda (H21: (arity g x1 x2 x3)).(or_intror (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x1 x2 x3 
+(getl_intro i c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 
+(clear_bind Abbr x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) 
+H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) 
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
+(Flat f) t)) \to (\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: 
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda 
+(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H9: (csuba g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
+C).(csuba g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) u1) 
+(clear_gen_flat f x0 (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def 
+(csuba_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead 
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x2: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) 
+x2)).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst g d1 x2 u1 
+H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead 
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
+(H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 
+(Bind Abst) u1))).(\lambda (H17: (csuba g d1 x3)).(let H18 \def (eq_ind C x2 
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) u1) H16) in 
+(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u1) c2 
+(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(eq C x2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity 
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
+A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) x4))).(\lambda (H17: (csuba 
+g d1 x3)).(\lambda (H18: (arity g d1 u1 (asucc g x5))).(\lambda (H19: (arity 
+g x3 x4 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 
+(CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (getl_intro O c2 (CHead 
+x3 (Bind Abbr) x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) 
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1: 
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g x1 c2) 
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) 
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat 
+f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x1 c2)).(let H11 \def 
+(drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) 
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat 
+f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
+(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 
+(Flat f) t))).(let H14 \def (csuba_clear_conf g x1 c2 H10 (CHead x3 (Bind x2) 
+x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x3 (Bind x2) x4) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g (CHead x3 (Bind x2) x4) 
+x5)).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind g x2 x3 x5 
+x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda 
+(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g x3 e2)))) (or (ex2 C (\lambda 
+(d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g 
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl 
+(S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x6: B).(\lambda (x7: 
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) 
+x8))).(\lambda (H19: (csuba g x3 x7)).(let H20 \def (eq_ind C x5 (\lambda (c: 
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 
+x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or 
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda 
+(H22: (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u1))) (\lambda 
+(d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: 
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(a: A).(arity g d2 u2 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7 
+(CHead x9 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x9)).(or_introl (ex2 C 
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: 
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda 
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 
+d2)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) u1) n H23) 
+H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 
+u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a)))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: 
+(getl n x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H24: (csuba g d1 
+x9)).(\lambda (H25: (arity g d1 u1 (asucc g x11))).(\lambda (H26: (arity g x9 
+x10 x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g 
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x9 x10 x11 (getl_clear_bind x6 
+c2 x7 x8 H20 (CHead x9 (Bind Abbr) x10) n H23) H24 H25 H26))))))))) H22)) 
+H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 
+H2)))) H0))))))).
+
+lemma csuba_getl_abst_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g 
+c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u))) 
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (x: 
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 
+(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) 
+(\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear 
+(CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1 (Bind Abst) u) 
+n H4 (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: 
+C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) u)) 
+\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i 
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: 
+(clear (CHead x0 k t) (CHead d1 (Bind Abst) u))).(K_ind (\lambda (k0: 
+K).((drop i O c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind 
+Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: 
+B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear 
+(CHead x0 (Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
+\Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abst | 
+(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in (\lambda (H10: (eq B 
+Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
+g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
+(Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: B).(drop 
+i O c1 (CHead x0 (Bind b0) u))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 
+(\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u))) H14 d1 H11) in (let 
+H16 \def (csuba_drop_abst_rev i c1 d1 u H15 g c2 H12) in (or_ind (ex2 C 
+(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O 
+c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (H18: (drop 
+i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H19: (csuba g x1 d1)).(or_introl 
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 (Bind Abst) 
+u) (CHead x1 (Bind Abst) u) H18 (clear_bind Abst x1 u)) H19))))) H17)) 
+(\lambda (H17: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or 
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead 
+x1 (Bind Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or_intror (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x1 
+x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead x1 (Bind Void) x2) H18 
+(clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) H8)) H7))))) (\lambda 
+(f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) t))).(\lambda (H6: 
+(clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abst) u))).(let H7 \def H5 in 
+(unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 (Flat f) t)) \to 
+(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (nat_ind (\lambda 
+(n: nat).(\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall 
+(c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: 
+(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g 
+c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead 
+x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def 
+(clear_flat x0 (CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind 
+Abst) u) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) 
+t) c2 H10 (CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba 
+g e2 (CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 
+(CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst) 
+u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2 
+u H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H15: 
+(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: 
+C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda (H17: (csuba g 
+x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead 
+x3 (Bind Abst) u) H16) in (or_introl (ex2 C (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C (\lambda 
+(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H18) 
+H17)))))) H15)) (\lambda (H15: (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C 
+x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C 
+x2 (CHead x3 (Bind Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def 
+(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) 
+in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 (Bind Void) x4) c2 
+(drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) (\lambda (n: 
+nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 (Flat f) t)) \to 
+(\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))).(\lambda (x1: 
+C).(\lambda (H9: (drop (S n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H10: (csuba g c2 x1)).(let H11 \def (drop_clear x1 (CHead x0 
+(Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 (Flat f) t))))) (or 
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl 
+(S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba 
+g d2 d1))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
+(H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead 
+x0 (Flat f) t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind 
+x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) 
+x4))) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) 
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x5: 
+C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear 
+c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def 
+H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda (d2: C).(getl (S n) 
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x6: 
+B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind 
+x6) x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda 
+(c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 
+H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or (ex2 C (\lambda (d2: C).(getl (S n) 
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C 
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind 
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) 
+u))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C (\lambda (d2: C).(getl 
+(S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abst) 
+u) n H23) H24))))) H22)) (\lambda (H22: (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(getl 
+n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g 
+d2 d1))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: 
+(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 
+d1)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u))) (\lambda (d2: C).(csuba g d2 d1))) (ex2_2 C T (\lambda (d2: C).(\lambda 
+(u2: T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda 
+(_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))) x9 x10 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind 
+Void) x10) n H23) H24)))))) H22)) H21)))))))) H17)))))) H14))))))) 
+H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
+lemma csuba_getl_abbr_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall 
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba 
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda 
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u1))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1))) 
+(\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) 
+(\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead 
+d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c 
+(CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or3 
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (n: nat).(\lambda (_: 
+(drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) 
+u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4 (\forall (c2: C).((csuba 
+g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))))) (\lambda (x0: 
+C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0 (CHead d1 (Bind Abbr) u1)) 
+\to (\forall (c2: C).((csuba g c2 c1) \to (or3 (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 
+d1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop i O c1 
+(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
+u1))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to ((clear 
+(CHead x0 k0 t) (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 
+c1) \to (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (b: 
+B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear 
+(CHead x0 (Bind b) t) (CHead d1 (Bind Abbr) u1))).(let H7 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
+\Rightarrow c])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H8 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in ((let H9 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) 
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B 
+Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba 
+g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0 
+(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: 
+B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def 
+(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1 
+H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in 
+(or3_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: 
+C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr) u1))).(\lambda (H19: 
+(csuba g x1 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1 (Bind Abbr) u1) (CHead x1 
+(Bind Abbr) u1) H18 (clear_bind Abbr x1 u1)) H19))))) H17)) (\lambda (H17: 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C 
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abst) 
+x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1 x2 (asucc g 
+x3))).(\lambda (H21: (arity g d1 u1 x3)).(or3_intro1 (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3 
+(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18 
+(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) (\lambda (H17: (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
+(d2: C).(\lambda (u2: T).(drop i O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl i c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (x1: C).(\lambda (x2: T).(\lambda (H18: (drop i O c2 (CHead x1 (Bind 
+Void) x2))).(\lambda (H19: (csuba g x1 d1)).(or3_intro2 (ex2 C (\lambda (d2: 
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))) x1 x2 (getl_intro i c2 (CHead x1 (Bind Void) x2) (CHead 
+x1 (Bind Void) x2) H18 (clear_bind Void x1 x2)) H19)))))) H17)) H16)))))))))) 
+H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f) 
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) 
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0 
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or3 (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl i c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop 
+n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 
+(ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))))))))) (\lambda (x1: C).(\lambda (H8: 
+(drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g 
+c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead 
+x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def 
+(clear_flat x0 (CHead d1 (Bind Abbr) u1) (clear_gen_flat f x0 (CHead d1 (Bind 
+Abbr) u1) t H6) f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) 
+t) c2 H10 (CHead d1 (Bind Abbr) u1) H_y) in (ex2_ind C (\lambda (e2: 
+C).(csuba g e2 (CHead d1 (Bind Abbr) u1))) (\lambda (e2: C).(clear c2 e2)) 
+(or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: 
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x2: 
+C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr) u1))).(\lambda (H13: 
+(clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2 u1 H12) in (let H14 
+\def H_x in (or3_ind (ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x2 (CHead 
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C (\lambda 
+(d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1)) (or3 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (x3: 
+C).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abbr) u1))).(\lambda (H17: (csuba 
+g x3 d1)).(let H18 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead 
+x3 (Bind Abbr) u1) H16) in (or3_intro0 (ex2 C (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
+(ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda 
+(d2: C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) u1) c2 
+(drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) 
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
+A).(\lambda (H16: (eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba 
+g x3 d1)).(\lambda (H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity 
+g d1 u1 x5)).(let H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 
+(CHead x3 (Bind Abst) x4) H16) in (or3_intro1 (ex2 C (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T 
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst) 
+x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) (\lambda (H15: (ex2_2 
+C T (\lambda (d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T (\lambda 
+(d2: C).(\lambda (u2: T).(eq C x2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind 
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 
+(Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) 
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind 
+Void) x4))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C x2 
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Void) x4) H16) in 
+(or3_intro2 (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) 
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda 
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda 
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro C T (\lambda 
+(d2: C).(\lambda (u2: T).(getl O c2 (CHead d2 (Bind Void) u2)))) (\lambda 
+(d2: C).(\lambda (_: T).(csuba g d2 d1))) x3 x4 (getl_intro O c2 (CHead x3 
+(Bind Void) x4) c2 (drop_refl c2) H18) H17))))))) H15)) H14)))))) H11)))))))) 
+(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0 
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or3 (ex2 C (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n 
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl n c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O x1 
+(CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2 x1)).(let 
+H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B C T 
+(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind b) 
+v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead x0 
+(Flat f) t))))) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 
+(Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 
+\def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind 
+C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2: 
+C).(clear c2 e2)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind 
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: 
+(clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let 
+H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda 
+(v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csuba g e2 x3)))) (or3 (ex2 C (\lambda (d2: C).(getl (S 
+n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: 
+T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6) x8))).(\lambda (H19: (csuba g 
+x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead 
+x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13 x7 H19) in (or3_ind (ex2 C 
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda 
+(_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) (\lambda (d2: 
+C).(\lambda (_: T).(csuba g d2 d1)))) (or3 (ex2 C (\lambda (d2: C).(getl (S 
+n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C 
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead 
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: 
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x7 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind C 
+(\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) (or3 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda 
+(x9: C).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abbr) u1))).(\lambda (H24: 
+(csuba g x9 d1)).(or3_intro0 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex_intro2 C 
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: 
+C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) 
+u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) 
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda 
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: 
+A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: 
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: 
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or3 (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: 
+A).(\lambda (H23: (getl n x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: 
+(csuba g x9 d1)).(\lambda (H25: (arity g x9 x10 (asucc g x11))).(\lambda 
+(H26: (arity g d1 u1 x11)).(or3_intro1 (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g 
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 
+u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead 
+d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) 
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20 
+(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) (\lambda (H22: 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))).(ex2_2_ind C T 
+(\lambda (d2: C).(\lambda (u2: T).(getl n x7 (CHead d2 (Bind Void) u2)))) 
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) (or3 (ex2 C (\lambda (d2: 
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S 
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda 
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: 
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(arity g d1 u1 a))))) (ex2_2 C T (\lambda (d2: C).(\lambda (u2: 
+T).(getl (S n) c2 (CHead d2 (Bind Void) u2)))) (\lambda (d2: C).(\lambda (_: 
+T).(csuba g d2 d1))))) (\lambda (x9: C).(\lambda (x10: T).(\lambda (H23: 
+(getl n x7 (CHead x9 (Bind Void) x10))).(\lambda (H24: (csuba g x9 
+d1)).(or3_intro2 (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) 
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: 
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) 
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) 
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g 
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) 
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind 
+Void) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) (ex2_2_intro 
+C T (\lambda (d2: C).(\lambda (u2: T).(getl (S n) c2 (CHead d2 (Bind Void) 
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) x9 x10 
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Void) x10) n H23) H24)))))) 
+H22)) H21)))))))) H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) 
+x H1 H2)))) H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csuba/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csuba/props.ma
new file mode 100644 (file)
index 0000000..145b0f1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csuba/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csuba/defs.ma".
+
+include "basic_1A/C/fwd.ma".
+
+lemma csuba_refl:
+ \forall (g: G).(\forall (c: C).(csuba g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0)) 
+(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/arity.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/arity.ma
new file mode 100644 (file)
index 0000000..7871555
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/arity.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/csuba.ma".
+
+lemma csubc_arity_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
+(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t 
+a)).(csuba_arity g c1 t a H0 c2 (csubc_csuba g c1 c2 H)))))))).
+
+lemma csubc_arity_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to 
+((csubv c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c2 t a) \to 
+(arity g c1 t a))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(\lambda (H0: (csubv c1 c2)).(\lambda (t: T).(\lambda (a: A).(\lambda 
+(H1: (arity g c2 t a)).(csuba_arity_rev g c2 t a H1 c1 (csubc_csuba g c1 c2 
+H) H0)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/clear.ma
new file mode 100644 (file)
index 0000000..5650cc9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/clear.ma
@@ -0,0 +1,167 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma csubc_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall 
+(c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda 
+(e2: C).(csubc g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (H: (clear c1 
+e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c 
+c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0 
+e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2: 
+C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def 
+(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or3_ind (ex2 
+C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g 
+e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq 
+C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda 
+(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
+g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda (b0: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) 
+(\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C 
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e 
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda 
+(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
+C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 
+(CHead x (Bind b) u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x 
+(Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda 
+(e2: C).(csubc g (CHead e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: 
+C).(clear (CHead x (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind 
+b) u) e2)) (CHead x (Bind b) u) (clear_bind b x u) (csubc_head g e x H4 (Bind 
+b) u)) c2 H3)))) H2)) (\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda 
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind 
+b) (Bind Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda 
+(H5: (csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: 
+(sc3 g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 
+C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) 
+u) e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 with [(Bind 
+b0) \Rightarrow b0 | (Flat _) \Rightarrow b])) (Bind b) (Bind Abst) H3) in 
+(eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x0 
+(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda 
+(e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead x0 (Bind Abbr) x1) 
+(clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2 H6 x1 H7)) b H8)) c2 
+H4))))))))) H2)) (\lambda (H2: (ex4_3 B C T (\lambda (b0: B).(\lambda (c3: 
+C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) (Bind Void))))) (\lambda 
+(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))))).(ex4_3_ind B C T 
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind 
+b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind b) 
+(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B 
+b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e 
+c3)))) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g 
+(CHead e (Bind b) u) e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H3: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda (H4: (eq K (Bind 
+b) (Bind Void))).(\lambda (H5: (not (eq B x0 Void))).(\lambda (H6: (csubc g e 
+x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: 
+C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))) (let 
+H7 \def (f_equal K B (\lambda (e0: K).(match e0 with [(Bind b0) \Rightarrow 
+b0 | (Flat _) \Rightarrow b])) (Bind b) (Bind Void) H4) in (eq_ind_r B Void 
+(\lambda (b0: B).(ex2 C (\lambda (e2: C).(clear (CHead x1 (Bind x0) x2) e2)) 
+(\lambda (e2: C).(csubc g (CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda 
+(e2: C).(clear (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g (CHead 
+e (Bind Void) u) e2)) (CHead x1 (Bind x0) x2) (clear_bind x0 x1 x2) 
+(csubc_void g e x1 H6 x0 H5 u x2)) b H7)) c2 H3)))))))) H2)) H1)))))))) 
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: 
+((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) 
+(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H_x 
+\def (csubc_gen_head_l g e c2 u (Flat f) H2) in (let H3 \def H_x in (or3_ind 
+(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: 
+C).(csubc g e c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
+A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: 
+C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3))))) (ex2 C (\lambda (e2: 
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (H4: (ex2 C 
+(\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e 
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u))) (\lambda 
+(c3: C).(csubc g e c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: 
+C).(csubc g c e2))) (\lambda (x: C).(\lambda (H5: (eq C c2 (CHead x (Flat f) 
+u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C (CHead x (Flat f) u) (\lambda 
+(c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c 
+e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda 
+(e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c e2)) (ex2 C (\lambda (e2: 
+C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2: C).(csubc g c e2))) 
+(\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda (H9: (csubc g c 
+x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda 
+(e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9)))) H7))) c2 H5)))) 
+H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: 
+A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda 
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda 
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2 
+e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind Abst))).(\lambda (H6: 
+(eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (csubc g e x0)).(\lambda 
+(_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C 
+(CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 
+e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10 \def (eq_ind K (Flat f) 
+(\lambda (ee: K).(match ee with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])) I (Bind Abst) H5) in (False_ind (ex2 C (\lambda (e2: 
+C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g c e2))) 
+H10)) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda (b: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind Void))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))))).(ex4_3_ind B C T 
+(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Flat f) (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g e c3)))) 
+(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) 
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H5: (eq C c2 
+(CHead x1 (Bind x0) x2))).(\lambda (H6: (eq K (Flat f) (Bind Void))).(\lambda 
+(_: (not (eq B x0 Void))).(\lambda (_: (csubc g e x1)).(eq_ind_r C (CHead x1 
+(Bind x0) x2) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2)) 
+(\lambda (e2: C).(csubc g c e2)))) (let H9 \def (eq_ind K (Flat f) (\lambda 
+(ee: K).(match ee with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])) I (Bind Void) H6) in (False_ind (ex2 C (\lambda (e2: C).(clear (CHead 
+x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g c e2))) H9)) c2 H5)))))))) 
+H4)) H3))))))))))) c1 e1 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/csuba.ma
new file mode 100644 (file)
index 0000000..48b71ed
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/csuba.ma
@@ -0,0 +1,37 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/fwd.ma".
+
+include "basic_1A/sc3/props.ma".
+
+lemma csubc_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba 
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1 
+c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
+(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
+(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: 
+B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v: 
+T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w: 
+T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g 
+c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/defs.ma
new file mode 100644 (file)
index 0000000..e28634e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/defs.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sc3/defs.ma".
+
+inductive csubc (g: G): C \to (C \to Prop) \def
+| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
+| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
+| csubc_void: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubc g 
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall 
+(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g 
+a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) 
+w))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop.ma
new file mode 100644 (file)
index 0000000..995cfee
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop.ma
@@ -0,0 +1,466 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/fwd.ma".
+
+include "basic_1A/sc3/props.ma".
+
+lemma csubc_drop_conf_O:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h 
+O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
+C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1: 
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H: 
+(drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n) 
+c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda 
+(e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1: 
+(eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c: 
+C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c 
+e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: 
+C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2)))) 
+(drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1: 
+C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2) 
+\to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall 
+(c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c 
+k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind 
+C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) 
+(\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O 
+c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1 
+(drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0: 
+(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t) 
+c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g 
+e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2: 
+C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l 
+g c c2 t k H2) in (let H3 \def H_x in (or3_ind (ex2 C (\lambda (c3: C).(eq C 
+c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda 
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c c3))))) 
+(ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 
+e2))) (\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) 
+(\lambda (c3: C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 
+(CHead c3 k t))) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: 
+C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: 
+C).(\lambda (H5: (eq C c2 (CHead x k t))).(\lambda (H6: (csubc g c 
+x)).(eq_ind_r C (CHead x k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop 
+(S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k 
+n) (drop_gen_drop k c e1 t n H1) x H6) in (let H7 \def H_x0 in (ex2_ind C 
+(\lambda (e2: C).(drop (r k n) O x e2)) (\lambda (e2: C).(csubc g e1 e2)) 
+(ex2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda (e2: 
+C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H8: (drop (r k n) O x 
+x0)).(\lambda (H9: (csubc g e1 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) 
+O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 
+H8 t) H9)))) H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c 
+t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 
+C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K k 
+(Bind Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda 
+(H7: (csubc g c x0)).(\lambda (_: (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 
+g x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C 
+(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) 
+(let H10 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) 
+(drop_gen_drop k c e1 t n H1) (Bind Abst) H5) in (let H11 \def (eq_ind K k 
+(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g 
+(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda 
+(e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H5) in (let H_x0 \def (H e1 (r 
+(Bind Abst) n) H10 x0 H7) in (let H12 \def H_x0 in (ex2_ind C (\lambda (e2: 
+C).(drop n O x0 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: 
+C).(drop (S n) O (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 
+e2))) (\lambda (x: C).(\lambda (H13: (drop n O x0 x)).(\lambda (H14: (csubc g 
+e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) 
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x H13 
+x1) H14)))) H12))))) c2 H6))))))))) H4)) (\lambda (H4: (ex4_3 B C T (\lambda 
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c 
+c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c c3)))) (ex2 C (\lambda (e2: C).(drop (S n) O c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: B).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(\lambda 
+(H6: (eq K k (Bind Void))).(\lambda (_: (not (eq B x0 Void))).(\lambda (H8: 
+(csubc g c x1)).(eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex2 C 
+(\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2: C).(csubc g e1 e2)))) 
+(let H9 \def (eq_ind K k (\lambda (k0: K).(drop (r k0 n) O c e1)) 
+(drop_gen_drop k c e1 t n H1) (Bind Void) H6) in (let H10 \def (eq_ind K k 
+(\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall (c3: C).((csubc g 
+(CHead c k0 t) c3) \to (ex2 C (\lambda (e2: C).(drop n O c3 e2)) (\lambda 
+(e2: C).(csubc g e1 e2))))))) H0 (Bind Void) H6) in (let H_x0 \def (H e1 (r 
+(Bind Void) n) H9 x1 H8) in (let H11 \def H_x0 in (ex2_ind C (\lambda (e2: 
+C).(drop n O x1 e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: 
+C).(drop (S n) O (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: C).(csubc g e1 
+e2))) (\lambda (x: C).(\lambda (H12: (drop n O x1 x)).(\lambda (H13: (csubc g 
+e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x1 (Bind x0) x2) 
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind x0) n x1 x H12 x2) 
+H13)))) H11))))) c2 H5)))))))) H4)) H3)))))))) h))))))) c1)).
+
+lemma drop_csubc_trans:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g 
+(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c 
+c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 
+e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k 
+t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) 
+c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k 
+x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: 
+C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r 
+T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def 
+(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C 
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g 
+(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g 
+a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g x0 c3))))) (ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) 
+(\lambda (H10: (ex2 C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda 
+(c3: C).(csubc g x0 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k 
+x1))) (\lambda (c3: C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) 
+(\lambda (x: C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: 
+(csubc g x0 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r 
+k n) x1)) c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def 
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: 
+C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) 
+(\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: 
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c 
+x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r 
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k 
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 
+x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) 
+(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g 
+(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C 
+e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14: 
+(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C 
+(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S 
+n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) 
+(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n 
+(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: 
+C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))))))) 
+H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r 
+k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: 
+K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) 
+(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0 
+\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind 
+C (\lambda (c1: C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c c1)) (ex2 C 
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: 
+C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c1))) 
+(\lambda (x: C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g c 
+x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) 
+x3))) (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) 
+n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3)) (drop_skip_bind h n x x2 H19 
+Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind Abst) n) x1) x4 (sc3_lift g 
+(asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17) (lift h n x3) (sc3_lift g 
+x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda 
+(H10: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C e1 
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g x0 c3)))))).(ex4_3_ind B C T (\lambda (b: B).(\lambda (c3: 
+C).(\lambda (v2: T).(eq C e1 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g x0 c3)))) (ex2 C (\lambda (c1: 
+C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) 
+x1)) c1))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: 
+(eq C e1 (CHead x3 (Bind x2) x4))).(\lambda (H12: (eq K k (Bind 
+Void))).(\lambda (H13: (not (eq B x2 Void))).(\lambda (H14: (csubc g x0 
+x3)).(eq_ind_r C (CHead x3 (Bind x2) x4) (\lambda (c0: C).(ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r 
+k n) x1)) c1)))) (let H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: 
+nat).((drop h0 n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to 
+(\forall (e3: C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: 
+C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) 
+x1)) c1)))))))) H8 (Bind Void) H12) in (let H16 \def (eq_ind K k (\lambda 
+(k0: K).(drop h (r k0 n) c x0)) H5 (Bind Void) H12) in (eq_ind_r K (Bind 
+Void) (\lambda (k0: K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 
+(Bind x2) x4))) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) 
+c1)))) (let H_x0 \def (H x0 (r (Bind Void) n) h H16 x3 H14) in (let H17 \def 
+H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 x3)) (\lambda (c1: C).(csubc 
+g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) x4))) 
+(\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) n) x1)) 
+c1))) (\lambda (x: C).(\lambda (H18: (drop h n x x3)).(\lambda (H19: (csubc g 
+c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x3 (Bind x2) 
+x4))) (\lambda (c1: C).(csubc g (CHead c (Bind Void) (lift h (r (Bind Void) 
+n) x1)) c1)) (CHead x (Bind x2) (lift h n x4)) (drop_skip_bind h n x x3 H18 
+x2 x4) (csubc_void g c x H19 x2 H13 (lift h (r (Bind Void) n) x1) (lift h n 
+x4)))))) H17))) k H12))) e1 H11)))))))) H10)) H9))) t H4))))))))) 
+(drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
+lemma csubc_drop_conf_rev:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: 
+C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda 
+(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda 
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda 
+(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat 
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: 
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: 
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 
+(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: 
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def 
+(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C 
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) 
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 
+c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) 
+e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h 
+n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall 
+(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O 
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 
+\def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t) 
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O 
+O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) 
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to 
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop 
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 
+e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in 
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) 
+(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda 
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C 
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c 
+k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) 
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda 
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k 
+t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) 
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: 
+C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) 
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda 
+(c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 
+(\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to 
+(\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 
+e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) 
+H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 
+n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 
+k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc 
+g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h 
+(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) 
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def 
+(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or3_ind (ex2 C 
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
+x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k 
+(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
+x0 x1))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq 
+C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: 
+T).(csubc g c1 x0))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C 
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1 
+x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: 
+C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x: 
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x 
+x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: 
+C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k 
+n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in 
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g 
+c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2: 
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2 
+c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda 
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r 
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k 
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v))))) 
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda 
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2 
+C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead 
+c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2 
+(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g 
+(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2 
+(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 
+c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let 
+H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c 
+k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 
+(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) 
+H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 
+(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda 
+(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc 
+g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind 
+Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1: 
+C).(drop h n c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: 
+C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 
+(CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: 
+C).(\lambda (H19: (drop h n x x2)).(\lambda (H20: (csubc g x c)).(ex_intro2 C 
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: 
+C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x 
+(Bind Abst) (lift h n x3)) (drop_skip_bind h n x x2 H19 Abst x3) (csubc_abst 
+g x c H20 (lift h n x3) x4 (sc3_lift g (asucc g x4) x2 x3 H14 x h n H19) 
+(lift h (r (Bind Abbr) n) x1) (sc3_lift g x4 x0 x1 H15 c h (r (Bind Abbr) n) 
+H17)))))) H18))) k H11))) e1 H12))))))))) H10)) (\lambda (H10: (ex4_3 B C T 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C e1 (CHead c1 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
+x0)))))).(ex4_3_ind B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: 
+T).(eq C e1 (CHead c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c1: 
+C).(\lambda (_: T).(csubc g c1 x0)))) (ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) 
+(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H11: (eq C e1 
+(CHead x3 (Bind Void) x4))).(\lambda (H12: (eq K k (Bind x2))).(\lambda (H13: 
+(not (eq B x2 Void))).(\lambda (H14: (csubc g x3 x0)).(eq_ind_r C (CHead x3 
+(Bind Void) x4) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1 
+c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let 
+H15 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c 
+k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 
+(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda 
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind x2) 
+H12) in (let H16 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 
+(Bind x2) H12) in (eq_ind_r K (Bind x2) (\lambda (k0: K).(ex2 C (\lambda (c1: 
+C).(drop h (S n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 
+(CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind x2) n) h 
+H16 x3 H14) in (let H17 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop h n c1 
+x3)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) 
+c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind 
+x2) (lift h (r (Bind x2) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h n 
+x x3)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S 
+n) c1 (CHead x3 (Bind Void) x4))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind 
+x2) (lift h (r (Bind x2) n) x1)))) (CHead x (Bind Void) (lift h n x4)) 
+(drop_skip_bind h n x x3 H18 Void x4) (csubc_void g x c H19 x2 H13 (lift h n 
+x4) (lift h (r (Bind x2) n) x1)))))) H17))) k H12))) e1 H11)))))))) H10)) 
+H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop1.ma
new file mode 100644 (file)
index 0000000..18a783c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/drop1.ma
@@ -0,0 +1,86 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/drop.ma".
+
+lemma drop1_csubc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 
+e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def 
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: 
+C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: 
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) 
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n 
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def 
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda 
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))) 
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x 
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C 
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C 
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 
+c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g 
+x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def 
+H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: 
+C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) 
+(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0 
+x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1 
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0 
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
+
+lemma csubc_drop1_conf_rev:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: 
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C 
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 
+e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def 
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c: 
+C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 
+e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: 
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) 
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n 
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def 
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda 
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))) 
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x 
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C 
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C 
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 
+c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g 
+x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8 
+\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1: 
+C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) 
+(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0 
+x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1 
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0 
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/fwd.ma
new file mode 100644 (file)
index 0000000..8725216
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/fwd.ma
@@ -0,0 +1,664 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/defs.ma".
+
+implied rec lemma csubc_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubc 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (v: T).(P (CHead c1 k v) 
+(CHead c2 k v))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubc g c1 
+c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
+T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
+u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to ((P 
+c1 c2) \to (\forall (v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to 
+(\forall (w: T).((sc3 g a c2 w) \to (P (CHead c1 (Bind Abst) v) (CHead c2 
+(Bind Abbr) w)))))))))))) (c: C) (c0: C) (c1: csubc g c c0) on c1: P c c0 
+\def match c1 with [(csubc_sort n) \Rightarrow (f n) | (csubc_head c2 c3 c4 k 
+v) \Rightarrow (f0 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) k v) | 
+(csubc_void c2 c3 c4 b n u1 u2) \Rightarrow (f1 c2 c3 c4 ((csubc_ind g P f f0 
+f1 f2) c2 c3 c4) b n u1 u2) | (csubc_abst c2 c3 c4 v a s0 w s1) \Rightarrow 
+(f2 c2 c3 c4 ((csubc_ind g P f f0 f1 f2) c2 c3 c4) v a s0 w s1)].
+
+lemma csubc_gen_sort_l:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g 
+(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e with [(CSort n1) \Rightarrow n1 | 
+(CHead _ _ _) \Rightarrow n0])) (CSort n0) (CSort n) H1) in (eq_ind_r nat n 
+(\lambda (n1: nat).(eq C (CSort n1) (CSort n1))) (refl_equal C (CSort n)) n0 
+H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 
+c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 c1)))).(\lambda (k: 
+K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v) (CSort n))).(let H4 
+\def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
+(False_ind (eq C (CHead c2 k v) (CHead c1 k v)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 
+(CSort n)) \to (eq C c2 c1)))).(\lambda (b: B).(\lambda (_: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind 
+Void) u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c2 (Bind b) 
+u2) (CHead c1 (Bind Void) u1)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C 
+c2 c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
+c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c2 (Bind Abbr) 
+w) (CHead c1 (Bind Abst) v)) H6)))))))))))) y x H0))) H)))).
+
+lemma csubc_gen_head_l:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k: 
+K).((csubc g (CHead c1 k v) x) \to (or3 (ex2 C (\lambda (c2: C).(eq C x 
+(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w))))) 
+(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
+(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead c2 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
+c2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k: 
+K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v) 
+(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or3 (ex2 C (\lambda (c2: 
+C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind 
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C x (CHead 
+c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda 
+(c2: C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind 
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C c0 
+(CHead c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: 
+T).(csubc g c1 c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) 
+(CHead c1 k v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I 
+(CHead c1 k v) H1) in (False_ind (or3 (ex2 C (\lambda (c2: C).(eq C (CSort n) 
+(CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2 (Bind Abbr) 
+w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
+(\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B 
+C T (\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(eq C (CSort n) (CHead 
+c2 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c2: C).(\lambda (_: T).(csubc g c1 
+c2)))))) H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: 
+(eq C (CHead c0 k0 v0) (CHead c1 k v))).(let H4 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow 
+c])) (CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v0 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: 
+(eq K k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or3 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k0 t) (CHead c3 
+(Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k 
+(Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3))))))) (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C 
+(CHead c2 k1 v) (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead 
+c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc 
+g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g 
+a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 
+w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+(CHead c2 k1 v) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))))) (let H9 \def (eq_ind C c0 (\lambda 
+(c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 
+(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c 
+c2)) H1 c1 H8) in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
+(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr) 
+w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B 
+C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 k v) 
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) 
+(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 
+k v)) H10)))) k0 H7) v0 H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda 
+(c2: C).(\lambda (H1: (csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k 
+v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind Void))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3))))))))).(\lambda (b: 
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c0 (Bind Void) u1) (CHead c1 k v))).(let H5 
+\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Void) u1) (CHead c1 k v) H4) 
+in ((let H6 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind 
+Void) u1) (CHead c1 k v) H4) in ((let H7 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c0 (Bind Void) u1) (CHead c1 k v) H4) in (\lambda (H8: (eq K (Bind 
+Void) k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead 
+c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind 
+b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H2 c1 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c 
+c2)) H1 c1 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1 
+(CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda 
+(b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b0) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
+Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H10 (Bind Void) H8) in (eq_ind K (Bind Void) (\lambda (k0: K).(or3 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) u2) (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (_: A).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T 
+(\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) 
+u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k0 (Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))))) (or3_intro2 (ex2 C (\lambda (c3: C).(eq C (CHead c2 
+(Bind b) u2) (CHead c3 (Bind Void) v))) (\lambda (c3: C).(csubc g c1 c3))) 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind 
+Void) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C 
+(CHead c2 (Bind b) u2) (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda 
+(_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: 
+T).(\lambda (a: A).(sc3 g a c3 w))))) (ex4_3 B C T (\lambda (b0: B).(\lambda 
+(c3: C).(\lambda (v2: T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) 
+(Bind Void))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B 
+b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3))))) (ex4_3_intro B C T (\lambda (b0: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C (CHead c2 (Bind b) u2) (CHead c3 (Bind b0) v2))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Void) (Bind Void))))) (\lambda 
+(b0: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 c3)))) b c2 u2 (refl_equal C 
+(CHead c2 (Bind b) u2)) (refl_equal K (Bind Void)) H3 H11)) k H8))))))) H6)) 
+H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0 
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 
+(CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: 
+(sc3 g (asucc g a) c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 
+w)).(\lambda (H5: (eq C (CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 
+\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | 
+(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) 
+in ((let H7 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind 
+Abst) v0) (CHead c1 k v) H5) in ((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind 
+Abst) k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda 
+(t: T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 
+(\lambda (c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind 
+C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) 
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind 
+Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
+c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 
+w0))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+c2 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c1 c3)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda 
+(c: C).(csubc g c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda 
+(k0: K).((eq C c1 (CHead c1 k0 v)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c2 
+(CHead c3 k0 v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda 
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda 
+(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) (ex4_3 B C T 
+(\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C c2 (CHead c3 (Bind b) 
+v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K k0 (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c1 
+c3)))))))) H13 (Bind Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or3 
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) 
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
+w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead 
+c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k0 (Bind Void))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))))) (or3_intro1 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst) v))) (\lambda (c3: 
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: 
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) 
+w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) 
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))) 
+(ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C (CHead 
+c2 (Bind Abbr) w) (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c1 c3))))) (ex5_3_intro C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) 
+(CHead c3 (Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g 
+(asucc g a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 
+g a0 c3 w0)))) c2 w a (refl_equal K (Bind Abst)) (refl_equal C (CHead c2 
+(Bind Abbr) w)) H14 H12 H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) 
+H)))))).
+
+lemma csubc_gen_sort_r:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to 
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x 
+(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda 
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0)))) 
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def 
+(f_equal C nat (\lambda (e: C).(match e with [(CSort n1) \Rightarrow n1 | 
+(CHead _ _ _) \Rightarrow n0])) (CSort n0) (CSort n) H1) in (eq_ind_r nat n 
+(\lambda (n1: nat).(eq C (CSort n1) (CSort n1))) (refl_equal C (CSort n)) n0 
+H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubc g c1 
+c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1 c2)))).(\lambda (k: 
+K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v) (CSort n))).(let H4 
+\def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
+(False_ind (eq C (CHead c1 k v) (CHead c2 k v)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 
+(CSort n)) \to (eq C c1 c2)))).(\lambda (b: B).(\lambda (_: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c2 (Bind 
+b) u2) (CSort n))).(let H5 \def (eq_ind C (CHead c2 (Bind b) u2) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H4) in (False_ind (eq C (CHead c1 (Bind Void) 
+u1) (CHead c2 (Bind b) u2)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C 
+c1 c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1 
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead 
+c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H5) in (False_ind (eq C (CHead c1 (Bind Abst) 
+v) (CHead c2 (Bind Abbr) w)) H6)))))))))))) x y H0))) H)))).
+
+lemma csubc_gen_head_r:
+ \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k: 
+K).((csubc g x (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c1: C).(eq C x 
+(CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v))))) 
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda 
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k: 
+K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w) 
+(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or3 (ex2 C (\lambda (c1: 
+C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead 
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
+c2))))))) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda 
+(c1: C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C 
+T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind 
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead 
+c1 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 
+c2))))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k 
+w))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c2 k w) H1) 
+in (False_ind (or3 (ex2 C (\lambda (c1: C).(eq C (CSort n) (CHead c1 k w))) 
+(\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C (CSort n) (CHead c1 (Bind Abst) v))))) (\lambda (c1: 
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda (c1: 
+C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda 
+(_: B).(\lambda (c1: C).(\lambda (v1: T).(eq C (CSort n) (CHead c1 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c1: C).(\lambda (_: T).(csubc g c1 c2)))))) H2)))) 
+(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda 
+(H2: (((eq C c0 (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 
+(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T 
+(\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 
+c2))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0 
+v) (CHead c2 k w))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 v) 
+(CHead c2 k w) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
+k0 v) (CHead c2 k w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 
+v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 
+c2)).(eq_ind_r T w (\lambda (t: T).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead 
+c1 k0 t) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k0 t) 
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 
+g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 
+g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: 
+T).(eq C (CHead c1 k0 t) (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))) (eq_ind_r K k 
+(\lambda (k1: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3 
+k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind 
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 k1 w) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (let H9 \def (eq_ind C c0 (\lambda 
+(c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 
+(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind 
+Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind 
+b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) 
+(\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 
+H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) 
+in (or3_intro0 (ex2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) 
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (_: A).(eq C (CHead c1 k w) (CHead c3 (Bind Abst) v0))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 k w) 
+(CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2))))) (ex_intro2 C (\lambda (c3: C).(eq C (CHead c1 k w) (CHead c3 k 
+w))) (\lambda (c3: C).(csubc g c3 c2)) c1 (refl_equal C (CHead c1 k w)) 
+H10)))) k0 H7) v H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c0: 
+C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) 
+\to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (b: 
+B).(\lambda (H3: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c0 (Bind b) u2) (CHead c2 k w))).(let H5 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead 
+c _ _) \Rightarrow c])) (CHead c0 (Bind b) u2) (CHead c2 k w) H4) in ((let H6 
+\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind 
+b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind b) u2) (CHead c2 k w) 
+H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind b) u2) (CHead 
+c2 k w) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c0 
+c2)).(let H10 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to 
+(or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b0))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H9) in (let 
+H11 \def (eq_ind C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H9) in (let H12 
+\def (eq_ind_r K k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 
+C (\lambda (c3: C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 
+c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 
+(CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g 
+(asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a 
+c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq 
+C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda 
+(_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: 
+T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g c3 c2)))))))) H10 (Bind b) H8) in (eq_ind K (Bind b) (\lambda 
+(k0: K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Void) u1) (CHead 
+c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: 
+C).(\lambda (v: T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 
+(Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g 
+c3 c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) 
+c3 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 (Bind Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda 
+(_: C).(\lambda (_: T).(eq K k0 (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro2 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind b) w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K (Bind b) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Void) u1) (CHead c3 (Bind Abst) 
+v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C 
+T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind 
+Void) u1) (CHead c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex4_3_intro B C T (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead c1 (Bind Void) u1) (CHead 
+c3 (Bind Void) v1))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(eq K 
+(Bind b) (Bind b0))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b0 Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2)))) b c1 u1 (refl_equal C (CHead c1 (Bind Void) u1)) (refl_equal K 
+(Bind b)) H3 H11)) k H8))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda 
+(c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda (H2: (((eq C c0 (CHead c2 k 
+w)) \to (or3 (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v: T).(\lambda (_: 
+A).(eq C c1 (CHead c3 (Bind Abst) v))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v: 
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda 
+(c3: C).(\lambda (v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: 
+B).(\lambda (c3: C).(\lambda (_: T).(csubc g c3 c2))))))))).(\lambda (v: 
+T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0: 
+T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr) 
+w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind 
+Abbr) w0) (CHead c2 k w) H5) in ((let H7 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) 
+\Rightarrow k0])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow w0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) 
+in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq C c0 c2)).(let H11 
+\def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8) in (let H12 \def 
+(eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in (let H13 \def 
+(eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or3 (ex2 C 
+(\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) 
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind 
+Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead 
+c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
+A).(sc3 g a0 c2 w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda 
+(v1: T).(eq C c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2)))))))) H2 c2 H10) in (let H14 \def (eq_ind 
+C c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K 
+k (\lambda (k0: K).((eq C c2 (CHead c2 k0 w)) \to (or3 (ex2 C (\lambda (c3: 
+C).(eq C c1 (CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A 
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind 
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) 
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 
+w))))) (ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C 
+c1 (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not 
+(eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g 
+c3 c2)))))))) H13 (Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: 
+K).(or3 (ex2 C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 
+w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda 
+(_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda 
+(v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K k0 (Bind b))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))))) (or3_intro1 (ex2 C (\lambda (c3: 
+C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w))) (\lambda (c3: 
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda 
+(_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: 
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) 
+v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) 
+(\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 
+v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))) 
+(ex4_3 B C T (\lambda (_: B).(\lambda (c3: C).(\lambda (v1: T).(eq C (CHead 
+c1 (Bind Abst) v) (CHead c3 (Bind Void) v1))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abbr) (Bind b))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g c3 c2))))) (ex5_3_intro C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda 
+(c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) 
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: 
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: 
+A).(sc3 g a0 c2 w)))) c1 v a (refl_equal K (Bind Abbr)) (refl_equal C (CHead 
+c1 (Bind Abst) v)) H14 H3 H12)) k H9))))))))) H7)) H6)))))))))))) x y H0))) 
+H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/getl.ma
new file mode 100644 (file)
index 0000000..16c59d2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/getl.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/drop.ma".
+
+include "basic_1A/csubc/clear.ma".
+
+lemma csubc_getl_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i 
+c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2: 
+C).(getl i c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (e1: C).(\lambda (i: nat).(\lambda 
+(H: (getl i c1 e1)).(\lambda (c2: C).(\lambda (H0: (csubc g c1 c2)).(let H1 
+\def (getl_gen_all c1 e1 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) 
+(\lambda (e: C).(clear e e1)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H2: (drop i O c1 
+x)).(\lambda (H3: (clear x e1)).(let H_x \def (csubc_drop_conf_O g c1 x i H2 
+c2 H0) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(drop i O c2 e2)) 
+(\lambda (e2: C).(csubc g x e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
+c2 x0)).(\lambda (H6: (csubc g x x0)).(let H_x0 \def (csubc_clear_conf g x e1 
+H3 x0 H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(clear x0 e2)) 
+(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(getl i c2 e2)) 
+(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x1: C).(\lambda (H8: (clear x0 
+x1)).(\lambda (H9: (csubc g e1 x1)).(ex_intro2 C (\lambda (e2: C).(getl i c2 
+e2)) (\lambda (e2: C).(csubc g e1 e2)) x1 (getl_intro i c2 x1 x0 H5 H8) 
+H9)))) H7)))))) H4)))))) H1)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubc/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubc/props.ma
new file mode 100644 (file)
index 0000000..8daf402
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubc/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/defs.ma".
+
+include "basic_1A/sc3/props.ma".
+
+lemma csubc_refl:
+ \forall (g: G).(\forall (c: C).(csubc g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0)) 
+(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/clear.ma
new file mode 100644 (file)
index 0000000..ab3e38c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/clear.ma
@@ -0,0 +1,1170 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/props.ma".
+
+include "basic_1A/csubst0/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma csubst0_clear_O:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
+(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: 
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) 
+c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda 
+(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: 
+C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: 
+T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) 
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(let H2 \def 
+(csubst0_gen_head k c c2 t v O H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear c2 c0) 
+(\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq 
+nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (H6: 
+(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(clear c3 
+c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 
+x1)) \to (clear (CHead c k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear 
+(CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x1))).(let H9 
+\def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S 
+_) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead c (Bind b) 
+x0) c0) H9))))) (\lambda (f: F).(\lambda (H7: (clear (CHead c (Flat f) t) 
+c0)).(\lambda (H8: (eq nat O (s (Flat f) x1))).(let H9 \def (eq_ind_r nat x1 
+(\lambda (n: nat).(subst0 n v t x0)) H6 O H8) in (clear_flat c c0 
+(clear_gen_flat f c c0 t H7) f x0))))) k H1 H4) c2 H5)))))) H3)) (\lambda 
+(H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq 
+nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: 
+(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 
+c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 
+x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear 
+(CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x1))).(let H9 
+\def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S 
+_) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead x0 (Bind b) 
+t) c0) H9))))) (\lambda (f: F).(\lambda (H7: (clear (CHead c (Flat f) t) 
+c0)).(\lambda (H8: (eq nat O (s (Flat f) x1))).(let H9 \def (eq_ind_r nat x1 
+(\lambda (n: nat).(csubst0 n v c x0)) H6 O H8) in (clear_flat x0 c0 (H x0 v 
+H9 c0 (clear_gen_flat f c c0 t H7)) f t))))) k H1 H4) c2 H5)))))) H3)) 
+(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear c2 c0) (\lambda (x0: 
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat O (s k 
+x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t 
+x0)).(\lambda (H7: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda 
+(c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to 
+((eq nat O (s k0 x2)) \to (clear (CHead x1 k0 x0) c0)))) (\lambda (b: 
+B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H9: (eq nat O (s 
+(Bind b) x2))).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee with 
+[O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H9) in (False_ind 
+(clear (CHead x1 (Bind b) x0) c0) H10))))) (\lambda (f: F).(\lambda (H8: 
+(clear (CHead c (Flat f) t) c0)).(\lambda (H9: (eq nat O (s (Flat f) 
+x2))).(let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 
+O H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) 
+H6 O H9) in (clear_flat x1 c0 (H x1 v H10 c0 (clear_gen_flat f c c0 t H8)) f 
+x0)))))) k H1 H4) c2 H5)))))))) H3)) H2))))))))))) c1).
+
+lemma csubst0_clear_O_back:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to 
+(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c 
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: 
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 
+c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: 
+C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to 
+(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O 
+v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(let H2 
+\def (csubst0_gen_head k c c2 t v O H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear (CHead c 
+k t) c0) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) 
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda 
+(x1: nat).(\lambda (H4: (eq nat O (s k x1))).(\lambda (H5: (eq C c2 (CHead c 
+k x0))).(\lambda (H6: (subst0 x1 v t x0)).(let H7 \def (eq_ind C c2 (\lambda 
+(c3: C).(clear c3 c0)) H1 (CHead c k x0) H5) in (K_ind (\lambda (k0: K).((eq 
+nat O (s k0 x1)) \to ((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) 
+c0)))) (\lambda (b: B).(\lambda (H8: (eq nat O (s (Bind b) x1))).(\lambda (_: 
+(clear (CHead c (Bind b) x0) c0)).(let H10 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
+H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda (f: 
+F).(\lambda (H8: (eq nat O (s (Flat f) x1))).(\lambda (H9: (clear (CHead c 
+(Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n 
+v t x0)) H6 O H8) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H9) f t))))) 
+k H4 H7))))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) 
+(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat O (s k 
+x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c 
+x0)).(let H7 \def (eq_ind C c2 (\lambda (c3: C).(clear c3 c0)) H1 (CHead x0 k 
+t) H5) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 
+k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8: (eq 
+nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let 
+H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True 
+| (S _) \Rightarrow False])) I (S x1) H8) in (False_ind (clear (CHead c (Bind 
+b) t) c0) H10))))) (\lambda (f: F).(\lambda (H8: (eq nat O (s (Flat f) 
+x1))).(\lambda (H9: (clear (CHead x0 (Flat f) t) c0)).(let H10 \def (eq_ind_r 
+nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H6 O H8) in (clear_flat c c0 (H 
+x0 v H10 c0 (clear_gen_flat f x0 c0 t H9)) f t))))) k H4 H7))))))) H3)) 
+(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear (CHead c k t) c0) (\lambda 
+(x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat O (s k 
+x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t 
+x0)).(\lambda (H7: (csubst0 x2 v c x1)).(let H8 \def (eq_ind C c2 (\lambda 
+(c3: C).(clear c3 c0)) H1 (CHead x1 k x0) H5) in (K_ind (\lambda (k0: K).((eq 
+nat O (s k0 x2)) \to ((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) 
+c0)))) (\lambda (b: B).(\lambda (H9: (eq nat O (s (Bind b) x2))).(\lambda (_: 
+(clear (CHead x1 (Bind b) x0) c0)).(let H11 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) 
+H9) in (False_ind (clear (CHead c (Bind b) t) c0) H11))))) (\lambda (f: 
+F).(\lambda (H9: (eq nat O (s (Flat f) x2))).(\lambda (H10: (clear (CHead x1 
+(Flat f) x0) c0)).(let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n 
+v c x1)) H7 O H9) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 
+n v t x0)) H6 O H9) in (clear_flat c c0 (H x1 v H11 c0 (clear_gen_flat f x1 
+c0 x0 H10)) f t)))))) k H4 H8))))))))) H3)) H2))))))))))) c1).
+
+lemma csubst0_clear_S:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
+(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 
+B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq 
+C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v e1 e2))))))))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
+T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c 
+c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: 
+C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 
+(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: 
+nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 
+(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) 
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(let H2 \def 
+(csubst0_gen_head k c c2 t v (S i) H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (clear c2 
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H3: (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 
+(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda 
+(x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat (S i) (s k x1))).(\lambda 
+(H5: (eq C c2 (CHead c k x0))).(\lambda (H6: (subst0 x1 v t x0)).(eq_ind_r C 
+(CHead c k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda 
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+i v e1 e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to 
+((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))))))) (\lambda (b: B).(\lambda (H7: (clear (CHead c (Bind b) t) 
+c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) x1))).(let H9 \def (f_equal nat 
+nat (\lambda (e: nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) 
+(S i) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 
+n v t x0)) H6 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 
+(clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) 
+(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
+T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 
+(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro1 
+(clear (CHead c (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda 
+(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind 
+b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) 
+u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind 
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) 
+u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) 
+(ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) 
+(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 
+(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H10)) c0 
+(clear_gen_bind b c c0 t H7))))))) (\lambda (f: F).(\lambda (H7: (clear 
+(CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) x1))).(let 
+H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H8) in 
+(let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H6 (S i) 
+H9) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda 
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) 
+(clear_flat c c0 (clear_gen_flat f c c0 t H7) f x0))))))) k H1 H4) c2 
+H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 
+(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: 
+nat).(\lambda (H4: (eq nat (S i) (s k x1))).(\lambda (H5: (eq C c2 (CHead x0 
+k t))).(\lambda (H6: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda 
+(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat 
+(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda 
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) 
+u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) 
+(\lambda (b: B).(\lambda (H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: 
+(eq nat (S i) (s (Bind b) x1))).(let H9 \def (f_equal nat nat (\lambda (e: 
+nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H8) 
+in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H6 i 
+H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead 
+x0 (Bind b) t) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) 
+t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind 
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro2 (clear (CHead x0 
+(Bind b) t) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e 
+(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) 
+t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 
+(Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) 
+t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2))))) b c x0 t (refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) 
+H10)) c0 (clear_gen_bind b c c0 t H7))))))) (\lambda (f: F).(\lambda (H7: 
+(clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) 
+x1))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) 
+x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c 
+x0)) H6 (S i) H9) in (let H11 \def (H x0 v i H10 c0 (clear_gen_flat f c c0 t 
+H7)) in (or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (H12: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat 
+f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) 
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H12 f t))) 
+(\lambda (H12: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) 
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e 
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: 
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C c0 (CHead x3 (Bind 
+x2) x4))).(\lambda (H14: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H15: 
+(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T 
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) 
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) 
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H13 (clear_flat x0 
+(CHead x3 (Bind x2) x5) H14 f t) H15))))))))) H12)) (\lambda (H12: (ex3_4 B C 
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T 
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) 
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (H13: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H14: (clear x0 
+(CHead x4 (Bind x2) x5))).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro2 
+(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C 
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2))))) x2 x3 x4 x5 H13 (clear_flat x0 (CHead x4 (Bind x2) x5) H14 f t) 
+H15))))))))) H12)) (\lambda (H12: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H13: (eq C c0 (CHead x3 (Bind x2) 
+x5))).(\lambda (H14: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H15: 
+(subst0 i v x5 x6)).(\lambda (H16: (csubst0 i v x3 x4)).(or4_intro3 (clear 
+(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H13 (clear_flat x0 
+(CHead x4 (Bind x2) x6) H14 f t) H15 H16))))))))))) H12)) H11))))))) k H1 H4) 
+c2 H5)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind 
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda 
+(x1: C).(\lambda (x2: nat).(\lambda (H4: (eq nat (S i) (s k x2))).(\lambda 
+(H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda 
+(H7: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 
+(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind 
+(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to 
+(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda 
+(H8: (clear (CHead c (Bind b) t) c0)).(\lambda (H9: (eq nat (S i) (s (Bind b) 
+x2))).(let H10 \def (f_equal nat nat (\lambda (e: nat).(match e with [O 
+\Rightarrow i | (S n) \Rightarrow n])) (S i) (S x2) H9) in (let H11 \def 
+(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 i H10) in (let H12 
+\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H6 i H10) in 
+(eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x1 (Bind 
+b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) 
+x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind 
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 (clear (CHead x1 
+(Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e 
+(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
+(CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
+C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
+C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) b c x1 t x0 
+(refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) H12 H11)) c0 
+(clear_gen_bind b c c0 t H8)))))))) (\lambda (f: F).(\lambda (H8: (clear 
+(CHead c (Flat f) t) c0)).(\lambda (H9: (eq nat (S i) (s (Flat f) x2))).(let 
+H10 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x2) H9) in 
+(let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H7 (S i) 
+H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) 
+H6 (S i) H10) in (let H13 \def (H x1 v i H11 c0 (clear_gen_flat f c c0 t H8)) 
+in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (H14: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat 
+f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) 
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H14 f x0))) 
+(\lambda (H14: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind 
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) 
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e 
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C 
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: 
+C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C c0 (CHead x4 (Bind 
+x3) x5))).(\lambda (H16: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H17: 
+(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) 
+x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) 
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H15 (clear_flat x1 
+(CHead x4 (Bind x3) x6) H16 f x0) H17))))))))) H14)) (\lambda (H14: (ex3_4 B 
+C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
+c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v 
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) 
+x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
+T).(\lambda (H15: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H16: (clear x1 
+(CHead x5 (Bind x3) x6))).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro2 
+(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C 
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2))))) x3 x4 x5 x6 H15 (clear_flat x1 (CHead x5 (Bind x3) x6) H16 f x0) 
+H17))))))))) H14)) (\lambda (H14: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind 
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) 
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) 
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H15: (eq C c0 (CHead x4 (Bind x3) 
+x6))).(\lambda (H16: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H17: 
+(subst0 i v x6 x7)).(\lambda (H18: (csubst0 i v x4 x5)).(or4_intro3 (clear 
+(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda 
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C 
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H15 (clear_flat x1 
+(CHead x5 (Bind x3) x7) H16 f x0) H17 H18))))))))))) H14)) H13)))))))) k H1 
+H4) c2 H5)))))))) H3)) H2)))))))))))) c1).
+
+lemma csubst0_clear_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 
+i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C 
+(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: 
+T).(\forall (i: nat).((csubst0 i v c c2) \to (\forall (e2: C).((clear c2 e2) 
+\to (or (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda 
+(e1: C).(clear c e1))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CSort n) c2)).(\lambda 
+(e2: C).(\lambda (_: (clear c2 e2)).(csubst0_gen_sort c2 v i n H (or (clear 
+(CSort n) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: 
+C).(clear (CSort n) e1)))))))))))) (\lambda (c: C).(\lambda (H: ((\forall 
+(c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 i v c c2) \to (\forall 
+(e2: C).((clear c2 e2) \to (or (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 
+i v e1 e2)) (\lambda (e1: C).(clear c e1)))))))))))).(\lambda (k: K).(\lambda 
+(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: 
+(csubst0 i v (CHead c k t) c2)).(\lambda (e2: C).(\lambda (H1: (clear c2 
+e2)).(let H2 \def (csubst0_gen_head k c c2 t v i H0) in (or3_ind (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
+nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat 
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3))))) (or (clear (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 
+e2)) (\lambda (e1: C).(clear (CHead c k t) e1)))) (\lambda (H3: (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or (clear 
+(CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: 
+C).(clear (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda 
+(H4: (eq nat i (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda 
+(H6: (subst0 x1 v t x0)).(eq_ind_r nat (s k x1) (\lambda (n: nat).(or (clear 
+(CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 n v e1 e2)) (\lambda (e1: 
+C).(clear (CHead c k t) e1))))) (let H7 \def (eq_ind C c2 (\lambda (c0: 
+C).(clear c0 e2)) H1 (CHead c k x0) H5) in (K_ind (\lambda (k0: K).((clear 
+(CHead c k0 x0) e2) \to (or (clear (CHead c k0 t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (s k0 x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c k0 t) 
+e1)))))) (\lambda (b: B).(\lambda (H8: (clear (CHead c (Bind b) x0) 
+e2)).(eq_ind_r C (CHead c (Bind b) x0) (\lambda (c0: C).(or (clear (CHead c 
+(Bind b) t) c0) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 c0)) 
+(\lambda (e1: C).(clear (CHead c (Bind b) t) e1))))) (or_intror (clear (CHead 
+c (Bind b) t) (CHead c (Bind b) x0)) (ex2 C (\lambda (e1: C).(csubst0 (s 
+(Bind b) x1) v e1 (CHead c (Bind b) x0))) (\lambda (e1: C).(clear (CHead c 
+(Bind b) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 
+(CHead c (Bind b) x0))) (\lambda (e1: C).(clear (CHead c (Bind b) t) e1)) 
+(CHead c (Bind b) t) (csubst0_snd (Bind b) x1 v t x0 H6 c) (clear_bind b c 
+t))) e2 (clear_gen_bind b c e2 x0 H8)))) (\lambda (f: F).(\lambda (H8: (clear 
+(CHead c (Flat f) x0) e2)).(or_introl (clear (CHead c (Flat f) t) e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear 
+(CHead c (Flat f) t) e1))) (clear_flat c e2 (clear_gen_flat f c e2 x0 H8) f 
+t)))) k H7)) i H4)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or (clear (CHead c k t) e2) 
+(ex2 C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear (CHead c 
+k t) e1)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat i (s k 
+x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c 
+x0)).(eq_ind_r nat (s k x1) (\lambda (n: nat).(or (clear (CHead c k t) e2) 
+(ex2 C (\lambda (e1: C).(csubst0 n v e1 e2)) (\lambda (e1: C).(clear (CHead c 
+k t) e1))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(clear c0 e2)) H1 
+(CHead x0 k t) H5) in (K_ind (\lambda (k0: K).((clear (CHead x0 k0 t) e2) \to 
+(or (clear (CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 x1) v e1 
+e2)) (\lambda (e1: C).(clear (CHead c k0 t) e1)))))) (\lambda (b: B).(\lambda 
+(H8: (clear (CHead x0 (Bind b) t) e2)).(eq_ind_r C (CHead x0 (Bind b) t) 
+(\lambda (c0: C).(or (clear (CHead c (Bind b) t) c0) (ex2 C (\lambda (e1: 
+C).(csubst0 (s (Bind b) x1) v e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind 
+b) t) e1))))) (or_intror (clear (CHead c (Bind b) t) (CHead x0 (Bind b) t)) 
+(ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) x1) v e1 (CHead x0 (Bind b) t))) 
+(\lambda (e1: C).(clear (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: 
+C).(csubst0 (s (Bind b) x1) v e1 (CHead x0 (Bind b) t))) (\lambda (e1: 
+C).(clear (CHead c (Bind b) t) e1)) (CHead c (Bind b) t) (csubst0_fst (Bind 
+b) x1 c x0 v H6 t) (clear_bind b c t))) e2 (clear_gen_bind b x0 e2 t H8)))) 
+(\lambda (f: F).(\lambda (H8: (clear (CHead x0 (Flat f) t) e2)).(let H_x \def 
+(H x0 v x1 H6 e2 (clear_gen_flat f x0 e2 t H8)) in (let H9 \def H_x in 
+(or_ind (clear c e2) (ex2 C (\lambda (e1: C).(csubst0 x1 v e1 e2)) (\lambda 
+(e1: C).(clear c e1))) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda 
+(e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c 
+(Flat f) t) e1)))) (\lambda (H10: (clear c e2)).(or_introl (clear (CHead c 
+(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) 
+(\lambda (e1: C).(clear (CHead c (Flat f) t) e1))) (clear_flat c e2 H10 f 
+t))) (\lambda (H10: (ex2 C (\lambda (e1: C).(csubst0 x1 v e1 e2)) (\lambda 
+(e1: C).(clear c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 x1 v e1 e2)) 
+(\lambda (e1: C).(clear c e1)) (or (clear (CHead c (Flat f) t) e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear 
+(CHead c (Flat f) t) e1)))) (\lambda (x: C).(\lambda (H11: (csubst0 x1 v x 
+e2)).(\lambda (H12: (clear c x)).(or_intror (clear (CHead c (Flat f) t) e2) 
+(ex2 C (\lambda (e1: C).(csubst0 (s (Flat f) x1) v e1 e2)) (\lambda (e1: 
+C).(clear (CHead c (Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 
+(s (Flat f) x1) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)) x 
+H11 (clear_flat c x H12 f t)))))) H10)) H9))))) k H7)) i H4)))))) H3)) 
+(\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (or (clear (CHead c k t) e2) (ex2 
+C (\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear (CHead c k t) 
+e1)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H4: (eq 
+nat i (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 k x0))).(\lambda (H6: 
+(subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c x1)).(eq_ind_r nat (s k x2) 
+(\lambda (n: nat).(or (clear (CHead c k t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 n v e1 e2)) (\lambda (e1: C).(clear (CHead c k t) e1))))) (let H8 
+\def (eq_ind C c2 (\lambda (c0: C).(clear c0 e2)) H1 (CHead x1 k x0) H5) in 
+(K_ind (\lambda (k0: K).((clear (CHead x1 k0 x0) e2) \to (or (clear (CHead c 
+k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 x2) v e1 e2)) (\lambda (e1: 
+C).(clear (CHead c k0 t) e1)))))) (\lambda (b: B).(\lambda (H9: (clear (CHead 
+x1 (Bind b) x0) e2)).(eq_ind_r C (CHead x1 (Bind b) x0) (\lambda (c0: C).(or 
+(clear (CHead c (Bind b) t) c0) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) 
+x2) v e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind b) t) e1))))) (or_intror 
+(clear (CHead c (Bind b) t) (CHead x1 (Bind b) x0)) (ex2 C (\lambda (e1: 
+C).(csubst0 (s (Bind b) x2) v e1 (CHead x1 (Bind b) x0))) (\lambda (e1: 
+C).(clear (CHead c (Bind b) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 
+(s (Bind b) x2) v e1 (CHead x1 (Bind b) x0))) (\lambda (e1: C).(clear (CHead 
+c (Bind b) t) e1)) (CHead c (Bind b) t) (csubst0_both (Bind b) x2 v t x0 H6 c 
+x1 H7) (clear_bind b c t))) e2 (clear_gen_bind b x1 e2 x0 H9)))) (\lambda (f: 
+F).(\lambda (H9: (clear (CHead x1 (Flat f) x0) e2)).(let H_x \def (H x1 v x2 
+H7 e2 (clear_gen_flat f x1 e2 x0 H9)) in (let H10 \def H_x in (or_ind (clear 
+c e2) (ex2 C (\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: C).(clear c 
+e1))) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (s 
+(Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)))) 
+(\lambda (H11: (clear c e2)).(or_introl (clear (CHead c (Flat f) t) e2) (ex2 
+C (\lambda (e1: C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear 
+(CHead c (Flat f) t) e1))) (clear_flat c e2 H11 f t))) (\lambda (H11: (ex2 C 
+(\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: C).(clear c 
+e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 x2 v e1 e2)) (\lambda (e1: 
+C).(clear c e1)) (or (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c (Flat 
+f) t) e1)))) (\lambda (x: C).(\lambda (H12: (csubst0 x2 v x e2)).(\lambda 
+(H13: (clear c x)).(or_intror (clear (CHead c (Flat f) t) e2) (ex2 C (\lambda 
+(e1: C).(csubst0 (s (Flat f) x2) v e1 e2)) (\lambda (e1: C).(clear (CHead c 
+(Flat f) t) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (s (Flat f) x2) v e1 
+e2)) (\lambda (e1: C).(clear (CHead c (Flat f) t) e1)) x H12 (clear_flat c x 
+H13 f t)))))) H11)) H10))))) k H8)) i H4)))))))) H3)) H2)))))))))))) c1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/defs.ma
new file mode 100644 (file)
index 0000000..c25dafe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/defs.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/defs.ma".
+
+include "basic_1A/C/defs.ma".
+
+inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
+| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: 
+T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) 
+v (CHead c k u1) (CHead c k u2))))))))
+| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s 
+k i) v (CHead c1 k u) (CHead c2 k u))))))))
+| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall 
+(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall 
+(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 
+k u2)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/drop.ma
new file mode 100644 (file)
index 0000000..4e2337b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/drop.ma
@@ -0,0 +1,6291 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/fwd.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+lemma csubst0_drop_gt:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
+c1 e) \to (drop n O c2 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) 
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
+\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda 
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
+O c1 e)).(lt_x_O i H (drop O O c2 e)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall 
+(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop 
+n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda 
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
+c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) 
+(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v 
+(CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) 
+e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) 
+O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) 
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop 
+(S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
+(drop (S n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e 
+H2)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: 
+T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S 
+n0) O c2 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda 
+(H3: (drop (S n0) O (CHead c k t) e)).(let H4 \def (csubst0_gen_head k c c2 t 
+v i H2) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i 
+(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) 
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda 
+(_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H5: (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S 
+n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k 
+x1))).(\lambda (H7: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t 
+x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let 
+H9 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop 
+(S n0) O c3 e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda 
+(n1: nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop 
+(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) 
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) 
+e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda 
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c 
+e H11 x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c 
+e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) 
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) 
+(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
+(drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 
+O)).(drop_drop (Flat f) n0 c e H11 x0)) (\lambda (H14: (ex2 nat (\lambda (m: 
+nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
+(CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H11 x0)))) H14)) 
+(lt_gen_xS x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2 
+H7)))))) H5)) (\lambda (H5: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda 
+(x0: C).(\lambda (x1: nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7: 
+(eq C c2 (CHead x0 k t))).(\lambda (H8: (csubst0 x1 v c x0)).(eq_ind_r C 
+(CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) H1 (s k x1) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).((drop (r k0 
+n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) 
+e))))) (\lambda (b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda 
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H13: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 
+x0 e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H11) t))))) (\lambda (f: 
+F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt 
+(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq 
+nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat 
+f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8 
+e H11) t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) 
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
+n0)).(drop_drop (Flat f) n0 x0 e (H12 x0 v H8 e H11) t)))) H14)) (lt_gen_xS 
+x1 n0 H13)))))) k (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H7)))))) H5)) 
+(\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: 
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H6: (eq nat i (s k 
+x2))).(\lambda (H7: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t 
+x0)).(\lambda (H9: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda 
+(c0: C).(drop (S n0) O c0 e)) (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H6) 
+in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
+H6) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop 
+(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to 
+(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H12: (drop 
+(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c 
+e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H14: (lt (s (Bind b) x2) (S 
+n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H12) 
+x0))))) (\lambda (f: F).(\lambda (H12: (drop (r (Flat f) n0) O c e)).(\lambda 
+(H13: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) 
+\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) 
+(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
+(drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 
+O)).(drop_drop (Flat f) n0 x1 e (H13 x1 v H9 e H12) x0)) (\lambda (H15: (ex2 
+nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m 
+n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: 
+nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: 
+nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat 
+f) n0 x1 e (H13 x1 v H9 e H12) x0)))) H15)) (lt_gen_xS x2 n0 H14)))))) k 
+(drop_gen_drop k c e t n0 H3) H10 H11))) c2 H7)))))))) H5)) H4))))))))))) 
+c1)))))) n).
+
+lemma csubst0_drop_gt_back:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
+c2 e) \to (drop n O c1 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) 
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
+\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda 
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
+O c2 e)).(lt_x_O i H (drop O O c1 e)))))))))) (\lambda (n0: nat).(\lambda (H: 
+((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall 
+(v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop 
+n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda 
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
+c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
+(\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i 
+v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 
+e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) 
+(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v 
+c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c 
+e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: 
+T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda 
+(H3: (drop (S n0) O c2 e)).(let H4 \def (csubst0_gen_head k c c2 t v i H2) in 
+(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) 
+(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: 
+T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) (\lambda (H5: 
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda 
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: 
+T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H6: (eq nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead c k 
+x0))).(\lambda (_: (subst0 x1 v t x0)).(let H9 \def (eq_ind C c2 (\lambda 
+(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H7) in (let H10 \def (eq_ind 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c 
+e0))))))) H1 (s k x1) H6) in (let H11 \def (eq_ind nat i (\lambda (n1: 
+nat).(lt n1 (S n0))) H0 (s k x1) H6) in (K_ind (\lambda (k0: K).(((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) 
+(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) 
+(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
+(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
+(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda 
+(H14: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H14 t))))) 
+(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
+(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
+(S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S n0))).(\lambda 
+(H14: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda 
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O 
+(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c 
+e H14 t)) (\lambda (H15: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) 
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
+n0)).(drop_drop (Flat f) n0 c e H14 t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k 
+H10 H11 (drop_gen_drop k c e x0 n0 H9)))))))))) H5)) (\lambda (H5: (ex3_2 C 
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S 
+n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H6: (eq 
+nat i (s k x1))).(\lambda (H7: (eq C c2 (CHead x0 k t))).(\lambda (H8: 
+(csubst0 x1 v c x0)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) 
+O c0 e)) H3 (CHead x0 k t) H7) in (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H6) 
+in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) 
+H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
+(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
+n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to 
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt 
+(s (Bind b) x1) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x0 
+e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H13) c x0 v H8 e H14) 
+t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x1) (S 
+n0))).(\lambda (H14: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) 
+(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop 
+(Flat f) n0 c e (H12 x0 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: 
+nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x0 v H8 e H14) 
+t)))) H15)) (lt_gen_xS x1 n0 H13)))))) k H10 H11 (drop_gen_drop k x0 e t n0 
+H9)))))))))) H5)) (\lambda (H5: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O 
+(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
+nat).(\lambda (H6: (eq nat i (s k x2))).(\lambda (H7: (eq C c2 (CHead x1 k 
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H9: (csubst0 x2 v c 
+x1)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 
+(CHead x1 k x0) H7) in (let H11 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H6) 
+in (let H12 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
+H6) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
+(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
+n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to 
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt 
+(s (Bind b) x2) (S n0))).(\lambda (H15: (drop (r (Bind b) n0) O x1 
+e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H14) c x1 v H9 e H15) 
+t))))) (\lambda (f: F).(\lambda (H13: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H14: (lt (s (Flat f) x2) (S 
+n0))).(\lambda (H15: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) 
+(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop 
+(Flat f) n0 c e (H13 x1 v H9 e H15) t)) (\lambda (H16: (ex2 nat (\lambda (m: 
+nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda 
+(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S 
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H13 x1 v H9 e H15) 
+t)))) H16)) (lt_gen_xS x2 n0 H14)))))) k H11 H12 (drop_gen_drop k x1 e x0 n0 
+H10)))))))))))) H5)) H4))))))))))) c1)))))) n).
+
+lemma csubst0_drop_lt:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O 
+c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k 
+w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) 
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k 
+n)) v e1 e2))))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) 
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
+\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T 
+T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) 
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
+O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: 
+nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: 
+T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O 
+c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T 
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) 
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: 
+nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) 
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) 
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
+c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: 
+K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r 
+nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) 
+(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s 
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C 
+(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: 
+K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
+T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 
+O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 
+O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda 
+(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) 
+in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s 
+k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead 
+e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 
+(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in 
+(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k 
+u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k 
+i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
+v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k 
+i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
+v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k 
+u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 
+c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 
+O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 
+e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 
+u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r 
+nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) 
+(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k 
+u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s 
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k 
+u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u 
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 
+u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 
+H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda 
+(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: 
+C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: 
+C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) 
+(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead 
+e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: 
+(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c 
+e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: 
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) 
+c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind 
+(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) 
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S 
+n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq 
+nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: 
+C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k 
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind 
+nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
+\Rightarrow True])) I O H3) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) 
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C (CSort n1) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 k u)))))) 
+(\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CSort n1) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (s k (S n0))) v e1 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) 
+O e H1)))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c2: C).(\forall (v: 
+T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 
+(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda 
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 
+(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (s k (S n0))) v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: 
+T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) 
+c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CHead c k t) e)).(let H3 
+\def (csubst0_gen_head k c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
+n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))))) 
+(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: 
+(eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: 
+(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S 
+n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let 
+H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 
+(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
+(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C 
+T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i 
+(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
+(\lambda (n1: nat).(or4 (drop (S n0) O (CHead c k x0) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c k x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead c k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c k x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) 
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) 
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 
+e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c 
+k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
+(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) 
+x1))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) x0) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2))))))) (drop_drop (Bind b) n0 c e H10 x0)))))) (\lambda (f: F).(\lambda 
+(H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall 
+(v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) 
+O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
+x1))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
+e2))))))) (drop_drop (Flat f) n0 c e H10 x0)))))) k (drop_gen_drop k c e t n0 
+H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O c2 e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: 
+(eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: 
+(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop 
+(S n0) O c0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k0 (S n0))) v e1 e2))))))))) (let 
+H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 
+(drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
+(CHead e1 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C 
+T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v0 e1 e2)))))))))))))) H0 (s k x1) H5) in (let H9 \def (eq_ind nat i 
+(\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
+(\lambda (n1: nat).(or4 (drop (S n0) O (CHead x0 k t) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 k t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead x0 k t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 k t) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) 
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 e2)))))) 
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v0 
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v0 e1 
+e2)))))))))))))) \to ((lt (S n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 
+k0 t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k0 x1) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k0 x1) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
+(b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v0 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) 
+x1))).(let H13 \def (IHn x1 (le_S_n (S n0) x1 H12) c x0 v H7 e H10) in 
+(or4_ind (drop n0 O x0 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O x0 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 
+n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))))) (or4 
+(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x1) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H14: (drop n0 O x0 
+e)).(or4_intro0 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2))))))) (drop_drop (Bind b) n0 x0 e H14 t))) (\lambda (H14: (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O x0 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e0 
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x1 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x0 
+(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x4))).(\lambda (H16: 
+(drop n0 O x0 (CHead x3 x2 x5))).(\lambda (H17: (subst0 (minus x1 (s x2 n0)) 
+v x4 x5)).(eq_ind_r C (CHead x3 x2 x4) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead x3 x2 x4)) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
+x4) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x3 x2 x4) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x4) (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 x4)) 
+(drop_drop (Bind b) n0 x0 (CHead x3 x2 x5) H16 t) (eq_ind_r nat (S (s x2 n0)) 
+(\lambda (n1: nat).(subst0 (minus (s (Bind b) x1) n1) v x4 x5)) H17 (s x2 (S 
+n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex3_4 K C C T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 O x0 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x1 (s k0 
+n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 
+k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus x1 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x0 
+(Bind b) t) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: 
+C).(\lambda (x5: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
+(drop n0 O x0 (CHead x4 x2 x5))).(\lambda (H17: (csubst0 (minus x1 (s x2 n0)) 
+v x3 x4)).(eq_ind_r C (CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x0 (Bind b) t) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x1) (s k0 (S n0))) v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 x2 
+x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x5) H16 t) (eq_ind_r nat (S (s x2 
+n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H17 (s 
+x2 (S n0)) (s_S x2 n0)))) e H15)))))))) H14)) (\lambda (H14: (ex4_5 K C C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 
+k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus x1 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x1 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 K C T 
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2)))))))) (\lambda (x2: K).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x3 x2 x5))).(\lambda (H16: 
+(drop n0 O x0 (CHead x4 x2 x6))).(\lambda (H17: (subst0 (minus x1 (s x2 n0)) 
+v x5 x6)).(\lambda (H18: (csubst0 (minus x1 (s x2 n0)) v x3 x4)).(eq_ind_r C 
+(CHead x3 x2 x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) 
+c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x0 (Bind b) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 
+(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x1) (s k0 (S n0))) v e1 
+e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Bind b) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) x1) (s k0 (S n0))) v e1 e2)))))) x2 x3 x4 x5 x6 
+(refl_equal C (CHead x3 x2 x5)) (drop_drop (Bind b) n0 x0 (CHead x4 x2 x6) 
+H16 t) (eq_ind_r nat (S (s x2 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind 
+b) x1) n1) v x5 x6)) H17 (s x2 (S n0)) (s_S x2 n0)) (eq_ind_r nat (S (s x2 
+n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x1) n1) v x3 x4)) H18 (s 
+x2 (S n0)) (s_S x2 n0)))) e H15)))))))))) H14)) H13)))))) (\lambda (f: 
+F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x1) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v0 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x1) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
+x1))).(let H13 \def (H11 x0 v H7 e H10) in (or4_ind (drop (S n0) O x0 e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x1 (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus x1 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 
+(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
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+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
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+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
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+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 x5) (CHead e1 k0 
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+x5)) (drop_drop (Flat f) n0 x0 (CHead x4 x2 x5) H16 t) H17)) e H15)))))))) 
+H14)) (\lambda (H14: (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
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+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x1 (s k0 
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+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
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+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
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+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
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+C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) x1) (s k0 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead x0 (Flat f) t) (CHead x3 x2 x5)) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x1) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 x2 
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Flat f) x1) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x3 x2 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Flat f) t) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 x2 x5) 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x1) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x1) (s k0 (S n0))) v e1 
+e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x3 x2 x5)) (drop_drop (Flat f) 
+n0 x0 (CHead x4 x2 x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))) k 
+(drop_gen_drop k c e t n0 H2) H8 H9) i H5))) c2 H6)))))) H4)) (\lambda (H4: 
+(ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s 
+k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
+nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k 
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c 
+x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop (S n0) O c0 e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2))))))))) (let H9 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k0 (S 
+n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus n1 (s k0 (S n0))) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus n1 (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
+(s k0 (S n0))) v0 e1 e2)))))))))))))) H0 (s k x2) H5) in (let H10 \def 
+(eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r 
+nat (s k x2) (\lambda (n1: nat).(or4 (drop (S n0) O (CHead x1 k x0) e) (ex3_4 
+K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 k x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead x1 k x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 k x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus n1 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k0 
+(S n0))) v e1 e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to 
+(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) 
+(s k1 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 e2)))))) 
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v0 
+u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v0 e1 
+e2)))))))))))))) \to ((lt (S n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 
+k0 x0) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) (ex3_4 
+K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 
+k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k0 x2) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k0 x2) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
+(b: B).(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v0 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v0 e1 e2))))))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b) 
+x2))).(let H14 \def (IHn x2 (le_S_n (S n0) x2 H13) c x1 v H8 e H11) in 
+(or4_ind (drop n0 O x1 e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (ex3_4 K C C T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 
+n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))))) (or4 
+(drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (H15: (drop n0 O x1 
+e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2))))))) (drop_drop (Bind b) n0 x1 e H15 x0))) (\lambda (H15: (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O x1 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
+k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x2 (s k0 n0)) v u w))))) (or4 (drop (S n0) O (CHead x1 
+(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x5))).(\lambda (H17: 
+(drop n0 O x1 (CHead x4 x3 x6))).(\lambda (H18: (subst0 (minus x2 (s x3 n0)) 
+v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead x4 x3 x5)) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 
+x5) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x4 x3 x5) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) 
+(drop_drop (Bind b) n0 x1 (CHead x4 x3 x6) H17 x0) (eq_ind_r nat (S (s x3 
+n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) x2) n1) v x5 x6)) H18 (s 
+x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C C T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 O x1 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 
+n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 
+k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus x2 (s k0 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead x1 
+(Bind b) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
+C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
+(drop n0 O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3 n0)) 
+v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x1 (Bind b) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) x2) (s k0 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 
+x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x4 x3 x6) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x2) (s k0 (S n0))) v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 
+x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 x6) H17 x0) (eq_ind_r nat (S (s 
+x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H18 
+(s x3 (S n0)) (s_S x3 n0)))) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead 
+e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus x2 (s k0 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x2 (s k0 n0)) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 K C T 
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 
+u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
+(drop n0 O x1 (CHead x5 x3 x7))).(\lambda (H18: (subst0 (minus x2 (s x3 n0)) 
+v x6 x7)).(\lambda (H19: (csubst0 (minus x2 (s x3 n0)) v x4 x5)).(eq_ind_r C 
+(CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) 
+c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Bind b) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 
+(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead x4 x3 x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 
+e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Bind b) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 
+x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Bind b) n0 x1 (CHead x5 x3 
+x7) H17 x0) (eq_ind_r nat (S (s x3 n0)) (\lambda (n1: nat).(subst0 (minus (s 
+(Bind b) x2) n1) v x6 x7)) H18 (s x3 (S n0)) (s_S x3 n0)) (eq_ind_r nat (S (s 
+x3 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) x2) n1) v x4 x5)) H19 
+(s x3 (S n0)) (s_S x3 n0)))) e H16)))))))))) H15)) H14)))))) (\lambda (f: 
+F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x2) (s k0 (S n0))) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v0 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
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+(S n0))) v0 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
+x2))).(let H14 \def (H12 x1 v H8 e H11) in (or4_ind (drop (S n0) O x1 e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x2 (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus x2 
+(s k0 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
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+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))))) (\lambda (H15: (drop (S n0) O x1 e)).(or4_intro0 (drop (S n0) O 
+(CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) x2) (s k0 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H15 
+x0))) (\lambda (H15: (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead 
+e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus x2 (s k0 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O x1 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u 
+w))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: 
+C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 
+x5))).(\lambda (H17: (drop (S n0) O x1 (CHead x4 x3 x6))).(\lambda (H18: 
+(subst0 (minus x2 (s x3 (S n0))) v x5 x6)).(eq_ind_r C (CHead x4 x3 x5) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T 
+T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 
+k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 
+x5)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
+(ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 x3 x5) (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
+w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x5)) (drop_drop (Flat f) n0 x1 
+(CHead x4 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex3_4 K C 
+C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus x2 (s k0 
+(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead 
+e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead 
+x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 
+(S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: C).(\lambda (x5: 
+C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x4 x3 x6))).(\lambda (H17: 
+(drop (S n0) O x1 (CHead x5 x3 x6))).(\lambda (H18: (csubst0 (minus x2 (s x3 
+(S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 x6) (\lambda (c0: C).(or4 (drop 
+(S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K C T T (\lambda (k0: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 k0 u)))))) 
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 
+(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 k0 u)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))))) 
+(or4_intro2 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 x6)) (ex3_4 K 
+C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
+(ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 
+(CHead x5 x3 x6) H17 x0) H18)) e H16)))))))) H15)) (\lambda (H15: (ex4_5 K C 
+C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S n0))) v u 
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 
+e2)))))))).(ex4_5_ind K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O x1 (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus x2 (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus x2 (s k0 (S n0))) v e1 e2)))))) (or4 
+(drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k0 u))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) x2) (s k0 (S n0))) v e1 e2)))))))) (\lambda (x3: K).(\lambda (x4: 
+C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H16: (eq C e 
+(CHead x4 x3 x6))).(\lambda (H17: (drop (S n0) O x1 (CHead x5 x3 
+x7))).(\lambda (H18: (subst0 (minus x2 (s x3 (S n0))) v x6 x7)).(\lambda 
+(H19: (csubst0 (minus x2 (s x3 (S n0))) v x4 x5)).(eq_ind_r C (CHead x4 x3 
+x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 K 
+C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+c0 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 k0 w)))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 
+k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) x2) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S 
+n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 x3 
+x6)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e0 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2))))))) 
+(ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 x3 x6) (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 k0 
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) x2) (s k0 (S n0))) v u w)))))) 
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) x2) (s k0 (S n0))) v e1 e2)))))) x3 x4 x5 
+x6 x7 (refl_equal C (CHead x4 x3 x6)) (drop_drop (Flat f) n0 x1 (CHead x5 x3 
+x7) H17 x0) H18 H19)) e H16)))))))))) H15)) H14)))))) k (drop_gen_drop k c e 
+t n0 H2) H9 H10) i H5))) c2 H6)))))))) H4)) H3))))))))))) c1)))))) n).
+
+lemma csubst0_drop_eq:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
+n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: 
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 
+e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
+c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 
+e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) 
+(\lambda (n0: nat).(or4 (drop n0 n0 c2 c1) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 n0 c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 n0 c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 n0 c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 
+y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: 
+C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c0 c) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 n0 c0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 
+t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 n0 c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 t u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: 
+K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: 
+T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 
+(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
+c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (s k0 i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) 
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s 
+k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c k0 u2) 
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda 
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 
+u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat 
+(S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
+\Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c 
+(Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (S i) 
+v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S i) 
+(S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S i) v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 
+e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 
+u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i 
+(\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O (\lambda 
+(n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u2) (CHead c (Flat f) u1)) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c 
+(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 (CHead c (Flat f) u2) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c 
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro1 (drop O O (CHead c (Flat f) 
+u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat 
+f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
+(CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) 
+u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) 
+(drop_refl (CHead c (Flat f) u2)) H4)) i H3)))))))))) k)) (\lambda (k: 
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: 
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop 
+i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to (\forall 
+(u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c4 k0 u) 
+(CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 
+i) (s k0 i) (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 
+i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) 
+u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (s k0 i) v0 u0 w)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 
+(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
+(CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) 
+u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (S i) v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) 
+u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat 
+f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 
+(Bind b) u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S i) (S i) (CHead 
+c4 (Bind b) u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (S i) v0 u0 
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: 
+F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
+T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 
+(drop i i c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e0 
+(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i i 
+c4 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda 
+(u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0: 
+nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 c3) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat 
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop n0 n0 c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u0))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H3 O H4) in (let H6 \def 
+(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r 
+nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) u) (CHead c3 
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) (ex3_4 F C C 
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 n0 (CHead c4 (Flat f) u) 
+(CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 (CHead c4 
+(Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 n0 v0 u0 w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c4 (Flat f) 
+u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O 
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F 
+C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq 
+C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 
+(Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u 
+(refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4 (Flat f) u)) H6)) i 
+H4)))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: 
+nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) 
+\to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i 
+O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 i v0 u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop i i c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T 
+T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i i c4 (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 i v0 u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
+e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead 
+c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e0 (Flat f) 
+u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (s k0 
+i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) (s k0 
+i) (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F 
+C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k0 i) (s k0 i) (CHead c4 k0 u2) (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (s k0 i) v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
+e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop i i c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False 
+| (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i) 
+(CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) 
+u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat 
+f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (S i) v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
+i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (S i) v0 u w)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (S i) v0 e1 
+e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c4 c3) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop i i c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop i i c4 (CHead e2 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop i i c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 i v0 u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
+\def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c4 
+c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e0 (Flat f0) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 c4 (CHead e2 
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop n0 n0 c4 (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))))))) H4 O H5) in 
+(let H7 \def (eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) 
+in (let H8 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O 
+H5) in (eq_ind_r nat O (\lambda (n0: nat).(or4 (drop n0 n0 (CHead c4 (Flat f) 
+u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 n0 v0 
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 n0 
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F 
+C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 n0 (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 n0 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 
+(drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) 
+u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) 
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) 
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) 
+(drop_refl (CHead c4 (Flat f) u2)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 
+H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: 
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to 
+(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda 
+(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall 
+(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: 
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort 
+n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) 
+e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop 
+(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) 
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 
+(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with 
+[O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (or4 
+(drop (S n0) O c2 (CSort n1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 (Flat f) 
+u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort 
+n1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop 
+(S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e 
+H0)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: 
+T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 
+(drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: 
+C).(\lambda (H1: (drop (S n0) O (CHead c k t) e)).(let H2 \def 
+(csubst0_gen_head k c c2 t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
+n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
+nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T 
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat 
+(S n0) (s k x1))).(\lambda (H5: (eq C c2 (CHead c k x0))).(\lambda (H6: 
+(subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(or4 (drop (S 
+n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
+n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead c k0 x0) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k0 x0) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c 
+e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
+n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: 
+nat).(subst0 n1 v t x0)) H6 n0 H9) in (or4_intro0 (drop (S n0) O (CHead c 
+(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c 
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(drop_drop (Bind b) n0 c e H7 x0))))))) (\lambda (f: F).(\lambda (H7: (drop 
+(r (Flat f) n0) O c e)).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(let 
+H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in 
+(let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S 
+n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) x0) e) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H7 
+x0))))))) k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3: 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3))) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0) 
+(s k x1))).(\lambda (H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 
+v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(or4 (drop (S n0) O c0 
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
+n0) (s k0 x1)) \to (or4 (drop (S n0) O (CHead x0 k0 t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+k0 t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 k0 t) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))) (\lambda (b: B).(\lambda (H7: (drop (r (Bind b) n0) O c 
+e)).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(let H9 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
+n1])) (S n0) (S x1) H8) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: 
+nat).(csubst0 n1 v c x0)) H6 n0 H9) in (let H11 \def (IHn c x0 v H10 e H7) in 
+(or4_ind (drop n0 O x0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
+x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n0 O x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(or4 (drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (H12: (drop n0 O x0 e)).(or4_intro0 (drop (S n0) O 
+(CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(drop_drop (Bind b) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O x0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 
+(drop (S n0) O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop 
+n0 O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4 
+x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x0 (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat 
+x2) x4)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
+x3 (Flat x2) x4) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
+x4) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x4)) 
+(drop_drop (Bind b) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) 
+H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x0 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
+x0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x0 
+(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O 
+x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C 
+(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind 
+b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat 
+x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
+x3 (Flat x2) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) 
+x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) 
+(drop_drop (Bind b) n0 x0 (CHead x4 (Flat x2) x5) H14 t) H15)) e H13)))))))) 
+H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x0 (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) 
+O (CHead x0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda 
+(H14: (drop n0 O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v x5 
+x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
+x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat 
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat 
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 
+x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) 
+(\lambda (f: F).(\lambda (H7: (drop (r (Flat f) n0) O c e)).(\lambda (H8: (eq 
+nat (S n0) (s (Flat f) x1))).(let H9 \def (f_equal nat nat (\lambda (e0: 
+nat).e0) (S n0) (s (Flat f) x1) H8) in (let H10 \def (eq_ind_r nat x1 
+(\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H9) in (let H11 \def (H x0 
+v H10 e H7) in (or4_ind (drop (S n0) O x0 e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead x0 (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: 
+(drop (S n0) O x0 e)).(or4_intro0 (drop (S n0) O (CHead x0 (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat 
+f) n0 x0 e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O x0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) 
+O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x4))).(\lambda (H14: (drop 
+(S n0) O x0 (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x4 
+x5)).(eq_ind_r C (CHead x3 (Flat x2) x4) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 
+(Flat x2) x4)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u))))))) (\lambda 
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w))))) x2 x3 x4 x5 (refl_equal C (CHead x3 
+(Flat x2) x4)) (drop_drop (Flat f) n0 x0 (CHead x3 (Flat x2) x5) H14 t) H15)) 
+e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
+C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O x0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
+(or4 (drop (S n0) O (CHead x0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda 
+(x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop 
+(S n0) O x0 (CHead x4 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 
+x4)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x0 (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 
+(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 
+(Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u))))))) (\lambda 
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
+C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C 
+(CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead 
+x3 (Flat x2) x5)) (drop_drop (Flat f) n0 x0 (CHead x4 (Flat x2) x5) H14 t) 
+H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O x0 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead x0 
+(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda 
+(H14: (drop (S n0) O x0 (CHead x4 (Flat x2) x6))).(\lambda (H15: (subst0 O v 
+x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x3 (Flat x2) 
+x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x0 (Flat f) t) c0) (ex3_4 F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) 
+x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat 
+f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat 
+f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x3 (Flat x2) x5) (CHead e1 
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat 
+f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+x2 x3 x4 x5 x6 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 
+x0 (CHead x4 (Flat x2) x6) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) 
+k (drop_gen_drop k c e t n0 H1) H4) c2 H5)))))) H3)) (\lambda (H3: (ex4_3 T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
+(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O c2 e) (ex3_4 F 
+C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: 
+C).(\lambda (x2: nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5: 
+(eq C c2 (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7: 
+(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(or4 (drop 
+(S n0) O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to ((eq nat (S 
+n0) (s k0 x2)) \to (or4 (drop (S n0) O (CHead x1 k0 x0) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+k0 x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 k0 x0) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))) (\lambda (b: B).(\lambda (H8: (drop (r (Bind b) n0) O c 
+e)).(\lambda (H9: (eq nat (S n0) (s (Bind b) x2))).(let H10 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow 
+n1])) (S n0) (S x2) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: 
+nat).(csubst0 n1 v c x1)) H7 n0 H10) in (let H12 \def (eq_ind_r nat x2 
+(\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H10) in (let H13 \def (IHn c x1 
+v H11 e H8) in (or4_ind (drop n0 O x1 e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 O x1 (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
+O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O x1 
+e)).(or4_intro0 (drop (S n0) O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 x1 e H14 
+x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
+x1 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) O (CHead x1 (Bind 
+b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop n0 O 
+x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C 
+(CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind 
+b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat 
+x3) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
+x4 (Flat x3) x5) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) 
+x5) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x5)) 
+(drop_drop (Bind b) n0 x1 (CHead x4 (Flat x3) x6) H16 x0) H17)) e H15)))))))) 
+H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O x1 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
+x1 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead x1 
+(Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
+T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O 
+x1 (CHead x5 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C 
+(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind 
+b) x0) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
+f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat 
+x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
+x4 (Flat x3) x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) 
+x6) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) 
+(drop_drop (Bind b) n0 x1 (CHead x5 (Flat x3) x6) H16 x0) H17)) e H15)))))))) 
+H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O x1 (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) 
+O (CHead x1 (Bind b) x0) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda 
+(H16: (drop n0 O x1 (CHead x5 (Flat x3) x7))).(\lambda (H17: (subst0 O v x6 
+x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Bind b) x0) c0) (ex3_4 F C T 
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) 
+O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) 
+x6) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat 
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat 
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+x3 x4 x5 x6 x7 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 
+x1 (CHead x5 (Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) 
+H13)))))))) (\lambda (f: F).(\lambda (H8: (drop (r (Flat f) n0) O c 
+e)).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(let H10 \def (f_equal nat 
+nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x2) H9) in (let H11 \def 
+(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H10) in 
+(let H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S 
+n0) H10) in (let H13 \def (H x1 v H11 e H8) in (or4_ind (drop (S n0) O x1 e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead 
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop (S n0) O 
+x1 e)).(or4_intro0 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 x1 e H14 
+x0))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O x1 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (S n0) 
+O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda 
+(x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x5))).(\lambda (H16: (drop 
+(S n0) O x1 (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x5 
+x6)).(eq_ind_r C (CHead x4 (Flat x3) x5) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead 
+x4 (Flat x3) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) 
+u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u))))))) (\lambda 
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w))))) x3 x4 x5 x6 (refl_equal C (CHead x4 
+(Flat x3) x5)) (drop_drop (Flat f) n0 x1 (CHead x4 (Flat x3) x6) H16 x0) 
+H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind 
+F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O x1 (CHead e2 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))) (or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: 
+C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat 
+x3) x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x6))).(\lambda 
+(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: 
+C).(or4 (drop (S n0) O (CHead x1 (Flat f) x0) c0) (ex3_4 F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c0 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead x1 (Flat 
+f) x0) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) 
+u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
+x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 
+(CHead x5 (Flat x3) x6) H16 x0) H17)) e H15)))))))) H14)) (\lambda (H14: 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O x1 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O x1 (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(or4 (drop (S n0) O (CHead x1 (Flat f) x0) e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) 
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) 
+x6))).(\lambda (H16: (drop (S n0) O x1 (CHead x5 (Flat x3) x7))).(\lambda 
+(H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C 
+(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead x1 (Flat 
+f) x0) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c0 (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c0 (CHead e1 (Flat 
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat 
+x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead x1 
+(Flat f) x0) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) (\lambda 
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C 
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 
+(refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 x1 (CHead x5 
+(Flat x3) x7) H16 x0) H17 H18)) e H15)))))))))) H14)) H13)))))))) k 
+(drop_gen_drop k c e t n0 H1) H4) c2 H5)))))))) H3)) H2))))))))))) c1)))) n).
+
+lemma csubst0_drop_eq_back:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
+n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: 
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 
+e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
+c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 
+e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O 
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) 
+(\lambda (n0: nat).(or4 (drop n0 n0 c1 c2) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop n0 n0 c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop n0 n0 c1 (CHead e1 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 n0 c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v u1 
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 n0 v e1 e2))))))))) (\lambda (y: nat).(\lambda 
+(H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: 
+T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop n0 n0 c c0) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 
+t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 n0 t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop n0 n0 c (CHead e1 (Flat f) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 n0 t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 t e1 e2))))))))))))) (\lambda 
+(k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall 
+(u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s 
+k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead c k0 u2)) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: 
+T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 
+u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c k0 u1) (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c k0 
+u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (s k0 i) v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: 
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def 
+(eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S i) (S i) (CHead c 
+(Bind b) u1) (CHead c (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead 
+e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e0 (Flat f) 
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
+T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Bind b) u2) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S i) (S i) (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
+i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead 
+e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c (Bind b) u1) 
+(CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (S i) v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: 
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind 
+nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (eq_ind_r nat O 
+(\lambda (n0: nat).(or4 (drop n0 n0 (CHead c (Flat f) u1) (CHead c (Flat f) 
+u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 
+n0 (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 
+u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 
+(CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F 
+C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop n0 n0 (CHead c (Flat f) u1) (CHead e1 (Flat f0) u3))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) 
+(or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C 
+(CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) 
+(CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) 
+u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) 
+u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) 
+(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 
+(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) 
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
+T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) 
+(drop_refl (CHead c (Flat f) u1)) H4)) i H3)))))))))) k)) (\lambda (k: 
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: 
+C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop 
+i i c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 e2)))))))))) \to 
+(\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop (s k0 i) (s k0 i) (CHead 
+c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) 
+u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (s 
+k0 i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop (s k0 
+i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (s k0 i) v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) 
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u) (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 (s k0 i) v0 u1 u2)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (s k0 i) v0 
+e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop i i c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat 
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 
+(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) 
+u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 (S i) v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) 
+u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(drop (S i) (S i) (CHead c3 (Bind b) u) (CHead e1 (Flat 
+f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (S i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 
+(Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S i) (S i) (CHead 
+c3 (Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (S i) v0 u1 
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (S i) v0 e1 e2)))))))) H5))))))))))) (\lambda (f: 
+F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
+T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 
+(drop i i c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e0 
+(Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 i v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i 
+c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 i v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
+e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def 
+(eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c4 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 (CHead e2 (Flat f0) u0)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop n0 
+n0 c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
+C c4 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: 
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (eq_ind_r nat O (\lambda (n0: 
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) 
+u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(drop n0 n0 (CHead c3 (Flat f) u) (CHead e1 (Flat f0) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) 
+u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 n0 (CHead c3 (Flat f) u) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 n0 v0 u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+n0 v0 e1 e2))))))))) (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 
+(Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O 
+(CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C 
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C 
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) 
+(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 
+(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) 
+u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) 
+(drop_refl (CHead c3 (Flat f) u)) H6)) i H4)))))))))))) k)) (\lambda (k: 
+K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: 
+T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: 
+C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
+C c4 (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop 
+(s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 
+u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e0 
+(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 u1) (CHead e1 (Flat 
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (s k0 i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 
+u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop (s k0 i) (s k0 i) (CHead c3 k0 
+u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (s k0 i) v0 u3 u4)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (s k0 i) v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: 
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 
+(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop i i c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 i v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O \Rightarrow False 
+| (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop (S i) (S i) 
+(CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) 
+u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e0 
+(Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(u4: T).(subst0 (S i) v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead 
+e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (S 
+i) v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead 
+e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop (S i) (S i) (CHead c3 (Bind b) u1) 
+(CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 (S i) v0 u3 u4)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (S i) v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda 
+(i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: 
+(subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 
+i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop i i c3 c4) (ex3_4 F 
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
+C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u3: T).(\lambda (_: T).(drop i i c3 (CHead e0 (Flat f0) u3)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 i v0 u3 
+u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i i c3 (CHead e1 
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 i v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop i i c3 (CHead e1 (Flat f0) 
+u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (u4: T).(subst0 i v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v0 e1 
+e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda 
+(n0: nat).((eq nat n0 O) \to (or4 (drop n0 n0 c3 c4) (ex3_4 F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e0 
+(Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop n0 n0 c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 
+u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 c3 (CHead e1 
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 c3 (CHead e1 (Flat f0) 
+u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 n0 v0 e1 
+e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: 
+nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda 
+(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (eq_ind_r nat O (\lambda (n0: 
+nat).(or4 (drop n0 n0 (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F 
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq 
+C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 n0 (CHead c3 
+(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 n0 v0 u3 u4)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 n0 (CHead c3 (Flat f) u1) 
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 n0 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C 
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop n0 
+n0 (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 
+n0 v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 n0 v0 e1 e2))))))))) (or4_intro3 (drop O O 
+(CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) 
+u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) 
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
+T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead 
+e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) 
+u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) 
+(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) 
+(drop_refl (CHead c3 (Flat f) u1)) H8 H7)) i H5))))))))))))))) k)) y v c1 c2 
+H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: 
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to 
+(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind 
+(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to 
+(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: 
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort 
+n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort 
+c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat 
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S 
+n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O 
+c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: 
+C).(\lambda (H1: (drop (S n0) O c2 e)).(let H2 \def (csubst0_gen_head k c c2 
+t v (S n0) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
+nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda 
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: 
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
+n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H3: (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 
+(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda 
+(x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda (H5: (eq C c2 
+(CHead c k x0))).(\lambda (H6: (subst0 x1 v t x0)).(let H7 \def (eq_ind C c2 
+(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H5) in (K_ind 
+(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to 
+(or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: 
+B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9: (drop (r 
+(Bind b) n0) O c e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match 
+e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H8) in 
+(let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 
+H10) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e 
+H9 t))))))) (\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) 
+x1))).(\lambda (H9: (drop (r (Flat f) n0) O c e)).(let H10 \def (f_equal nat 
+nat (\lambda (e0: nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def 
+(eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H10) in 
+(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e 
+H9 t))))))) k H4 (drop_gen_drop k c e x0 n0 H7)))))))) H3)) (\lambda (H3: 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j 
+v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x0: C).(\lambda (x1: nat).(\lambda (H4: (eq nat (S n0) (s k x1))).(\lambda 
+(H5: (eq C c2 (CHead x0 k t))).(\lambda (H6: (csubst0 x1 v c x0)).(let H7 
+\def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead x0 k t) 
+H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 
+n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) 
+(\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H9: 
+(drop (r (Bind b) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0: 
+nat).(match e0 with [O \Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S 
+x1) H8) in (let H11 \def (eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c 
+x0)) H6 n0 H10) in (let H12 \def (IHn c x0 v H11 e H9) in (or4_ind (drop n0 O 
+c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c 
+(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (H13: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead 
+c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(drop_drop (Bind b) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 (Flat x2) 
+x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H16: 
+(subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0 
+c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C (CHead x3 (Flat x2) x5) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead c (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat 
+x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
+x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c 
+(CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda (H13: 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H14: (eq 
+C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) 
+x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let H17 \def (eq_ind C e (\lambda 
+(c0: C).(drop n0 O x0 c0)) H9 (CHead x4 (Flat x2) x5) H14) in (eq_ind_r C 
+(CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind 
+b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat 
+x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
+C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
+(CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead 
+x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H15 t) 
+H16)) e H14))))))))) H13)) (\lambda (H13: (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind 
+F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 
+O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda 
+(x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq 
+C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop n0 O c (CHead x3 (Flat x2) 
+x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda (H17: (csubst0 O v x3 
+x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x0 c0)) H9 (CHead 
+x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: 
+C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind 
+b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) 
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 
+c (CHead x3 (Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12))))))) 
+(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H9: 
+(drop (r (Flat f) n0) O x0 e)).(let H10 \def (f_equal nat nat (\lambda (e0: 
+nat).e0) (S n0) (s (Flat f) x1) H8) in (let H11 \def (eq_ind_r nat x1 
+(\lambda (n1: nat).(csubst0 n1 v c x0)) H6 (S n0) H10) in (let H12 \def (H x0 
+v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(H13: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
+(Flat f) n0 c e H13 t))) (\lambda (H13: (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda 
+(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq C e (CHead x3 
+(Flat x2) x5))).(\lambda (H15: (drop (S n0) O c (CHead x3 (Flat x2) 
+x4))).(\lambda (H16: (subst0 O v x4 x5)).(let H17 \def (eq_ind C e (\lambda 
+(c0: C).(drop (S n0) O x0 c0)) H9 (CHead x3 (Flat x2) x5) H14) in (eq_ind_r C 
+(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat 
+f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
+f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x3 (Flat 
+x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) 
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) 
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) 
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat 
+f) n0 c (CHead x3 (Flat x2) x4) H15 t) H16)) e H14))))))))) H13)) (\lambda 
+(H13: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) 
+O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda 
+(x5: T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H15: (drop 
+(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (csubst0 O v x3 x4)).(let 
+H17 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x0 c0)) H9 (CHead x4 
+(Flat x2) x5) H14) in (eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: 
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat 
+f) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop 
+(Flat f) n0 c (CHead x3 (Flat x2) x5) H15 t) H16)) e H14))))))))) H13)) 
+(\lambda (H13: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) 
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
+C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (H14: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H15: (drop (S 
+n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H16: (subst0 O v x5 x6)).(\lambda 
+(H17: (csubst0 O v x3 x4)).(let H18 \def (eq_ind C e (\lambda (c0: C).(drop 
+(S n0) O x0 c0)) H9 (CHead x4 (Flat x2) x6) H14) in (eq_ind_r C (CHead x4 
+(Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) 
+(or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x6)) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) u2)))))) (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O 
+(CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C 
+T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
+(CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda 
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 
+(refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Flat f) n0 c (CHead x3 
+(Flat x2) x5) H15 t) H16 H17)) e H14))))))))))) H13)) H12))))))) k H4 
+(drop_gen_drop k x0 e t n0 H7)))))))) H3)) (\lambda (H3: (ex4_3 T C nat 
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) 
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
+(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O (CHead c k t) 
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k 
+t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
+nat).(\lambda (H4: (eq nat (S n0) (s k x2))).(\lambda (H5: (eq C c2 (CHead x1 
+k x0))).(\lambda (H6: (subst0 x2 v t x0)).(\lambda (H7: (csubst0 x2 v c 
+x1)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 
+(CHead x1 k x0) H5) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to 
+((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead 
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H9: (eq nat (S n0) (s (Bind b) 
+x2))).(\lambda (H10: (drop (r (Bind b) n0) O x1 e)).(let H11 \def (f_equal 
+nat nat (\lambda (e0: nat).(match e0 with [O \Rightarrow n0 | (S n1) 
+\Rightarrow n1])) (S n0) (S x2) H9) in (let H12 \def (eq_ind_r nat x2 
+(\lambda (n1: nat).(csubst0 n1 v c x1)) H7 n0 H11) in (let H13 \def (eq_ind_r 
+nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 n0 H11) in (let H14 \def 
+(IHn c x1 v H12 e H10) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 
+O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H15: (drop n0 O c 
+e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e 
+H15 t))) (\lambda (H15: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
+c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) 
+O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead 
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda 
+(x6: T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop 
+n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19 
+\def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 c0)) H10 (CHead x4 (Flat x3) 
+x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop 
+(S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead 
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 
+(Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead 
+x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H17 t) 
+H18)) e H16))))))))) H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
+C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda 
+(x5: C).(\lambda (x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) 
+x6))).(\lambda (H17: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H18: 
+(csubst0 O v x4 x5)).(let H19 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 
+c0)) H10 (CHead x5 (Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
+x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 
+(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 
+(Flat x3) x6) H17 t) H18)) e H16))))))))) H15)) (\lambda (H15: (ex4_5 F C C T 
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop n0 O 
+c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda (H19: 
+(csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop n0 O x1 
+c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5 (Flat x3) x7) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead c (Bind b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
+x3) x7) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 
+c (CHead x4 (Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) 
+(\lambda (f: F).(\lambda (H9: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H10: 
+(drop (r (Flat f) n0) O x1 e)).(let H11 \def (f_equal nat nat (\lambda (e0: 
+nat).e0) (S n0) (s (Flat f) x2) H9) in (let H12 \def (eq_ind_r nat x2 
+(\lambda (n1: nat).(csubst0 n1 v c x1)) H7 (S n0) H11) in (let H13 \def 
+(eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H6 (S n0) H11) in 
+(let H14 \def (H x1 v H12 e H10) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T 
+T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S 
+n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (H15: (drop (S n0) O c e)).(or4_intro0 (drop (S 
+n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (drop_drop (Flat f) n0 c e H15 t))) (\lambda (H15: (ex3_4 F 
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq 
+C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead 
+e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat 
+f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
+f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (H16: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H17: (drop (S 
+n0) O c (CHead x4 (Flat x3) x5))).(\lambda (H18: (subst0 O v x5 x6)).(let H19 
+\def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x4 (Flat 
+x3) x6) H16) in (eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 
+(drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat 
+f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) 
+(drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H17 t) H18)) e H16))))))))) 
+H15)) (\lambda (H15: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) 
+O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda 
+(x6: T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H17: (drop 
+(S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (csubst0 O v x4 x5)).(let 
+H19 \def (eq_ind C e (\lambda (c0: C).(drop (S n0) O x1 c0)) H10 (CHead x5 
+(Flat x3) x6) H16) in (eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: 
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat 
+f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop 
+(Flat f) n0 c (CHead x4 (Flat x3) x6) H17 t) H18)) e H16))))))))) H15)) 
+(\lambda (H15: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) 
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
+C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (H16: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H17: (drop (S 
+n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H18: (subst0 O v x6 x7)).(\lambda 
+(H19: (csubst0 O v x4 x5)).(let H20 \def (eq_ind C e (\lambda (c0: C).(drop 
+(S n0) O x1 c0)) H10 (CHead x5 (Flat x3) x7) H16) in (eq_ind_r C (CHead x5 
+(Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) 
+(or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x7)) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) u2)))))) (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O 
+(CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C 
+T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
+(CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda 
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 
+(refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 
+(Flat x3) x6) H17 t) H18 H19)) e H16))))))))))) H15)) H14)))))))) k H4 
+(drop_gen_drop k x1 e x0 n0 H8)))))))))) H3)) H2))))))))))) c1)))) n).
+
+lemma csubst0_drop_lt_back:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((drop n O 
+c2 e2) \to (or (drop n O c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) 
+v e1 e2)) (\lambda (e1: C).(drop n O c1 e1))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) 
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) 
+\to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O 
+c1 e1))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
+c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(eq_ind C c2 (\lambda 
+(c: C).(or (drop O O c1 c) (ex2 C (\lambda (e1: C).(csubst0 (minus i O) v e1 
+c)) (\lambda (e1: C).(drop O O c1 e1))))) (eq_ind nat i (\lambda (n0: 
+nat).(or (drop O O c1 c2) (ex2 C (\lambda (e1: C).(csubst0 n0 v e1 c2)) 
+(\lambda (e1: C).(drop O O c1 e1))))) (or_intror (drop O O c1 c2) (ex2 C 
+(\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O c1 e1))) 
+(ex_intro2 C (\lambda (e1: C).(csubst0 i v e1 c2)) (\lambda (e1: C).(drop O O 
+c1 e1)) c1 H0 (drop_refl c1))) (minus i O) (minus_n_O i)) e2 (drop_gen_refl 
+c2 e2 H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt 
+n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 
+c2) \to (\forall (e2: C).((drop n0 O c2 e2) \to (or (drop n0 O c1 e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus i n0) v e1 e2)) (\lambda (e1: C).(drop n0 O 
+c1 e1)))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda 
+(c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v 
+c c2) \to (\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c 
+e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: 
+C).(drop (S n0) O c e1)))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H0: (csubst0 i v (CSort n1) c2)).(\lambda (e2: C).(\lambda 
+(_: (drop (S n0) O c2 e2)).(csubst0_gen_sort c2 v i n1 H0 (or (drop (S n0) O 
+(CSort n1) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) 
+(\lambda (e1: C).(drop (S n0) O (CSort n1) e1))))))))))) (\lambda (c: 
+C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to 
+(\forall (e2: C).((drop (S n0) O c2 e2) \to (or (drop (S n0) O c e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O c e1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
+C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda 
+(e2: C).(\lambda (H2: (drop (S n0) O c2 e2)).(let H3 \def (csubst0_gen_head k 
+c c2 t v i H1) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
+nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k 
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))) (or (drop (S n0) O (CHead c k t) 
+e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: 
+C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (H4: (ex3_2 T nat (\lambda 
+(_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda 
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or (drop (S n0) O (CHead c k 
+t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k 
+x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda 
+(c0: C).(drop (S n0) O c0 e2)) H2 (CHead c k x0) H6) in (let H9 \def (eq_ind 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
+c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) 
+(ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: 
+C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let H10 \def (eq_ind nat 
+i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in (eq_ind_r nat (s k x1) 
+(\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
+(CHead c k t) e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall 
+(v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 
+e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
+k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to 
+((lt (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e2) \to (or (drop (S n0) O 
+(CHead c k0 t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) 
+v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda 
+(b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) 
+x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) 
+O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 
+e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) 
+(s (Bind b) x1))).(\lambda (H13: (drop (r (Bind b) n0) O c e2)).(or_introl 
+(drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 
+(minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
+(CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H13 t)))))) (\lambda 
+(f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) 
+x1) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) 
+O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1 
+e3)) (\lambda (e1: C).(drop (S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) 
+(s (Flat f) x1))).(\lambda (H13: (drop (r (Flat f) n0) O c e2)).(or_introl 
+(drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 
+(minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O 
+(CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H13 t)))))) k H9 H10 
+(drop_gen_drop k c e2 x0 n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex3_2 C 
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or (drop 
+(S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i (S n0)) 
+v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) e1)))) (\lambda (x0: 
+C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C 
+c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(let H8 \def (eq_ind C 
+c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 (CHead x0 k t) H6) in (let H9 
+\def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
+T).((csubst0 n1 v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or 
+(drop (S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v0 e1 
+e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) H0 (s k x1) H5) in (let 
+H10 \def (eq_ind nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k x1) H5) in 
+(eq_ind_r nat (s k x1) (\lambda (n1: nat).(or (drop (S n0) O (CHead c k t) 
+e2) (ex2 C (\lambda (e1: C).(csubst0 (minus n1 (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c k t) e1))))) (K_ind (\lambda (k0: 
+K).(((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to 
+(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus (s k0 x1) (S n0)) v0 e1 e3)) (\lambda (e1: 
+C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 x1)) \to ((drop (r k0 
+n0) O x0 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s k0 x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) 
+O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: ((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e3: 
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Bind b) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
+(S n0) O c e1))))))))))).(\lambda (H12: (lt (S n0) (s (Bind b) x1))).(\lambda 
+(H13: (drop (r (Bind b) n0) O x0 e2)).(let H_x \def (IHn x1 (lt_S_n n0 x1 
+H12) c x0 v H7 e2 H13) in (let H14 \def H_x in (or_ind (drop n0 O c e2) (ex2 
+C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 
+O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H15: (drop n0 O c 
+e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H15 t))) 
+(\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 n0) v e1 e2)) 
+(\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
+(minus x1 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O 
+(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) 
+x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) 
+e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 n0) v x 
+e2)).(\lambda (H17: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind 
+b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v 
+e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 
+C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x1) (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H16 (drop_drop (Bind b) n0 
+c x H17 t)))))) H15)) H14))))))) (\lambda (f: F).(\lambda (H11: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e3: 
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x1) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
+(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x1))).(\lambda 
+(H13: (drop (r (Flat f) n0) O x0 e2)).(let H_x \def (H11 x0 v H7 e2 H13) in 
+(let H14 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c 
+e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H15: (drop (S n0) O c 
+e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H15 t))) 
+(\lambda (H15: (ex2 C (\lambda (e1: C).(csubst0 (minus x1 (S n0)) v e1 e2)) 
+(\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
+(minus x1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop 
+(S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
+(Flat f) x1) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat 
+f) t) e1)))) (\lambda (x: C).(\lambda (H16: (csubst0 (minus x1 (S n0)) v x 
+e2)).(\lambda (H17: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c 
+(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S 
+n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) 
+(ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x1) (S n0)) v e1 
+e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H16 
+(drop_drop (Flat f) n0 c x H17 t)))))) H15)) H14))))))) k H9 H10 
+(drop_gen_drop k x0 e2 t n0 H8)) i H5))))))))) H4)) (\lambda (H4: (ex4_3 T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))) (or (drop (S n0) O (CHead c k t) e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus i (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c k t) e1)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
+nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k 
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c 
+x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e2)) H2 
+(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda 
+(e1: C).(csubst0 (minus n1 (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop (S n0) O 
+c e1)))))))))) H0 (s k x2) H5) in (let H11 \def (eq_ind nat i (\lambda (n1: 
+nat).(lt (S n0) n1)) H (s k x2) H5) in (eq_ind_r nat (s k x2) (\lambda (n1: 
+nat).(or (drop (S n0) O (CHead c k t) e2) (ex2 C (\lambda (e1: C).(csubst0 
+(minus n1 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c k t) 
+e1))))) (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
+(s k0 x2) v0 c c3) \to (\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop 
+(S n0) O c e3) (ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v0 
+e1 e3)) (\lambda (e1: C).(drop (S n0) O c e1)))))))))) \to ((lt (S n0) (s k0 
+x2)) \to ((drop (r k0 n0) O x1 e2) \to (or (drop (S n0) O (CHead c k0 t) e2) 
+(ex2 C (\lambda (e1: C).(csubst0 (minus (s k0 x2) (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c k0 t) e1)))))))) (\lambda (b: B).(\lambda (_: 
+((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to 
+(\forall (e3: C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v0 e1 e3)) (\lambda 
+(e1: C).(drop (S n0) O c e1))))))))))).(\lambda (H13: (lt (S n0) (s (Bind b) 
+x2))).(\lambda (H14: (drop (r (Bind b) n0) O x1 e2)).(let H_x \def (IHn x2 
+(lt_S_n n0 x2 H13) c x1 v H8 e2 H14) in (let H15 \def H_x in (or_ind (drop n0 
+O c e2) (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) (\lambda 
+(e1: C).(drop n0 O c e1))) (or (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C 
+(\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)))) (\lambda (H16: (drop n0 O 
+c e2)).(or_introl (drop (S n0) O (CHead c (Bind b) t) e2) (ex2 C (\lambda 
+(e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda (e1: 
+C).(drop (S n0) O (CHead c (Bind b) t) e1))) (drop_drop (Bind b) n0 c e2 H16 
+t))) (\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 n0) v e1 e2)) 
+(\lambda (e1: C).(drop n0 O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
+(minus x2 n0) v e1 e2)) (\lambda (e1: C).(drop n0 O c e1)) (or (drop (S n0) O 
+(CHead c (Bind b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) 
+x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) 
+e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 n0) v x 
+e2)).(\lambda (H18: (drop n0 O c x)).(or_intror (drop (S n0) O (CHead c (Bind 
+b) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v 
+e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Bind b) t) e1))) (ex_intro2 
+C (\lambda (e1: C).(csubst0 (minus (s (Bind b) x2) (S n0)) v e1 e2)) (\lambda 
+(e1: C).(drop (S n0) O (CHead c (Bind b) t) e1)) x H17 (drop_drop (Bind b) n0 
+c x H18 t)))))) H16)) H15))))))) (\lambda (f: F).(\lambda (H12: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e3: 
+C).((drop (S n0) O c3 e3) \to (or (drop (S n0) O c e3) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x2) (S n0)) v0 e1 e3)) (\lambda (e1: C).(drop 
+(S n0) O c e1))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) x2))).(\lambda 
+(H14: (drop (r (Flat f) n0) O x1 e2)).(let H_x \def (H12 x1 v H8 e2 H14) in 
+(let H15 \def H_x in (or_ind (drop (S n0) O c e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c 
+e1))) (or (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Flat f) t) e1)))) (\lambda (H16: (drop (S n0) O c 
+e2)).(or_introl (drop (S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop 
+(S n0) O (CHead c (Flat f) t) e1))) (drop_drop (Flat f) n0 c e2 H16 t))) 
+(\lambda (H16: (ex2 C (\lambda (e1: C).(csubst0 (minus x2 (S n0)) v e1 e2)) 
+(\lambda (e1: C).(drop (S n0) O c e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 
+(minus x2 (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O c e1)) (or (drop 
+(S n0) O (CHead c (Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s 
+(Flat f) x2) (S n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat 
+f) t) e1)))) (\lambda (x: C).(\lambda (H17: (csubst0 (minus x2 (S n0)) v x 
+e2)).(\lambda (H18: (drop (S n0) O c x)).(or_intror (drop (S n0) O (CHead c 
+(Flat f) t) e2) (ex2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S 
+n0)) v e1 e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1))) 
+(ex_intro2 C (\lambda (e1: C).(csubst0 (minus (s (Flat f) x2) (S n0)) v e1 
+e2)) (\lambda (e1: C).(drop (S n0) O (CHead c (Flat f) t) e1)) x H17 
+(drop_drop (Flat f) n0 c x H18 t)))))) H16)) H15))))))) k H10 H11 
+(drop_gen_drop k x1 e2 x0 n0 H9)) i H5))))))))))) H4)) H3))))))))))) c1)))))) 
+n).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/fwd.ma
new file mode 100644 (file)
index 0000000..3a888e1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/fwd.ma
@@ -0,0 +1,473 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/defs.ma".
+
+include "basic_1A/C/fwd.ma".
+
+implied rec lemma csubst0_ind (P: (nat \to (T \to (C \to (C \to Prop))))) (f: 
+(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(P (s k i) v (CHead c k u1) 
+(CHead c k u2)))))))))) (f0: (\forall (k: K).(\forall (i: nat).(\forall (c1: 
+C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to ((P i v c1 c2) 
+\to (\forall (u: T).(P (s k i) v (CHead c1 k u) (CHead c2 k u))))))))))) (f1: 
+(\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i 
+v c1 c2) \to ((P i v c1 c2) \to (P (s k i) v (CHead c1 k u1) (CHead c2 k 
+u2))))))))))))) (n: nat) (t: T) (c: C) (c0: C) (c1: csubst0 n t c c0) on c1: 
+P n t c c0 \def match c1 with [(csubst0_snd k i v u1 u2 s0 c2) \Rightarrow (f 
+k i v u1 u2 s0 c2) | (csubst0_fst k i c2 c3 v c4 u) \Rightarrow (f0 k i c2 c3 
+v c4 ((csubst0_ind P f f0 f1) i v c2 c3 c4) u) | (csubst0_both k i v u1 u2 s0 
+c2 c3 c4) \Rightarrow (f1 k i v u1 u2 s0 c2 c3 c4 ((csubst0_ind P f f0 f1) i 
+v c2 c3 c4))].
+
+lemma csubst0_gen_sort:
+ \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 
+i v (CSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(insert_eq C (CSort n) 
+(\lambda (c: C).(csubst0 i v c x)) (\lambda (_: C).P) (\lambda (y: 
+C).(\lambda (H0: (csubst0 i v y x)).(csubst0_ind (\lambda (_: nat).(\lambda 
+(_: T).(\lambda (c: C).(\lambda (_: C).((eq C c (CSort n)) \to P))))) 
+(\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(\lambda (H2: (eq 
+C (CHead c k u1) (CSort n))).(let H3 \def (eq_ind C (CHead c k u1) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) 
+\Rightarrow True])) I (CSort n) H2) in (False_ind P H3)))))))))) (\lambda (k: 
+K).(\lambda (i0: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v0: 
+T).(\lambda (_: (csubst0 i0 v0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to 
+P))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CSort n))).(let H4 
+\def (eq_ind C (CHead c1 k u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in 
+(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csubst0 i0 v0 c1 
+c2)).(\lambda (_: (((eq C c1 (CSort n)) \to P))).(\lambda (H4: (eq C (CHead 
+c1 k u1) (CSort n))).(let H5 \def (eq_ind C (CHead c1 k u1) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort n) H4) in (False_ind P H5))))))))))))) i v y x H0))) H)))))).
+
+lemma csubst0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
+(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T 
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
+nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k 
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) 
+(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k 
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
+u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 
+c2))))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) 
+x)).(insert_eq C (CHead c1 k u1) (\lambda (c: C).(csubst0 i v c x)) (\lambda 
+(_: C).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda 
+(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: 
+nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j 
+v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: 
+nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
+nat).(csubst0 j v c1 c2))))))) (\lambda (y: C).(\lambda (H0: (csubst0 i v y 
+x)).(csubst0_ind (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda 
+(c0: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c0 (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+t u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k 
+j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda 
+(c2: C).(\lambda (j: nat).(csubst0 j t c1 c2)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: 
+T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j t u1 u2)))) (\lambda (_: 
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j t c1 c2))))))))))) (\lambda 
+(k0: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (c: C).(\lambda (H2: (eq C 
+(CHead c k0 u0) (CHead c1 k u1))).(let H3 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow c | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead c k0 u0) (CHead c1 k u1) H2) in ((let H4 \def (f_equal C K (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
+k1])) (CHead c k0 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H2) in (\lambda (H6: (eq K 
+k0 k)).(\lambda (H7: (eq C c c1)).(eq_ind_r C c1 (\lambda (c0: C).(or3 (ex3_2 
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda 
+(u3: T).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c1 k u3)))) (\lambda 
+(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda 
+(_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda 
+(j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c2: C).(\lambda (_: nat).(eq C (CHead c0 k0 u2) (CHead c2 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))))) (let H8 \def 
+(eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H5) in (eq_ind_r K k 
+(\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+(s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k1 
+u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 
+u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
+j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 u2) (CHead c2 k 
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k 
+j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k1 
+u2) (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: 
+nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c2))))))) (or3_intro0 (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda 
+(_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda 
+(j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C 
+(CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v0 c1 c2))))) (ex3_2_intro T 
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda 
+(u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda 
+(u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3))) u2 i0 (refl_equal nat (s k 
+i0)) (refl_equal C (CHead c1 k u2)) H8)) k0 H6)) c H7)))) H4)) H3)))))))))) 
+(\lambda (k0: K).(\lambda (i0: nat).(\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (v0: T).(\lambda (H1: (csubst0 i0 v0 c0 c2)).(\lambda (H2: (((eq 
+C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead 
+c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 
+C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda 
+(j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda (u: T).(\lambda 
+(H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H4 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H5 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
+_ k1 _) \Rightarrow k1])) (CHead c0 k0 u) (CHead c1 k u1) H3) in ((let H6 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) (CHead c1 k u1) H3) in 
+(\lambda (H7: (eq K k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T u1 
+(\lambda (t: T).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+(s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k0 t) 
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 t) (CHead c3 k u1)))) 
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat 
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k0 
+t) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: 
+nat).(subst0 j v0 u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c3))))))) (let H9 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat i0 (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) 
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda 
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H2 c1 H8) 
+in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H1 c1 H8) 
+in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq 
+C (CHead c2 k1 u1) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
+(CHead c2 k1 u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda 
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u1) (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro1 
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
+(\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) 
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v0 u1 u2)))) (ex3_2 C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
+(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k 
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
+c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) 
+(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))) c2 i0 
+(refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H10)) k0 H7))) u 
+H6)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (i0: nat).(\lambda (v0: 
+T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (subst0 i0 v0 u0 
+u2)).(\lambda (c0: C).(\lambda (c2: C).(\lambda (H2: (csubst0 i0 v0 c0 
+c2)).(\lambda (H3: (((eq C c0 (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
+(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: 
+nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j 
+v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))))).(\lambda 
+(H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k0 | (CHead 
+_ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H7 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in 
+(\lambda (H8: (eq K k0 k)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind 
+C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or3 (ex3_2 T nat (\lambda 
+(_: T).(\lambda (j: nat).(eq nat i0 (s k j)))) (\lambda (u3: T).(\lambda (_: 
+nat).(eq C c2 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j 
+v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i0 (s k j))))) (\lambda (u3: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))))))) H3 c1 H9) 
+in (let H11 \def (eq_ind C c0 (\lambda (c: C).(csubst0 i0 v0 c c2)) H2 c1 H9) 
+in (let H12 \def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H7) 
+in (eq_ind_r K k (\lambda (k1: K).(or3 (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq 
+C (CHead c2 k1 u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: 
+nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
+(CHead c2 k1 u2) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda 
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k1 u2) (CHead c3 k u3))))) (\lambda 
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 c3))))))) (or3_intro2 
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) 
+(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3)))) 
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (ex3_2 C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda 
+(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k 
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v0 u1 
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v0 c1 
+c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda 
+(_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v0 u1 u3)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v0 c1 c3)))) u2 c2 i0 (refl_equal nat (s k 
+i0)) (refl_equal C (CHead c2 k u2)) H12 H11)) k0 H8))))))) H6)) 
+H5))))))))))))) i v y x H0))) H))))))).
+
+lemma csubst0_gen_S_bind_2:
+ \forall (b: B).(\forall (x: C).(\forall (c2: C).(\forall (v: T).(\forall 
+(v2: T).(\forall (i: nat).((csubst0 (S i) v x (CHead c2 (Bind b) v2)) \to 
+(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C x 
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C x (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: 
+C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: 
+T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C x (CHead c1 
+(Bind b) v1))))))))))))
+\def
+ \lambda (b: B).(\lambda (x: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (v: T).(\forall (v2: T).(\forall (i: nat).((csubst0 (S i) v c 
+(CHead c2 (Bind b) v2)) \to (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
+(\lambda (v1: T).(eq C c (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C c (CHead c1 (Bind b) v2)))) 
+(ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda 
+(c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C c (CHead c1 (Bind b) v1)))))))))))) (\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (v: T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H: (csubst0 
+(S i) v (CSort n) (CHead c2 (Bind b) v2))).(csubst0_gen_sort (CHead c2 (Bind 
+b) v2) v (S i) n H (or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda 
+(v1: T).(eq C (CSort n) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CSort n) (CHead c1 (Bind b) 
+v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) 
+(\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
+C).(\lambda (v1: T).(eq C (CSort n) (CHead c1 (Bind b) v1))))))))))))) 
+(\lambda (c: C).(\lambda (_: ((\forall (c2: C).(\forall (v: T).(\forall (v2: 
+T).(\forall (i: nat).((csubst0 (S i) v c (CHead c2 (Bind b) v2)) \to (or3 
+(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C c (CHead 
+c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: 
+C).(eq C c (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
+T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C c (CHead c1 (Bind b) 
+v1))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: 
+T).(\lambda (v2: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c 
+k t) (CHead c2 (Bind b) v2))).(let H1 \def (csubst0_gen_head k c (CHead c2 
+(Bind b) v2) t v (S i) H0) in (or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda 
+(j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C 
+(CHead c2 (Bind b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
+nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind 
+b) v2) (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq 
+nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq 
+C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3))))) (or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1)))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq 
+nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind 
+b) v2) (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t 
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k 
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead 
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) 
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
+(x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C (CHead 
+c2 (Bind b) v2) (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 
+\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead c k x0) H4) in 
+((let H7 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) 
+v2) (CHead c k x0) H4) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 
+(Bind b) v2) (CHead c k x0) H4) in (\lambda (H9: (eq K (Bind b) k)).(\lambda 
+(H10: (eq C c2 c)).(let H11 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x1 v 
+t t0)) H5 v2 H8) in (eq_ind_r C c (\lambda (c0: C).(or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c0 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c0)) (\lambda (c1: C).(eq C 
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c0))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1))))))) (let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
+x1))) H3 (Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
+(CHead c (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c)) (\lambda 
+(c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda 
+(_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: 
+T).(csubst0 i v c1 c))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 
+t) (CHead c1 (Bind b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: 
+nat).(match e with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) 
+H12) in (let H14 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t v2)) 
+H11 i H13) in (or3_intro0 (ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) 
+(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1)))) (ex2 C 
+(\lambda (c1: C).(csubst0 i v c1 c)) (\lambda (c1: C).(eq C (CHead c (Bind b) 
+t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: 
+T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c))) 
+(\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind 
+b) v1))))) (ex_intro2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c (Bind b) v1))) t H14 (refl_equal C 
+(CHead c (Bind b) t)))))) k H9)) c2 H10))))) H7)) H6))))))) H2)) (\lambda 
+(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) 
+(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k 
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C 
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 v2))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind 
+b) v1)))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) 
+(s k x1))).(\lambda (H4: (eq C (CHead c2 (Bind b) v2) (CHead x0 k 
+t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H7 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 
+_) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in ((let H8 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v2 | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) (CHead x0 k t) H4) in 
+(\lambda (H9: (eq K (Bind b) k)).(\lambda (H10: (eq C c2 x0)).(let H11 \def 
+(eq_ind_r C x0 (\lambda (c0: C).(csubst0 x1 v c c0)) H5 c2 H10) in (eq_ind_r 
+T t (\lambda (t0: T).(or3 (ex2 T (\lambda (v1: T).(subst0 i v v1 t0)) 
+(\lambda (v1: T).(eq C (CHead c k t) (CHead c2 (Bind b) v1)))) (ex2 C 
+(\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c k t) 
+(CHead c1 (Bind b) t0)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 
+i v v1 t0))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda 
+(c1: C).(\lambda (v1: T).(eq C (CHead c k t) (CHead c1 (Bind b) v1))))))) 
+(let H12 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 x1))) H3 
+(Bind b) H9) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T (\lambda (v1: 
+T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C (CHead c k0 t) (CHead c2 (Bind 
+b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c k0 t) (CHead c1 (Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda 
+(v1: T).(subst0 i v v1 t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 
+c2))) (\lambda (c1: C).(\lambda (v1: T).(eq C (CHead c k0 t) (CHead c1 (Bind 
+b) v1))))))) (let H13 \def (f_equal nat nat (\lambda (e: nat).(match e with 
+[O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H12) in (let H14 \def 
+(eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c c2)) H11 i H13) in 
+(or3_intro1 (ex2 T (\lambda (v1: T).(subst0 i v v1 t)) (\lambda (v1: T).(eq C 
+(CHead c (Bind b) t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: 
+C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 
+(Bind b) t)))) (ex3_2 C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 
+t))) (\lambda (c1: C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: 
+C).(\lambda (v1: T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) 
+(ex_intro2 C (\lambda (c1: C).(csubst0 i v c1 c2)) (\lambda (c1: C).(eq C 
+(CHead c (Bind b) t) (CHead c1 (Bind b) t))) c H14 (refl_equal C (CHead c 
+(Bind b) t)))))) k H9)) v2 H8))))) H7)) H6))))))) H2)) (\lambda (H2: (ex4_3 T 
+C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 
+(Bind b) v2) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C (CHead c2 (Bind b) v2) (CHead c3 k u2))))) 
+(\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) 
+(\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or3 
+(ex2 T (\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k 
+t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c k t) (CHead c1 (Bind b) v1)))))) (\lambda (x0: T).(\lambda 
+(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda 
+(H4: (eq C (CHead c2 (Bind b) v2) (CHead x1 k x0))).(\lambda (H5: (subst0 x2 
+v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) in ((let H8 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) v2) (CHead x1 k x0) H4) 
+in ((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow v2 | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) v2) 
+(CHead x1 k x0) H4) in (\lambda (H10: (eq K (Bind b) k)).(\lambda (H11: (eq C 
+c2 x1)).(let H12 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 x2 v c c0)) H6 
+c2 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t0: T).(subst0 x2 v t t0)) 
+H5 v2 H9) in (let H14 \def (eq_ind_r K k (\lambda (k0: K).(eq nat (S i) (s k0 
+x2))) H3 (Bind b) H10) in (eq_ind K (Bind b) (\lambda (k0: K).(or3 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c k0 t) 
+(CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c k0 t) (CHead c1 (Bind b) v2)))) (ex3_2 C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c k0 t) (CHead c1 (Bind b) v1))))))) (let H15 \def (f_equal 
+nat nat (\lambda (e: nat).(match e with [O \Rightarrow i | (S n) \Rightarrow 
+n])) (S i) (S x2) H14) in (let H16 \def (eq_ind_r nat x2 (\lambda (n: 
+nat).(csubst0 n v c c2)) H12 i H15) in (let H17 \def (eq_ind_r nat x2 
+(\lambda (n: nat).(subst0 n v t v2)) H13 i H15) in (or3_intro2 (ex2 T 
+(\lambda (v1: T).(subst0 i v v1 v2)) (\lambda (v1: T).(eq C (CHead c (Bind b) 
+t) (CHead c2 (Bind b) v1)))) (ex2 C (\lambda (c1: C).(csubst0 i v c1 c2)) 
+(\lambda (c1: C).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v2)))) (ex3_2 
+C T (\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1))))) (ex3_2_intro C T 
+(\lambda (_: C).(\lambda (v1: T).(subst0 i v v1 v2))) (\lambda (c1: 
+C).(\lambda (_: T).(csubst0 i v c1 c2))) (\lambda (c1: C).(\lambda (v1: 
+T).(eq C (CHead c (Bind b) t) (CHead c1 (Bind b) v1)))) c t H17 H16 
+(refl_equal C (CHead c (Bind b) t))))))) k H10))))))) H8)) H7))))))))) H2)) 
+H1))))))))))) x)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/getl.ma
new file mode 100644 (file)
index 0000000..ae6794f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/getl.ma
@@ -0,0 +1,1145 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/clear.ma".
+
+include "basic_1A/csubst0/drop.ma".
+
+include "basic_1A/getl/fwd.ma".
+
+lemma csubst0_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all 
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: 
+C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1 
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5: 
+(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4)) 
+(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: 
+nat).(drop n0 O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: 
+nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e)) 
+(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9: 
+(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat 
+x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_: 
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
+(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3) 
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9)) 
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0: 
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x 
+(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0) 
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda 
+(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e 
+(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e 
+(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F 
+C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O 
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) 
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0: 
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O 
+c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: 
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) 
+H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4) 
+H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e 
+x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
+i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
+
+lemma csubst0_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 
+e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all 
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: 
+C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: 
+C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def 
+(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K 
+C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) 
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O 
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B 
+C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) 
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: 
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u 
+w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda 
+(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k 
+n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: 
+K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x 
+(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda 
+(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda 
+(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in (K_ind (\lambda (k: K).((drop 
+n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear 
+(CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: 
+B).(\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: 
+(subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 
+(Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 
+(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind 
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 
+(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead 
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 
+(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) 
+e (clear_gen_bind b x1 e x2 H13)))))) (\lambda (f: F).(\lambda (H11: (drop n 
+O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v 
+x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n 
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 
+(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))) 
+x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k 
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind 
+K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 
+e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: 
+K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x 
+(CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda 
+(H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda 
+(c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop 
+n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear 
+(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) (\lambda (b: 
+B).(\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: 
+(csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 
+(Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 
+(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind 
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 
+(CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead 
+e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n 
+c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) 
+H12)) e (clear_gen_bind b x1 e x3 H13)))))) (\lambda (f: F).(\lambda (H11: 
+(drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s 
+(Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let 
+H14 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(csubst0 n0 v x1 x2)) H12 
+(S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 
+x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear 
+x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear 
+x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 
+B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e 
+H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) 
+v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: 
+B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e 
+(CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4) 
+x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5 
+(Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 
+(getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead 
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
+x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) 
+u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 
+x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind 
+x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 
+f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda 
+(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: 
+(clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v 
+x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) 
+x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) 
+x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) 
+(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 
+(refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) 
+x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f 
+x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind 
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda 
+(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq 
+C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) 
+x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: 
+(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) 
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
+(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead 
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) 
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) 
+(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 
+(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) 
+H16)) H15))))))) x0 H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T 
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda 
+(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k 
+n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k 
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k 
+n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n 
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda 
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq 
+C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda 
+(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s 
+x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
+(CHead x1 x0 x3) H7) in (K_ind (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) 
+\to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 
+x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))))))))) (\lambda (b: B).(\lambda (H12: (drop n O c2 (CHead x2 
+(Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 
+x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda 
+(H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) 
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) 
+(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
+(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead 
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda 
+(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C 
+T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) 
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 
+(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) 
+(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind 
+b x1 e x3 H15))))))) (\lambda (f: F).(\lambda (H12: (drop n O c2 (CHead x2 
+(Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 
+x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda 
+(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) 
+(\lambda (n0: nat).(csubst0 n0 v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy 
+i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e 
+(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) 
+v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: 
+(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e 
+(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: 
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) 
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda 
+(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) 
+x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: 
+(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) 
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 
+(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead 
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) 
+u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 
+x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind 
+x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 
+f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind 
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda 
+(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: 
+(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v 
+x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5) 
+x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) 
+x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) 
+(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 
+(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) 
+x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f 
+x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind 
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda 
+(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq 
+C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) 
+x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: 
+(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) 
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 
+(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead 
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) 
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) 
+(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 
+(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) 
+H18)) H17)))))))) x0 H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
+
+lemma csubst0_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all 
+c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0: 
+C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2 
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5: 
+(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3) 
+H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n0: 
+nat).(drop n0 O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n0: 
+nat).(le i n0)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e)) 
+(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O 
+c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1 
+e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9: 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat 
+x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_: 
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 
+(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2) 
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9)) 
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0: 
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x 
+(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) 
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda 
+(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e 
+(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v 
+H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i 
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat 
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0: 
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O 
+c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13: 
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e)) 
+H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3) 
+H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0 
+x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n 
+i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
+
+lemma csubst0_getl_lt_back:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e2: C).((getl n c2 
+e2) \to (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
+e2)) (\lambda (e1: C).(getl n c1 e1))))))))))))
+\def
+ \lambda (n: nat).(\lambda (i: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 
+c2)).(\lambda (e2: C).(\lambda (H1: (getl n c2 e2)).(let H2 \def 
+(getl_gen_all c2 e2 n H1) in (ex2_ind C (\lambda (e: C).(drop n O c2 e)) 
+(\lambda (e: C).(clear e e2)) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
+(x: C).(\lambda (H3: (drop n O c2 x)).(\lambda (H4: (clear x e2)).(let H_x 
+\def (csubst0_drop_lt_back n i H c1 c2 v H0 x H3) in (let H5 \def H_x in 
+(or_ind (drop n O c1 x) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 x)) 
+(\lambda (e1: C).(drop n O c1 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
+(H6: (drop n O c1 x)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) 
+(getl_intro n c1 e2 x H6 H4))) (\lambda (H6: (ex2 C (\lambda (e1: C).(csubst0 
+(minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)))).(ex2_ind C (\lambda 
+(e1: C).(csubst0 (minus i n) v e1 x)) (\lambda (e1: C).(drop n O c1 e1)) (or 
+(getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) 
+(\lambda (e1: C).(getl n c1 e1)))) (\lambda (x0: C).(\lambda (H7: (csubst0 
+(minus i n) v x0 x)).(\lambda (H8: (drop n O c1 x0)).(let H_x0 \def 
+(csubst0_clear_trans x0 x v (minus i n) H7 e2 H4) in (let H9 \def H_x0 in 
+(or_ind (clear x0 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) 
+(\lambda (e1: C).(clear x0 e1))) (or (getl n c1 e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda 
+(H10: (clear x0 e2)).(or_introl (getl n c1 e2) (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(getl n c1 e1))) 
+(getl_intro n c1 e2 x0 H8 H10))) (\lambda (H10: (ex2 C (\lambda (e1: 
+C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 e1)))).(ex2_ind 
+C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: C).(clear x0 
+e1)) (or (getl n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
+e2)) (\lambda (e1: C).(getl n c1 e1)))) (\lambda (x1: C).(\lambda (H11: 
+(csubst0 (minus i n) v x1 e2)).(\lambda (H12: (clear x0 x1)).(or_intror (getl 
+n c1 e2) (ex2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 e2)) (\lambda (e1: 
+C).(getl n c1 e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (minus i n) v e1 
+e2)) (\lambda (e1: C).(getl n c1 e1)) x1 H11 (getl_intro n c1 x1 x0 H8 
+H12)))))) H10)) H9)))))) H6)) H5)))))) H2)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/props.ma
new file mode 100644 (file)
index 0000000..a681912
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst0/props.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/defs.ma".
+
+lemma csubst0_snd_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c 
+(Bind b) u1) (CHead c (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind 
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b) 
+u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S 
+i))))))))).
+
+lemma csubst0_fst_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall 
+(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 
+(Bind b) u) (CHead c2 (Bind b) u))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind 
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b) 
+u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))).
+
+theorem csubst0_both_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i 
+v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) 
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: 
+nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) 
+(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S 
+i))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/defs.ma
new file mode 100644 (file)
index 0000000..8a38449
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/defs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/defs.ma".
+
+inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
+| csubst1_refl: csubst1 i v c1 c1
+| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/fwd.ma
new file mode 100644 (file)
index 0000000..16c018a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/fwd.ma
@@ -0,0 +1,128 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst1/defs.ma".
+
+include "basic_1A/csubst0/fwd.ma".
+
+include "basic_1A/subst1/defs.ma".
+
+include "basic_1A/s/fwd.ma".
+
+implied lemma csubst1_ind:
+ \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (P: ((C \to 
+Prop))).((P c1) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2)))) \to 
+(\forall (c: C).((csubst1 i v c1 c) \to (P c))))))))
+\def
+ \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (P: ((C \to 
+Prop))).(\lambda (f: (P c1)).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1 
+c2) \to (P c2))))).(\lambda (c: C).(\lambda (c0: (csubst1 i v c1 c)).(match 
+c0 with [csubst1_refl \Rightarrow f | (csubst1_sing x x0) \Rightarrow (f0 x 
+x0)])))))))).
+
+lemma csubst1_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall 
+(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T 
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: 
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 i v c1 c2))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) 
+x)).(csubst1_ind (s k i) v (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C 
+(\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: 
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: 
+C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 
+c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl 
+i v c1)) (\lambda (c2: C).(\lambda (H0: (csubst0 (s k i) v (CHead c1 k u1) 
+c2)).(let H1 \def (csubst0_gen_head k c1 c2 u1 v (s k i) H0) in (or3_ind 
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) 
+(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: 
+T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 
+j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
+(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
+C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2: 
+T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
+c3)))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat 
+(s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k 
+u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T 
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda 
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c1 k u2)))) (\lambda (u2: 
+T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2: 
+T).(\lambda (c3: C).(eq C c2 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
+c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (s k i) (s k 
+x1))).(\lambda (H4: (eq C c2 (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1 
+x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: 
+T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 
+c3))))) (let H_y \def (s_inj k i x1 H3) in (let H6 \def (eq_ind_r nat x1 
+(\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in (ex3_2_intro T C (\lambda 
+(u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda 
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: 
+C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single 
+i v u1 x0 H6) (csubst1_refl i v c1)))) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 
+C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda 
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: 
+C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: 
+nat).(eq C c2 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 
+j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 
+k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: 
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: 
+nat).(\lambda (H3: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C c2 (CHead 
+x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) 
+(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead 
+c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda 
+(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x1 
+H3) in (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) 
+H5 i H_y) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead 
+x0 k u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 
+u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 
+(refl_equal C (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 
+H6)))) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda 
+(_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C 
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k 
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 
+c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c2 (CHead c3 k 
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: 
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: 
+C).(\lambda (x2: nat).(\lambda (H3: (eq nat (s k i) (s k x2))).(\lambda (H4: 
+(eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: 
+(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C 
+(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: 
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: 
+C).(csubst1 i v c1 c3))))) (let H_y \def (s_inj k i x2 H3) in (let H7 \def 
+(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c1 x1)) H6 i H_y) in (let H8 
+\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i H_y) in 
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) 
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) 
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C 
+(CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7))))) 
+c2 H4)))))))) H2)) H1)))) x H))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/getl.ma
new file mode 100644 (file)
index 0000000..4805e5f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/getl.ma
@@ -0,0 +1,269 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst1/props.ma".
+
+include "basic_1A/csubst0/getl.ma".
+
+include "basic_1A/subst1/props.ma".
+
+include "basic_1A/drop/props.ma".
+
+lemma csubst1_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to 
+(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda 
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
+(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+lemma csubst1_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 
+e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: 
+C).(getl n c2 e2)))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to 
+(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl 
+n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S 
+(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 
+e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1 
+(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H)))) 
+(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda 
+(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0: 
+nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n 
+c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind 
+(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: 
+B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: 
+T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n 
+c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2: 
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) 
+(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S 
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl 
+(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind 
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u 
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S 
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1 
+(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0) 
+x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1 
+(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S 
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2: 
+C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) 
+v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus 
+i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T 
+(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 
+(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: 
+C).(\lambda (u: T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+v e2 e3))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3 
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1 
+(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1 
+(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) 
+x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1 
+(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S 
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2: 
+C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n))) 
+v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus 
+i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T 
+T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e3 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) v e2 e3)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda 
+(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2 
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda 
+(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3)))))) 
+(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: 
+C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind 
+x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7: 
+(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1 
+x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: 
+C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind 
+x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4) 
+(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind 
+x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1 
+H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
+
+lemma csubst1_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1 
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to 
+(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda 
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2: 
+(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+lemma getl_csubst1:
+ \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c 
+(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: 
+C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 
+a))))))))
+\def
+ \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: 
+C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C 
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: 
+C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind 
+(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind 
+Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda 
+(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) 
+(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 
+C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda 
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda 
+(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to 
+(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda 
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(K_ind 
+(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O 
+(CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: 
+C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: 
+C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (b: B).(\lambda (t: 
+T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) 
+t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e 
+(Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) 
+H0))) in ((let H2 \def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort 
+_) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead e (Bind Abbr) u) 
+(CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t 
+(getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H3 
+\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) 
+t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind 
+b) t) (CHead e (Bind Abbr) u) H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda 
+(_: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(ex2_2 C C (\lambda (a0: 
+C).(\lambda (_: C).(csubst1 O t0 (CHead c0 (Bind b) t) a0))) (\lambda (a0: 
+C).(\lambda (a: C).(drop (S O) O a0 a))))) (eq_ind B Abbr (\lambda (b0: 
+B).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind 
+b0) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) 
+(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 
+(Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) 
+(CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead c0 (Bind Abbr) t)) 
+(drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u H3)))) H2)) 
+H1))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let 
+H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T (\lambda (t2: 
+T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: 
+C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
+C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: (subst1 O u t (lift 
+(S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead e (Bind Abbr) u) c0 
+(drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) u) t (getl_gen_O 
+(CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in (ex2_2_ind C C 
+(\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: 
+C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda 
+(_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
+C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H4: 
+(csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 x1)).(ex2_2_intro C C 
+(\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead x0 (Flat f) 
+(lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O x) H2 c0 x0 H4) 
+(drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) H1)))))))) k)))) 
+c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall (e: C).(\forall 
+(u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: 
+C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: 
+C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind (\lambda (c0: 
+C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) 
+\to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) (\lambda (n0: 
+nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) (CSort n0) 
+(CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind Abbr) u) H0 
+(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CSort n0) a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) (\lambda 
+(c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead 
+e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S 
+n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 
+a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall 
+(e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 t) (CHead e (Bind Abbr) u)) 
+\to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 k0 
+t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) 
+(\lambda (b: B).(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: 
+(getl (S n) (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H_x \def 
+(subst1_ex u t n) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 n 
+u t (lift (S O) n t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 
+(S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S 
+O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 n u t (lift (S O) n 
+x))).(let H4 \def (H c0 e u (getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t 
+n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C 
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: 
+C).(\lambda (x1: C).(\lambda (H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S 
+O) n x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) 
+u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S 
+n) a0 a))) (CHead x0 (Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) 
+(csubst1_bind b n u t (lift (S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n 
+x0 x1 H6 b x)))))) H4)))) H2)))))))) (\lambda (f: F).(\lambda (t: T).(\lambda 
+(e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead 
+e (Bind Abbr) u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in 
+(ex_ind T (\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C 
+C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) 
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda 
+(x: T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def 
+(H0 e u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in 
+(ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C 
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) 
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: 
+C).(\lambda (x1: C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: 
+(drop (S O) (S n) x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: 
+C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: 
+C).(drop (S O) (S n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead 
+x1 (Flat f) x) (csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) 
+(drop_skip_flat (S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))) k)))) c)))) d).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/props.ma
new file mode 100644 (file)
index 0000000..283eb28
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubst1/props.ma
@@ -0,0 +1,66 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst1/fwd.ma".
+
+include "basic_1A/subst1/fwd.ma".
+
+theorem csubst1_head:
+ \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
+v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: 
+T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k 
+i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c: 
+C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i) 
+v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 
+c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i 
+c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 
+t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1 
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1) 
+(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2) 
+(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v 
+c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both 
+k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))).
+
+theorem csubst1_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
+v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) 
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n: 
+nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2))) 
+(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S 
+i))))))))))).
+
+theorem csubst1_flat:
+ \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall 
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i 
+v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) 
+u2))))))))))
+\def
+ \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda 
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n: 
+nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2))) 
+(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/clear.ma
new file mode 100644 (file)
index 0000000..86e44ec
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/clear.ma
@@ -0,0 +1,71 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma csubt_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to 
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c0 
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) 
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (H0: (csubt g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear c4 
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: 
+(clear (CHead c3 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c3 k0 u) 
+e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear 
+(CHead c4 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c3 (Bind 
+b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csubt g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) 
+(ex_intro2 C (\lambda (e2: C).(csubt g (CHead c3 (Bind b) u) e2)) (\lambda 
+(e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csubt_head g 
+c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3)))) 
+(\lambda (f: F).(\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def 
+(H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csubt g 
+e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csubt g e1 
+e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: 
+C).(\lambda (H5: (csubt g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C 
+(\lambda (e2: C).(csubt g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) 
+u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3: 
+C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: ((\forall 
+(e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) (\lambda 
+(e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b 
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: 
+(clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) 
+(\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubt 
+g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) 
+u2) e2)) (CHead c4 (Bind b) u2) (csubt_void g c3 c4 H0 b H2 u1 u2) 
+(clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubt g c3 c4)).(\lambda (_: 
+((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csubt g e1 e2)) 
+(\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(H2: (ty3 g c3 u t)).(\lambda (H3: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda 
+(H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) 
+t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubt g c e2)) (\lambda (e2: 
+C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csubt g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 
+(Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csubt_abst g c3 c4 H0 u t H2 
+H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4)))))))))))) c1 
+c2 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/csuba.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/csuba.ma
new file mode 100644 (file)
index 0000000..965116e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/csuba.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/arity.ma".
+
+lemma csubt_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (csuba 
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda 
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda 
+(_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda 
+(u: T).(csuba_head g c3 c4 H1 k u))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (b: 
+B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(csuba_void g c3 c4 H1 b H2 u1 u2))))))))) (\lambda (c3: C).(\lambda (c4: 
+C).(\lambda (_: (csubt g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda (_: (ty3 g c4 u 
+t)).(let H_x \def (ty3_arity g c3 u t H2) in (let H4 \def H_x in (ex2_ind A 
+(\lambda (a1: A).(arity g c3 u a1)) (\lambda (a1: A).(arity g c3 t (asucc g 
+a1))) (csuba g (CHead c3 (Bind Abst) t) (CHead c4 (Bind Abbr) u)) (\lambda 
+(x: A).(\lambda (H5: (arity g c3 u x)).(\lambda (H6: (arity g c3 t (asucc g 
+x))).(csuba_abst g c3 c4 H1 t x H6 u (csuba_arity g c3 u x H5 c4 H1))))) 
+H4))))))))))) c1 c2 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/defs.ma
new file mode 100644 (file)
index 0000000..f129a74
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/defs.ma".
+
+inductive csubt (g: G): C \to (C \to Prop) \def
+| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
+| csubt_head: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(k: K).(\forall (u: T).(csubt g (CHead c1 k u) (CHead c2 k u))))))
+| csubt_void: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csubt g 
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csubt_abst: \forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall 
+(u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u t) \to (csubt g 
+(CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/drop.ma
new file mode 100644 (file)
index 0000000..1dc494d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/drop.ma
@@ -0,0 +1,579 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/fwd.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+lemma csubt_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall 
+(c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: 
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let 
+H_x \def (csubt_gen_flat g d1 c2 u f H1) in (let H2 \def H_x in (ex2_ind C 
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) u))) (\lambda (e2: C).(csubt g 
+d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O 
+c2 (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x 
+(Flat f) u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Flat f) u) 
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c (CHead d2 (Flat f) u))))) (ex_intro2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Flat f) u) (CHead d2 (Flat f) 
+u))) x H4 (drop_refl (CHead x (Flat f) u))) c2 H3)))) H2)))))))))) (\lambda 
+(n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) 
+\to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) 
+\to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 
+(CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(H0: (csubt g c1 c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).(\forall 
+(d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead 
+d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: 
+T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(and3_ind 
+(eq C (CHead d1 (Flat f) u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) 
+(CHead d2 (Flat f) u)))) (\lambda (_: (eq C (CHead d1 (Flat f) u) (CSort 
+n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 
+\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow 
+False | (S _) \Rightarrow True])) I O H3) in (False_ind (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 
+(Flat f) u)))) H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Flat f) u) 
+H1)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 
+c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (k: 
+K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: 
+T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) u0)) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda 
+(d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) 
+(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) 
+(CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
+x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
+u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) 
+u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Flat f) 
+u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: 
+C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) 
+(CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) 
+u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
+x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead 
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) 
+u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: 
+(csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x 
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead 
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) 
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
+t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda 
+(H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) 
+u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 
+O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) 
+u0)))) (\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O 
+c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) 
+u0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H7 u))))) (H c0 
+c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 
+H5)))))))))))))) c1 c2 H0)))))) n))).
+
+lemma csubt_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind 
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop 
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: 
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) 
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g c1 
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in 
+(let H2 \def (csubt_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (e2: C).(eq 
+C c2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt g d1 e2)) (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) 
+u))).(\lambda (H4: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) 
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c (CHead d2 (Bind Abbr) u))))) (ex_intro2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead 
+d2 (Bind Abbr) u))) x H4 (drop_refl (CHead x (Bind Abbr) u))) c2 H3)))) 
+H2))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csubt g c1 c2)).(csubt_ind g (\lambda 
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c 
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda 
+(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O 
+(CSort n1) (CHead d1 (Bind Abbr) u))).(and3_ind (eq C (CHead d1 (Bind Abbr) 
+u) (CSort n1)) (eq nat (S n0) O) (eq nat O O) (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) 
+u)))) (\lambda (_: (eq C (CHead d1 (Bind Abbr) u) (CSort n1))).(\lambda (H3: 
+(eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(let H5 \def (eq_ind nat (S n0) 
+(\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H3) in (False_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) 
+H5))))) (drop_gen_sort n1 (S n0) O (CHead d1 (Bind Abbr) u) H1)))))) (\lambda 
+(c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (H2: 
+((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) 
+u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (k: K).(K_ind (\lambda 
+(k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O 
+(CHead c0 k0 u) (CHead d1 (Bind Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind 
+Abbr) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda 
+(u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind 
+Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 x)).(\lambda (H5: 
+(drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 
+(Bind Abbr) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 
+u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) 
+u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda 
+(u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind 
+Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) 
+(CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csubt g d1 
+x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Flat f) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead 
+x (Bind Abbr) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 
+(Bind Abbr) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: 
+T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) 
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S 
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda 
+(x: C).(\lambda (H5: (csubt g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x 
+(Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u 
+(drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) 
+(\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: 
+((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) 
+u)) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: 
+C).(\lambda (u0: T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) 
+(CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind 
+Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H6: (csubt g 
+d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H6 (drop_drop (Bind Abbr) n0 c3 
+(CHead x (Bind Abbr) u0) H7 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind 
+Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
+
+lemma csubt_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csubt g 
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind 
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n 
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: 
+C).(\forall (c2: C).((csubt g c1 c2) \to (\forall (d1: C).(\forall (t: 
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) 
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
+g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubt g 
+c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 
+(Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c2)) H 
+(CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in 
+(let H2 \def (csubt_gen_abst g d1 c2 t H1) in (or_ind (ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2))) 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O 
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H3: (ex2 C 
+(\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt 
+g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) t))) 
+(\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) t))).(\lambda 
+(H5: (csubt g d1 x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c: C).(or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O c (CHead 
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c (CHead d2 (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_introl (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead 
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O (CHead x (Bind Abst) t) 
+(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x (Bind Abst) t) (CHead 
+d2 (Bind Abst) t))) x H5 (drop_refl (CHead x (Bind Abst) t)))) c2 H4)))) H3)) 
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+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) (or_intror (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop O O (CHead x0 (Bind 
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+\def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee with [O \Rightarrow 
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+d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x: C).(\lambda (H6: (csubt g d1 x)).(\lambda (H7: (drop n0 O c3 
+(CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) 
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind 
+Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g 
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 
+(Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 
+u2)))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex4_2_ind C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 
+u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) 
+u2) (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csubt g d1 x0)).(\lambda 
+(H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g d1 x1 
+t)).(\lambda (H9: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda 
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
+g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 
+u2) H8 H9)))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead 
+d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: 
+T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) 
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c0 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: 
+T).(\lambda (H5: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind 
+Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g 
+d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 
+(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
+g d2 u0 t0))))) (\lambda (H6: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 
+(Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H7: 
+(csubt g d1 x)).(\lambda (H8: (drop n0 O c3 (CHead x (Bind Abst) 
+t0))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) 
+(CHead d2 (Bind Abst) t0))) x H7 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind 
+Abst) t0) H8 u)))))) H6)) (\lambda (H6: (ex4_2 C T (\lambda (d2: C).(\lambda 
+(_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 
+(CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 
+t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))).(ex4_2_ind C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda 
+(u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 
+t0))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop (S 
+n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) 
+(\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda 
+(u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H7: 
+(csubt g d1 x0)).(\lambda (H8: (drop n0 O c3 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H9: (ty3 g d1 x1 t0)).(\lambda (H10: (ty3 g x0 x1 
+t0)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) 
+(ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind 
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda 
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex4_2_intro C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop 
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(ty3 g d1 u0 t0))) (\lambda (d2: C).(\lambda (u0: T).(ty3 
+g d2 u0 t0))) x0 x1 H7 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) 
+H8 u) H9 H10)))))))) H6)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 
+(CHead d1 (Bind Abst) t0) t n0 H5)))))))))))))) c1 c2 H0)))))) n)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/fwd.ma
new file mode 100644 (file)
index 0000000..e3f3529
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/fwd.ma
@@ -0,0 +1,386 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/defs.ma".
+
+implied rec lemma csubt_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubt 
+g c1 c2) \to ((P c1 c2) \to (\forall (k: K).(\forall (u: T).(P (CHead c1 k u) 
+(CHead c2 k u))))))))) (f1: (\forall (c1: C).(\forall (c2: C).((csubt g c1 
+c2) \to ((P c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: 
+T).(\forall (u2: T).(P (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) 
+u2))))))))))) (f2: (\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to ((P 
+c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to ((ty3 g c2 u 
+t) \to (P (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))))) (c: C) 
+(c0: C) (c1: csubt g c c0) on c1: P c c0 \def match c1 with [(csubt_sort n) 
+\Rightarrow (f n) | (csubt_head c2 c3 c4 k u) \Rightarrow (f0 c2 c3 c4 
+((csubt_ind g P f f0 f1 f2) c2 c3 c4) k u) | (csubt_void c2 c3 c4 b n u1 u2) 
+\Rightarrow (f1 c2 c3 c4 ((csubt_ind g P f f0 f1 f2) c2 c3 c4) b n u1 u2) | 
+(csubt_abst c2 c3 c4 u t t0 t1) \Rightarrow (f2 c2 c3 c4 ((csubt_ind g P f f0 
+f1 f2) c2 c3 c4) u t t0 t1)].
+
+lemma csubt_gen_abbr:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g 
+(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr) 
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2: 
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) 
+(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda 
+(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind 
+Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 
+(Bind Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) 
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C 
+c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abbr) 
+v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 
+(Bind Abbr) v) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k 
+u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in (\lambda (H7: (eq K k (Bind 
+Abbr))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C 
+(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda 
+(e2: C).(csubt g e1 e2)))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C 
+(\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda 
+(e2: C).(csubt g e1 e2)))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c 
+(CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))) H2 e1 H8) in (let H10 
+\def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in (ex_intro2 C 
+(\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) 
+(\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abbr) v)) 
+H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: 
+C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind 
+Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) 
+(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B 
+b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 
+(Bind Void) u1) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 
+(Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
+| (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match b0 
+with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
+True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H4) in 
+(False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
+c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u 
+t)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) 
+v))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) 
+I (CHead e1 (Bind Abbr) v) H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C 
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g 
+e1 e2))) H6))))))))))) y c2 H0))) H))))).
+
+lemma csubt_gen_abst:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g 
+(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead 
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda 
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda 
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind 
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda 
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 
+e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind 
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or 
+(ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: 
+(eq C (CSort n) (CHead e1 (Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or (ex2 C 
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2: 
+C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H2)))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
+(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g 
+e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead e1 (Bind Abst) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind Abst) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in (\lambda 
+(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 
+(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2: 
+C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 
+v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: 
+C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: 
+C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C 
+(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind 
+Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) 
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda 
+(e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3 (Bind Abst) v1)) H10)))) 
+k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda 
+(_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to 
+(or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda 
+(e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C 
+c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not 
+(eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead 
+c1 (Bind Void) u1) (CHead e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead 
+c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
+False | (CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow (match 
+b0 with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow 
+True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H4) in 
+(False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt 
+g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C 
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt 
+g e1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) 
+(\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: 
+(ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: (eq C (CHead c1 
+(Bind Abst) t) (CHead e1 (Bind Abst) v1))).(let H6 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow 
+c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H5) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow t | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) 
+H5) in (\lambda (H8: (eq C c1 e1)).(let H9 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c3 u t0)) H4 v1 H7) in (let H10 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c1 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c: 
+C).(ty3 g c u v1)) H10 e1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C 
+c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex4_2 C T 
+(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) 
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: C).(\lambda 
+(v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 
+v1))))))) H2 e1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H8) in (or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind 
+Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) 
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex4_2_intro C T (\lambda (e2: 
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g e1 v2 v1))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13 H11 
+H9))))))))) H6))))))))))) y c2 H0))) H))))).
+
+lemma csubt_gen_flat:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).(\forall 
+(f: F).((csubt g (CHead e1 (Flat f) v) c2) \to (ex2 C (\lambda (e2: C).(eq C 
+c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda 
+(f: F).(\lambda (H: (csubt g (CHead e1 (Flat f) v) c2)).(insert_eq C (CHead 
+e1 (Flat f) v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C 
+(\lambda (e2: C).(eq C c2 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
+e1 e2)))) (\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C (\lambda 
+(e2: C).(eq C c0 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Flat f) 
+v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Flat f) 
+v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Flat 
+f) v))) (\lambda (e2: C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda 
+(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 
+(Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) 
+(\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Flat f) v))).(let H4 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
+c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in ((let H5 
+\def (f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow k | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Flat f) v) H3) in 
+((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Flat 
+f) v) H3) in (\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k 
+t) (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K 
+(Flat f) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Flat f) v)) \to (ex2 C 
+(\lambda (e2: C).(eq C c3 (CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g 
+e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Flat f) v) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C 
+(CHead c3 (Flat f) v)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1 
+(CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Flat 
+f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (b: B).(\lambda (_: 
+(not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H4: (eq C 
+(CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v))).(let H5 \def (eq_ind C 
+(CHead c1 (Bind Void) u1) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
+H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead 
+e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H5))))))))))) (\lambda 
+(c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C 
+c1 (CHead e1 (Flat f) v)) \to (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 
+(Flat f) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (_: (ty3 g c3 u t)).(\lambda 
+(H5: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v))).(let H6 \def 
+(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e1 (Flat f) v) 
+H5) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) 
+(CHead e2 (Flat f) v))) (\lambda (e2: C).(csubt g e1 e2))) H6))))))))))) y c2 
+H0))) H)))))).
+
+lemma csubt_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall 
+(v1: T).((csubt g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda 
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C 
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: 
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: 
+T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y 
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq 
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
+e1 e2))))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T 
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2)))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) 
+(CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k 
+u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) 
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: 
+(eq K k (Bind b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: 
+T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: 
+K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda 
+(c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
+H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) 
+in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C (CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C 
+(CHead c3 (Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: 
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b: 
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) 
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) 
+(CHead e1 (Bind b1) v1) H4) in ((let H6 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) 
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) 
+in (\lambda (H8: (eq B Void b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def 
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C 
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind 
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 
+e2))))))) H2 e1 H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c 
+c3)) H1 e1 H9) in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 
+(CHead e1 (Bind b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: 
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: 
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in 
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C 
+(CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda 
+(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 
+(Bind b) u2)) H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: 
+C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind 
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (H4: (ty3 g c3 u t)).(\lambda (H5: 
+(eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(let H6 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
+c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) 
+in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) 
+(CHead e1 (Bind b1) v1) H5) in ((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B 
+Abst b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda 
+(t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H12 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c1 u t0)) H3 v1 H8) in (let H13 \def (eq_ind C c1 (\lambda (c: 
+C).(ty3 g c u v1)) H12 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: 
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) 
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 
+H10) in (let H15 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H10) 
+in (let H16 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) 
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq 
+C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda 
+(_: T).(csubt g e1 e2))))))) H14 Abst H9) in (ex2_3_intro B C T (\lambda (b2: 
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g 
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H15)))))))))) 
+H7)) H6))))))))))) y c2 H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/getl.ma
new file mode 100644 (file)
index 0000000..1406030
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/getl.ma
@@ -0,0 +1,417 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/clear.ma".
+
+include "basic_1A/csubt/drop.ma".
+
+include "basic_1A/getl/clear.ma".
+
+lemma csubt_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g 
+c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) 
+(\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: 
+C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind 
+Abbr) u))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 
+(Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 (CSort n0))).(\lambda 
+(H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 
+(Bind Abbr) u) n0 H4 (\forall (c2: C).((csubt g c1 c2) \to (ex2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u)))))))))) (\lambda (x0: C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 c2) \to (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
+Abbr) u)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3: (drop n O c1 
+(CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1 (Bind Abbr) 
+u))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t)) \to ((clear 
+(CHead x0 k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csubt g c1 
+c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abbr) u))))))))) (\lambda (b: B).(\lambda (H5: (drop n O c1 
+(CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 
+(Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind 
+Abbr) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) 
+t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
+((let H9 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind Abbr) u) 
+(CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr) u) t H6)) in 
+(\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: 
+C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t0: 
+T).(drop n O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def (eq_ind_r 
+B b (\lambda (b0: B).(drop n O c1 (CHead x0 (Bind b0) u))) H13 Abbr H10) in 
+(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop n O c1 (CHead c (Bind 
+Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (x1: C).(\lambda (H16: (csubt g d1 x1)).(\lambda (H17: (drop n O c2 
+(CHead x1 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x1 H16 (getl_intro n 
+c2 (CHead x1 (Bind Abbr) u) (CHead x1 (Bind Abbr) u) H17 (clear_bind Abbr x1 
+u)))))) (csubt_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7))))) 
+(\lambda (f: F).(\lambda (H5: (drop n O c1 (CHead x0 (Flat f) t))).(\lambda 
+(H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 
+in (unintro C c1 (\lambda (c: C).((drop n O c (CHead x0 (Flat f) t)) \to 
+(\forall (c2: C).((csubt g c c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda 
+(n0: nat).(\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x1: 
+C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H9: (csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: 
+C).(csubt g c c2)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat 
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u) 
+(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def 
+(csubt_clear_conf g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) 
+H_y) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead d1 (Bind Abbr) u) e2)) 
+(\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: 
+C).(\lambda (H12: (csubt g (CHead d1 (Bind Abbr) u) x2)).(\lambda (H13: 
+(clear c2 x2)).(let H14 \def (csubt_gen_abbr g d1 x2 u H12) in (ex2_ind C 
+(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csubt 
+g d1 e2)) (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x3: C).(\lambda (H15: (eq C x2 
+(CHead x3 (Bind Abbr) u))).(\lambda (H16: (csubt g d1 x3)).(let H17 \def 
+(eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u) H15) 
+in (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind Abbr) u))) x3 H16 (getl_intro O c2 (CHead x3 (Bind Abbr) u) 
+c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda 
+(H8: ((\forall (x1: C).((drop n0 O x1 (CHead x0 (Flat f) t)) \to (\forall 
+(c2: C).((csubt g x1 c2) \to (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x1: 
+C).(\lambda (H9: (drop (S n0) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: 
+C).(\lambda (H10: (csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 
+(Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: 
+C).(\lambda (v: T).(clear x1 (CHead e (Bind b) v))))) (\lambda (_: 
+B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead x0 (Flat f) t))))) 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
+(CHead d2 (Bind Abbr) u)))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: 
+T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 
+O x3 (CHead x0 (Flat f) t))).(let H14 \def (csubt_clear_conf g x1 c2 H10 
+(CHead x3 (Bind x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead 
+x3 (Bind x2) x4) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) 
+x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 
+x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g x3 e2)))) (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 
+(Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def (eq_ind C x5 
+(\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 
+\def (H8 x3 H13 x7 H19) in (ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (x9: C).(\lambda (H22: (csubt g d1 x9)).(\lambda (H23: (getl 
+n0 x7 (CHead x9 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x9 H22 
+(getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u) n0 H23))))) 
+H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))) k H3 H4))))))) x H1 
+H2)))) H0))))))).
+
+lemma csubt_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall 
+(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g 
+c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda 
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def 
+(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: 
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) 
+(\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) 
+(\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead 
+d1 (Bind Abst) t))).(C_ind (\lambda (c: C).((drop n O c1 c) \to ((clear c 
+(CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))))))))) (\lambda (n0: nat).(\lambda (_: (drop n O c1 
+(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) 
+t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csubt 
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) (\lambda (x0: 
+C).(\lambda (_: (((drop n O c1 x0) \to ((clear x0 (CHead d1 (Bind Abst) t)) 
+\to (\forall (c2: C).((csubt g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (H3: (drop n O c1 
+(CHead x0 k t0))).(\lambda (H4: (clear (CHead x0 k t0) (CHead d1 (Bind Abst) 
+t))).(K_ind (\lambda (k0: K).((drop n O c1 (CHead x0 k0 t0)) \to ((clear 
+(CHead x0 k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csubt g c1 
+c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n 
+c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (b: B).(\lambda (H5: 
+(drop n O c1 (CHead x0 (Bind b) t0))).(\lambda (H6: (clear (CHead x0 (Bind b) 
+t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) 
+(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
+d1 (Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
+(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
+d1 (Bind Abst) t) t0 H6)) in ((let H9 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) 
+(CHead d1 (Bind Abst) t) (CHead x0 (Bind b) t0) (clear_gen_bind b x0 (CHead 
+d1 (Bind Abst) t) t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq 
+C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csubt g c1 c2)).(let H13 \def 
+(eq_ind_r T t0 (\lambda (t1: T).(drop n O c1 (CHead x0 (Bind b) t1))) H5 t 
+H9) in (let H14 \def (eq_ind_r B b (\lambda (b0: B).(drop n O c1 (CHead x0 
+(Bind b0) t))) H13 Abst H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: 
+C).(drop n O c1 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (H16: (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) 
+(or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 
+(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H17: 
+(csubt g d1 x1)).(\lambda (H18: (drop n O c2 (CHead x1 (Bind Abst) 
+t))).(or_introl (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda 
+(d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+t))) x1 H17 (getl_intro n c2 (CHead x1 (Bind Abst) t) (CHead x1 (Bind Abst) 
+t) H18 (clear_bind Abst x1 t))))))) H16)) (\lambda (H16: (ex4_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u 
+t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H17: (csubt g d1 x1)).(\lambda (H18: (drop n O 
+c2 (CHead x1 (Bind Abbr) x2))).(\lambda (H19: (ty3 g d1 x2 t)).(\lambda (H20: 
+(ty3 g x1 x2 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex4_2_intro C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x1 x2 
+H17 (getl_intro n c2 (CHead x1 (Bind Abbr) x2) (CHead x1 (Bind Abbr) x2) H18 
+(clear_bind Abbr x1 x2)) H19 H20)))))))) H16)) (csubt_drop_abst g n c1 c2 H12 
+d1 t H15)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop n O c1 
+(CHead x0 (Flat f) t0))).(\lambda (H6: (clear (CHead x0 (Flat f) t0) (CHead 
+d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop n 
+O c (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g c c2) \to (or (ex2 
+C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x1: 
+C).((drop n0 O x1 (CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 
+c2) \to (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl 
+n0 c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x1: C).(\lambda 
+(H8: (drop O O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: 
+(csubt g x1 c2)).(let H10 \def (eq_ind C x1 (\lambda (c: C).(csubt g c c2)) 
+H9 (CHead x0 (Flat f) t0) (drop_gen_refl x1 (CHead x0 (Flat f) t0) H8)) in 
+(let H_y \def (clear_flat x0 (CHead d1 (Bind Abst) t) (clear_gen_flat f x0 
+(CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csubt_clear_conf g 
+(CHead x0 (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C 
+(\lambda (e2: C).(csubt g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: 
+C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O 
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u 
+t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: 
+C).(\lambda (H12: (csubt g (CHead d1 (Bind Abst) t) x2)).(\lambda (H13: 
+(clear c2 x2)).(let H14 \def (csubt_gen_abst g d1 x2 t H12) in (or_ind (ex2 C 
+(\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt 
+g d1 e2))) (ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) 
+(\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda 
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda 
+(d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: 
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x2 (CHead e2 (Bind Abst) t))) 
+(\lambda (e2: C).(csubt g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x2 
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csubt g d1 e2)) (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
+Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead x3 
+(Bind Abst) t))).(\lambda (H17: (csubt g d1 x3)).(let H18 \def (eq_ind C x2 
+(\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst) t) H16) in (or_introl 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead 
+d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x3 H17 (getl_intro O 
+c2 (CHead x3 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: 
+(ex4_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 (CHead e2 (Bind Abbr) 
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 e2))) (\lambda (_: 
+C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: C).(\lambda (v2: T).(ty3 
+g e2 v2 t))))).(ex4_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x2 
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g d1 
+e2))) (\lambda (_: C).(\lambda (v2: T).(ty3 g d1 v2 t))) (\lambda (e2: 
+C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (H16: (eq C x2 (CHead x3 (Bind 
+Abbr) x4))).(\lambda (H17: (csubt g d1 x3)).(\lambda (H18: (ty3 g d1 x4 
+t)).(\lambda (H19: (ty3 g x3 x4 t)).(let H20 \def (eq_ind C x2 (\lambda (c: 
+C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) x4) H16) in (or_intror (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
+Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t)))) (ex4_2_intro C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda 
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))) x3 x4 H17 (getl_intro O c2 (CHead x3 
+(Bind Abbr) x4) c2 (drop_refl c2) H20) H18 H19))))))))) H15)) H14))))) 
+H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x1: C).((drop n0 O x1 
+(CHead x0 (Flat f) t0)) \to (\forall (c2: C).((csubt g x1 c2) \to (or (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 
+(Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: 
+C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x1: C).(\lambda (H9: 
+(drop (S n0) O x1 (CHead x0 (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: 
+(csubt g x1 c2)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t0) n0 H9) 
+in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: 
+T).(drop n0 O e (CHead x0 (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 
+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: 
+(clear x1 (CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n0 O x3 (CHead x0 
+(Flat f) t0))).(let H14 \def (csubt_clear_conf g x1 c2 H10 (CHead x3 (Bind 
+x2) x4) H12) in (ex2_ind C (\lambda (e2: C).(csubt g (CHead x3 (Bind x2) x4) 
+e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x5: C).(\lambda (H15: (csubt g (CHead x3 (Bind x2) x4) 
+x5)).(\lambda (H16: (clear c2 x5)).(let H17 \def (csubt_gen_bind g x2 x3 x5 
+x4 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: 
+T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: 
+C).(\lambda (_: T).(csubt g x3 e2)))) (or (ex2 C (\lambda (d2: C).(csubt g d1 
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) 
+(\lambda (x6: B).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H18: (eq C x5 
+(CHead x7 (Bind x6) x8))).(\lambda (H19: (csubt g x3 x7)).(let H20 \def 
+(eq_ind C x5 (\lambda (c: C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) 
+in (let H21 \def (H8 x3 H13 x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: 
+T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or 
+(ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 
+(CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead 
+d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) 
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C 
+(\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: C).(getl n0 x7 (CHead d2 
+(Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl n0 x7 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csubt 
+g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 
+C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))) (\lambda (x9: C).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: 
+(getl n0 x7 (CHead x9 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: 
+C).(csubt g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) 
+t)))) (ex4_2 C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
+g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda (d2: 
+C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x9 H23 (getl_clear_bind x6 c2 
+x7 x8 H20 (CHead x9 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex4_2 C 
+T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: 
+C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g 
+d2 u t))))).(ex4_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) 
+(\lambda (d2: C).(\lambda (u: T).(getl n0 x7 (CHead d2 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda 
+(u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csubt g d1 d2)) (\lambda 
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl 
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(ty3 g 
+d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x9: 
+C).(\lambda (x10: T).(\lambda (H23: (csubt g d1 x9)).(\lambda (H24: (getl n0 
+x7 (CHead x9 (Bind Abbr) x10))).(\lambda (H25: (ty3 g d1 x10 t)).(\lambda 
+(H26: (ty3 g x9 x10 t)).(or_intror (ex2 C (\lambda (d2: C).(csubt g d1 d2)) 
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex4_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda (d2: C).(\lambda 
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (_: C).(\lambda 
+(u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) 
+(ex4_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubt g d1 d2))) (\lambda 
+(d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda 
+(_: C).(\lambda (u: T).(ty3 g d1 u t))) (\lambda (d2: C).(\lambda (u: T).(ty3 
+g d2 u t))) x9 x10 H23 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) 
+x10) n0 H24) H25 H26)))))))) H22)) H21)))))))) H17))))) H14))))))) 
+H11)))))))) n) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/pc3.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/pc3.ma
new file mode 100644 (file)
index 0000000..ee965e4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/pc3.ma
@@ -0,0 +1,56 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/getl.ma".
+
+include "basic_1A/pc3/left.ma".
+
+lemma csubt_pr2:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pr2 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csubt g c c2) \to (pr2 c2 t t0)))))) (\lambda (c: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: 
+C).(\lambda (_: (csubt g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abbr g c d u i H0 
+c2 H3) in (ex2_ind C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: C).(getl 
+i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_: 
+(csubt g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta 
+c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
+
+lemma csubt_pc3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((csubt g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t: 
+T).(\lambda (c2: C).(\lambda (_: (csubt g c1 c2)).(pc3_refl c2 t)))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4: 
+T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csubt g c1 
+c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csubt g c1 
+c2)).(pc3_t t3 c2 t0 (pc3_pr2_r c2 t0 t3 (csubt_pr2 g c1 t0 t3 H0 c2 H3)) t4 
+(H2 c2 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall 
+(c2: C).((csubt g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda 
+(H3: (csubt g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csubt_pr2 g c1 t0 
+t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/props.ma
new file mode 100644 (file)
index 0000000..c129b35
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/props.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/defs.ma".
+
+include "basic_1A/C/fwd.ma".
+
+lemma csubt_refl:
+ \forall (g: G).(\forall (c: C).(csubt g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubt g c0 c0)) 
+(\lambda (n: nat).(csubt_sort g n)) (\lambda (c0: C).(\lambda (H: (csubt g c0 
+c0)).(\lambda (k: K).(\lambda (t: T).(csubt_head g c0 c0 H k t))))) c)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubt/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubt/ty3.ma
new file mode 100644 (file)
index 0000000..e78356c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubt/ty3.ma
@@ -0,0 +1,98 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/pc3.ma".
+
+include "basic_1A/csubt/props.ma".
+
+include "basic_1A/ty3/fwd.ma".
+
+lemma csubt_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (c2: C).((csubt g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t t0)))))) (\lambda 
+(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda 
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
+C).((csubt g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3 
+t0)).(\lambda (c2: C).(\lambda (H5: (csubt g c c2)).(ty3_conv g c2 t0 t (H1 
+c2 H5) u t3 (H3 c2 H5) (csubt_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda 
+(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csubt g c 
+c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csubt g 
+d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csubt g c 
+c2)).(let H4 \def (csubt_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda 
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) 
+u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5: 
+(csubt g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g 
+n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: 
+((\forall (c2: C).((csubt g d c2) \to (ty3 g c2 u t))))).(\lambda (c2: 
+C).(\lambda (H3: (csubt g c c2)).(let H4 \def (csubt_getl_abst g c d u n H0 
+c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csubt g d d2)) (\lambda (d2: 
+C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 
+u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u)))) (ty3 g c2 (TLRef n) 
+(lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda (d2: C).(csubt g d d2)) 
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))))).(ex2_ind C (\lambda 
+(d2: C).(csubt g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) 
+u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: C).(\lambda (H6: 
+(csubt g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind Abst) u))).(ty3_abst g 
+n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex4_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n 
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(ty3 g d u0 
+u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex4_2_ind C T 
+(\lambda (d2: C).(\lambda (_: T).(csubt g d d2))) (\lambda (d2: C).(\lambda 
+(u0: T).(getl n c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: 
+T).(ty3 g d u0 u))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 u))) (ty3 
+g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(_: (csubt g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (ty3 g d x1 u)).(\lambda (H9: (ty3 g x0 x1 u)).(ty3_abbr g 
+n c2 x0 x1 H7 u H9))))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: 
+C).((csubt g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda 
+(H3: ((\forall (c2: C).((csubt g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 
+t3))))).(\lambda (c2: C).(\lambda (H4: (csubt g c c2)).(ty3_bind g c2 u t (H1 
+c2 H4) b t0 t3 (H3 (CHead c2 (Bind b) u) (csubt_head g c c2 H4 (Bind b) 
+u))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: 
+(ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 
+w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind 
+Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 v 
+(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
+(H1: ((\forall (c2: C).((csubt g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4: 
+T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csubt g c 
+c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csubt g c 
+c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+lemma csubt_ty3_ld:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u 
+v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 
+t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H: 
+(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead 
+c (Bind Abst) v) t1 t2)).(csubt_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead 
+c (Bind Abbr) u) (csubt_abst g c c (csubt_refl g c) u v H H))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/clear.ma
new file mode 100644 (file)
index 0000000..549995b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/clear.ma
@@ -0,0 +1,179 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubv/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma csubv_clear_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
+B).(\forall (d1: C).(\forall (v1: T).((clear c1 (CHead d1 (Bind b1) v1)) \to 
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(clear c2 (CHead d2 
+(Bind b2) v2))))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (b1: B).(\forall (d1: C).(\forall 
+(v1: T).((clear c (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
+B).(\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 (Bind b2) 
+v2)))))))))))) (\lambda (n: nat).(\lambda (b1: B).(\lambda (d1: C).(\lambda 
+(v1: T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind b1) 
+v1))).(clear_gen_sort (CHead d1 (Bind b1) v1) n H0 (ex2_3 B C T (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
+B).(\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead d2 (Bind b2) 
+v2)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
+c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear 
+c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
+C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: 
+T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 (Bind b1) 
+v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b1) v0) 
+(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 
+H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow b1 | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow b1])])) (CHead d1 (Bind b1) v0) (CHead 
+c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 H2)) in 
+((let H5 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b1) v0) 
+(CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind b1) v0) v1 
+H2)) in (\lambda (_: (eq B b1 Void)).(\lambda (H7: (eq C d1 c3)).(eq_ind_r C 
+c3 (\lambda (c: C).(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csubv c d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
+(CHead c4 (Bind Void) v2) (CHead d2 (Bind b2) v3))))))) (ex2_3_intro B C T 
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda 
+(b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) 
+(CHead d2 (Bind b2) v3))))) Void c4 v2 H0 (clear_bind Void c4 v2)) d1 H7)))) 
+H4)) H3)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
+c4)).(\lambda (_: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: T).((clear 
+c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
+C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) v2)))))))))))).(\lambda 
+(b1: B).(\lambda (_: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (v0: 
+T).(\lambda (H3: (clear (CHead c3 (Bind b1) v1) (CHead d1 (Bind b0) 
+v0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) v0) 
+(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) 
+in ((let H5 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow b0 | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) v0) (CHead 
+c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) in 
+((let H6 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) v0) 
+(CHead c3 (Bind b1) v1) (clear_gen_bind b1 c3 (CHead d1 (Bind b0) v0) v1 H3)) 
+in (\lambda (_: (eq B b0 b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 
+(\lambda (c: C).(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csubv c d2)))) (\lambda (b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
+(CHead c4 (Bind b2) v2) (CHead d2 (Bind b3) v3))))))) (ex2_3_intro B C T 
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv c3 d2)))) (\lambda 
+(b3: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind b2) v2) 
+(CHead d2 (Bind b3) v3))))) b2 c4 v2 H0 (clear_bind b2 c4 v2)) d1 H8)))) H5)) 
+H4))))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 
+c4)).(\lambda (H1: ((\forall (b1: B).(\forall (d1: C).(\forall (v1: 
+T).((clear c3 (CHead d1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: 
+B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: 
+B).(\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind b2) 
+v2)))))))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda 
+(v2: T).(\lambda (b1: B).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H2: 
+(clear (CHead c3 (Flat f1) v1) (CHead d1 (Bind b1) v0))).(let H_x \def (H1 b1 
+d1 v0 (clear_gen_flat f1 c3 (CHead d1 (Bind b1) v0) v1 H2)) in (let H3 \def 
+H_x in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear 
+c4 (CHead d2 (Bind b2) v3))))) (ex2_3 B C T (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
+C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) v2) (CHead d2 (Bind b2) 
+v3)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: 
+(csubv d1 x1)).(\lambda (H5: (clear c4 (CHead x1 (Bind x0) x2))).(ex2_3_intro 
+B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) 
+(\lambda (b2: B).(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat f2) 
+v2) (CHead d2 (Bind b2) v3))))) x0 x1 x2 H4 (clear_flat c4 (CHead x1 (Bind 
+x0) x2) H5 f2 v2))))))) H3))))))))))))))) c1 c2 H))).
+
+lemma csubv_clear_conf_void:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
+C).(\forall (v1: T).((clear c1 (CHead d1 (Bind Void) v1)) \to (ex2_2 C T 
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
+(v2: T).(clear c2 (CHead d2 (Bind Void) v2))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (v1: T).((clear c 
+(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c0 (CHead d2 
+(Bind Void) v2)))))))))) (\lambda (n: nat).(\lambda (d1: C).(\lambda (v1: 
+T).(\lambda (H0: (clear (CSort n) (CHead d1 (Bind Void) v1))).(clear_gen_sort 
+(CHead d1 (Bind Void) v1) n H0 (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear (CSort n) (CHead 
+d2 (Bind Void) v2)))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: 
+(csubv c3 c4)).(\lambda (_: ((\forall (d1: C).(\forall (v1: T).((clear c3 
+(CHead d1 (Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 
+(Bind Void) v2)))))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (d1: 
+C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Bind Void) v1) (CHead d1 
+(Bind Void) v0))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind 
+Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind 
+Void) v0) v1 H2)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind 
+Void) v0) (CHead c3 (Bind Void) v1) (clear_gen_bind Void c3 (CHead d1 (Bind 
+Void) v0) v1 H2)) in (\lambda (H5: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: 
+C).(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: 
+C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 (Bind Void) 
+v3)))))) (ex2_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csubv c3 d2))) 
+(\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Bind Void) v2) (CHead d2 
+(Bind Void) v3)))) c4 v2 H0 (clear_bind Void c4 v2)) d1 H5))) H3))))))))))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv c3 c4)).(\lambda (_: 
+((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 (Bind Void) v1)) \to 
+(ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: 
+C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) v2)))))))))).(\lambda 
+(b1: B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H3: (clear 
+(CHead c3 (Bind b1) v1) (CHead d1 (Bind Void) v0))).(let H4 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d1 | (CHead c _ _) 
+\Rightarrow c])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
+(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in ((let H5 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Void | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Void])])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
+(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v0 | (CHead 
+_ _ t) \Rightarrow t])) (CHead d1 (Bind Void) v0) (CHead c3 (Bind b1) v1) 
+(clear_gen_bind b1 c3 (CHead d1 (Bind Void) v0) v1 H3)) in (\lambda (H7: (eq 
+B Void b1)).(\lambda (H8: (eq C d1 c3)).(eq_ind_r C c3 (\lambda (c: C).(ex2_2 
+C T (\lambda (d2: C).(\lambda (_: T).(csubv c d2))) (\lambda (d2: C).(\lambda 
+(v3: T).(clear (CHead c4 (Bind b2) v2) (CHead d2 (Bind Void) v3)))))) (let H9 
+\def (eq_ind_r B b1 (\lambda (b: B).(not (eq B b Void))) H2 Void H7) in (let 
+H10 \def (match (H9 (refl_equal B Void)) in False with []) in H10)) d1 H8)))) 
+H5)) H4)))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubv 
+c3 c4)).(\lambda (H1: ((\forall (d1: C).(\forall (v1: T).((clear c3 (CHead d1 
+(Bind Void) v1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 
+d2))) (\lambda (d2: C).(\lambda (v2: T).(clear c4 (CHead d2 (Bind Void) 
+v2)))))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (d1: C).(\lambda (v0: T).(\lambda (H2: (clear (CHead c3 (Flat f1) 
+v1) (CHead d1 (Bind Void) v0))).(let H_x \def (H1 d1 v0 (clear_gen_flat f1 c3 
+(CHead d1 (Bind Void) v0) v1 H2)) in (let H3 \def H_x in (ex2_2_ind C T 
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
+(v3: T).(clear c4 (CHead d2 (Bind Void) v3)))) (ex2_2 C T (\lambda (d2: 
+C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear 
+(CHead c4 (Flat f2) v2) (CHead d2 (Bind Void) v3))))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H4: (csubv d1 x0)).(\lambda (H5: (clear c4 
+(CHead x0 (Bind Void) x1))).(ex2_2_intro C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v3: T).(clear (CHead c4 (Flat 
+f2) v2) (CHead d2 (Bind Void) v3)))) x0 x1 H4 (clear_flat c4 (CHead x0 (Bind 
+Void) x1) H5 f2 v2)))))) H3)))))))))))))) c1 c2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/defs.ma
new file mode 100644 (file)
index 0000000..35e9715
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/defs.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+inductive csubv: C \to (C \to Prop) \def
+| csubv_sort: \forall (n: nat).(csubv (CSort n) (CSort n))
+| csubv_void: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
+(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind Void) v1) (CHead c2 (Bind 
+Void) v2))))))
+| csubv_bind: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
+(b1: B).((not (eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall 
+(v2: T).(csubv (CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))
+| csubv_flat: \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall 
+(f1: F).(\forall (f2: F).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 
+(Flat f1) v1) (CHead c2 (Flat f2) v2)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/drop.ma
new file mode 100644 (file)
index 0000000..dacf29d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/drop.ma
@@ -0,0 +1,114 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubv/props.ma".
+
+include "basic_1A/csubv/fwd.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+lemma csubv_drop_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (e1: 
+C).(\forall (h: nat).((drop h O c1 e1) \to (ex2 C (\lambda (e2: C).(csubv e1 
+e2)) (\lambda (e2: C).(drop h O c2 e2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(csubv_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).(\forall (h: nat).((drop h 
+O c e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O 
+c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda 
+(H0: (drop h O (CSort n) e1)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq 
+nat O O) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O 
+(CSort n) e2))) (\lambda (H1: (eq C e1 (CSort n))).(\lambda (H2: (eq nat h 
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C 
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n0 O (CSort n) e2)))) 
+(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c e2)) 
+(\lambda (e2: C).(drop O O (CSort n) e2)))) (ex_intro2 C (\lambda (e2: 
+C).(csubv (CSort n) e2)) (\lambda (e2: C).(drop O O (CSort n) e2)) (CSort n) 
+(csubv_refl (CSort n)) (drop_refl (CSort n))) e1 H1) h H2)))) (drop_gen_sort 
+n h O e1 H0)))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
+c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to 
+(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 
+e2)))))))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: 
+nat).(\lambda (H2: (drop h O (CHead c3 (Bind Void) v1) e1)).(nat_ind (\lambda 
+(n: nat).((drop n O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: 
+C).(csubv e1 e2)) (\lambda (e2: C).(drop n O (CHead c4 (Bind Void) v2) 
+e2))))) (\lambda (H3: (drop O O (CHead c3 (Bind Void) v1) e1)).(eq_ind C 
+(CHead c3 (Bind Void) v1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubv c 
+e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind Void) v2) e2)))) (ex_intro2 C 
+(\lambda (e2: C).(csubv (CHead c3 (Bind Void) v1) e2)) (\lambda (e2: C).(drop 
+O O (CHead c4 (Bind Void) v2) e2)) (CHead c4 (Bind Void) v2) (csubv_bind_same 
+c3 c4 H0 Void v1 v2) (drop_refl (CHead c4 (Bind Void) v2))) e1 (drop_gen_refl 
+(CHead c3 (Bind Void) v1) e1 H3))) (\lambda (h0: nat).(\lambda (_: (((drop h0 
+O (CHead c3 (Bind Void) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) 
+(\lambda (e2: C).(drop h0 O (CHead c4 (Bind Void) v2) e2)))))).(\lambda (H3: 
+(drop (S h0) O (CHead c3 (Bind Void) v1) e1)).(let H_x \def (H1 e1 (r (Bind 
+Void) h0) (drop_gen_drop (Bind Void) c3 e1 v1 h0 H3)) in (let H4 \def H_x in 
+(ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O c4 
+e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O 
+(CHead c4 (Bind Void) v2) e2))) (\lambda (x: C).(\lambda (H5: (csubv e1 
+x)).(\lambda (H6: (drop h0 O c4 x)).(ex_intro2 C (\lambda (e2: C).(csubv e1 
+e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind Void) v2) e2)) x H5 
+(drop_drop (Bind Void) h0 c4 x H6 v2))))) H4)))))) h H2)))))))))) (\lambda 
+(c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 c4)).(\lambda (H1: ((\forall 
+(e1: C).(\forall (h: nat).((drop h O c3 e1) \to (ex2 C (\lambda (e2: 
+C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 e2)))))))).(\lambda (b1: 
+B).(\lambda (H2: (not (eq B b1 Void))).(\lambda (b2: B).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H3: (drop h 
+O (CHead c3 (Bind b1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead 
+c3 (Bind b1) v1) e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: 
+C).(drop n O (CHead c4 (Bind b2) v2) e2))))) (\lambda (H4: (drop O O (CHead 
+c3 (Bind b1) v1) e1)).(eq_ind C (CHead c3 (Bind b1) v1) (\lambda (c: C).(ex2 
+C (\lambda (e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind 
+b2) v2) e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Bind b1) v1) 
+e2)) (\lambda (e2: C).(drop O O (CHead c4 (Bind b2) v2) e2)) (CHead c4 (Bind 
+b2) v2) (csubv_bind c3 c4 H0 b1 H2 b2 v1 v2) (drop_refl (CHead c4 (Bind b2) 
+v2))) e1 (drop_gen_refl (CHead c3 (Bind b1) v1) e1 H4))) (\lambda (h0: 
+nat).(\lambda (_: (((drop h0 O (CHead c3 (Bind b1) v1) e1) \to (ex2 C 
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Bind 
+b2) v2) e2)))))).(\lambda (H4: (drop (S h0) O (CHead c3 (Bind b1) v1) 
+e1)).(let H_x \def (H1 e1 (r (Bind b1) h0) (drop_gen_drop (Bind b1) c3 e1 v1 
+h0 H4)) in (let H5 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) 
+(\lambda (e2: C).(drop h0 O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) 
+(\lambda (e2: C).(drop (S h0) O (CHead c4 (Bind b2) v2) e2))) (\lambda (x: 
+C).(\lambda (H6: (csubv e1 x)).(\lambda (H7: (drop h0 O c4 x)).(ex_intro2 C 
+(\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 
+(Bind b2) v2) e2)) x H6 (drop_drop (Bind b2) h0 c4 x H7 v2))))) H5)))))) h 
+H3))))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csubv c3 
+c4)).(\lambda (H1: ((\forall (e1: C).(\forall (h: nat).((drop h O c3 e1) \to 
+(ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop h O c4 
+e2)))))))).(\lambda (f1: F).(\lambda (f2: F).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H2: (drop h O (CHead c3 (Flat 
+f1) v1) e1)).(nat_ind (\lambda (n: nat).((drop n O (CHead c3 (Flat f1) v1) 
+e1) \to (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop n O 
+(CHead c4 (Flat f2) v2) e2))))) (\lambda (H3: (drop O O (CHead c3 (Flat f1) 
+v1) e1)).(eq_ind C (CHead c3 (Flat f1) v1) (\lambda (c: C).(ex2 C (\lambda 
+(e2: C).(csubv c e2)) (\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) 
+e2)))) (ex_intro2 C (\lambda (e2: C).(csubv (CHead c3 (Flat f1) v1) e2)) 
+(\lambda (e2: C).(drop O O (CHead c4 (Flat f2) v2) e2)) (CHead c4 (Flat f2) 
+v2) (csubv_flat c3 c4 H0 f1 f2 v1 v2) (drop_refl (CHead c4 (Flat f2) v2))) e1 
+(drop_gen_refl (CHead c3 (Flat f1) v1) e1 H3))) (\lambda (h0: nat).(\lambda 
+(_: (((drop h0 O (CHead c3 (Flat f1) v1) e1) \to (ex2 C (\lambda (e2: 
+C).(csubv e1 e2)) (\lambda (e2: C).(drop h0 O (CHead c4 (Flat f2) v2) 
+e2)))))).(\lambda (H3: (drop (S h0) O (CHead c3 (Flat f1) v1) e1)).(let H_x 
+\def (H1 e1 (r (Flat f1) h0) (drop_gen_drop (Flat f1) c3 e1 v1 h0 H3)) in 
+(let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv e1 e2)) (\lambda (e2: 
+C).(drop (S h0) O c4 e2)) (ex2 C (\lambda (e2: C).(csubv e1 e2)) (\lambda 
+(e2: C).(drop (S h0) O (CHead c4 (Flat f2) v2) e2))) (\lambda (x: C).(\lambda 
+(H5: (csubv e1 x)).(\lambda (H6: (drop (S h0) O c4 x)).(ex_intro2 C (\lambda 
+(e2: C).(csubv e1 e2)) (\lambda (e2: C).(drop (S h0) O (CHead c4 (Flat f2) 
+v2) e2)) x H5 (drop_drop (Flat f2) h0 c4 x H6 v2))))) H4)))))) h 
+H2)))))))))))) c1 c2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/fwd.ma
new file mode 100644 (file)
index 0000000..9c09f19
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/fwd.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubv/defs.ma".
+
+implied rec lemma csubv_ind (P: (C \to (C \to Prop))) (f: (\forall (n: 
+nat).(P (CSort n) (CSort n)))) (f0: (\forall (c1: C).(\forall (c2: C).((csubv 
+c1 c2) \to ((P c1 c2) \to (\forall (v1: T).(\forall (v2: T).(P (CHead c1 
+(Bind Void) v1) (CHead c2 (Bind Void) v2))))))))) (f1: (\forall (c1: 
+C).(\forall (c2: C).((csubv c1 c2) \to ((P c1 c2) \to (\forall (b1: B).((not 
+(eq B b1 Void)) \to (\forall (b2: B).(\forall (v1: T).(\forall (v2: T).(P 
+(CHead c1 (Bind b1) v1) (CHead c2 (Bind b2) v2)))))))))))) (f2: (\forall (c1: 
+C).(\forall (c2: C).((csubv c1 c2) \to ((P c1 c2) \to (\forall (f2: 
+F).(\forall (f3: F).(\forall (v1: T).(\forall (v2: T).(P (CHead c1 (Flat f2) 
+v1) (CHead c2 (Flat f3) v2))))))))))) (c: C) (c0: C) (c1: csubv c c0) on c1: 
+P c c0 \def match c1 with [(csubv_sort n) \Rightarrow (f n) | (csubv_void c2 
+c3 c4 v1 v2) \Rightarrow (f0 c2 c3 c4 ((csubv_ind P f f0 f1 f2) c2 c3 c4) v1 
+v2) | (csubv_bind c2 c3 c4 b1 n b2 v1 v2) \Rightarrow (f1 c2 c3 c4 
+((csubv_ind P f f0 f1 f2) c2 c3 c4) b1 n b2 v1 v2) | (csubv_flat c2 c3 c4 f3 
+f4 v1 v2) \Rightarrow (f2 c2 c3 c4 ((csubv_ind P f f0 f1 f2) c2 c3 c4) f3 f4 
+v1 v2)].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/getl.ma
new file mode 100644 (file)
index 0000000..42214eb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/getl.ma
@@ -0,0 +1,84 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubv/clear.ma".
+
+include "basic_1A/csubv/drop.ma".
+
+include "basic_1A/getl/fwd.ma".
+
+lemma csubv_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b1: 
+B).(\forall (d1: C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl 
+i c2 (CHead d2 (Bind b2) v2)))))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b1: 
+B).(\lambda (d1: C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i 
+c1 (CHead d1 (Bind b1) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind 
+b1) v1) i H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
+C).(clear e (CHead d1 (Bind b1) v1))) (ex2_3 B C T (\lambda (_: B).(\lambda 
+(d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
+C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) v2)))))) (\lambda (x: 
+C).(\lambda (H2: (drop i O c1 x)).(\lambda (H3: (clear x (CHead d1 (Bind b1) 
+v1))).(let H_x \def (csubv_drop_conf c1 c2 H x i H2) in (let H4 \def H_x in 
+(ex2_ind C (\lambda (e2: C).(csubv x e2)) (\lambda (e2: C).(drop i O c2 e2)) 
+(ex2_3 B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 
+d2)))) (\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead 
+d2 (Bind b2) v2)))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda 
+(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf x x0 H5 b1 d1 v1 H3) 
+in (let H7 \def H_x0 in (ex2_3_ind B C T (\lambda (_: B).(\lambda (d2: 
+C).(\lambda (_: T).(csubv d1 d2)))) (\lambda (b2: B).(\lambda (d2: 
+C).(\lambda (v2: T).(clear x0 (CHead d2 (Bind b2) v2))))) (ex2_3 B C T 
+(\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) (\lambda 
+(b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind b2) 
+v2)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H8: 
+(csubv d1 x2)).(\lambda (H9: (clear x0 (CHead x2 (Bind x1) x3))).(ex2_3_intro 
+B C T (\lambda (_: B).(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2)))) 
+(\lambda (b2: B).(\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind 
+b2) v2))))) x1 x2 x3 H8 (getl_intro i c2 (CHead x2 (Bind x1) x3) x0 H6 
+H9))))))) H7)))))) H4)))))) H1))))))))).
+
+lemma csubv_getl_conf_void:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (d1: 
+C).(\forall (v1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Void) v1)) 
+\to (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: 
+C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (d1: 
+C).(\lambda (v1: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead d1 
+(Bind Void) v1))).(let H1 \def (getl_gen_all c1 (CHead d1 (Bind Void) v1) i 
+H0) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e 
+(CHead d1 (Bind Void) v1))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 
+(Bind Void) v2))))) (\lambda (x: C).(\lambda (H2: (drop i O c1 x)).(\lambda 
+(H3: (clear x (CHead d1 (Bind Void) v1))).(let H_x \def (csubv_drop_conf c1 
+c2 H x i H2) in (let H4 \def H_x in (ex2_ind C (\lambda (e2: C).(csubv x e2)) 
+(\lambda (e2: C).(drop i O c2 e2)) (ex2_2 C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 
+(Bind Void) v2))))) (\lambda (x0: C).(\lambda (H5: (csubv x x0)).(\lambda 
+(H6: (drop i O c2 x0)).(let H_x0 \def (csubv_clear_conf_void x x0 H5 d1 v1 
+H3) in (let H7 \def H_x0 in (ex2_2_ind C T (\lambda (d2: C).(\lambda (_: 
+T).(csubv d1 d2))) (\lambda (d2: C).(\lambda (v2: T).(clear x0 (CHead d2 
+(Bind Void) v2)))) (ex2_2 C T (\lambda (d2: C).(\lambda (_: T).(csubv d1 
+d2))) (\lambda (d2: C).(\lambda (v2: T).(getl i c2 (CHead d2 (Bind Void) 
+v2))))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (H8: (csubv d1 
+x1)).(\lambda (H9: (clear x0 (CHead x1 (Bind Void) x2))).(ex2_2_intro C T 
+(\lambda (d2: C).(\lambda (_: T).(csubv d1 d2))) (\lambda (d2: C).(\lambda 
+(v2: T).(getl i c2 (CHead d2 (Bind Void) v2)))) x1 x2 H8 (getl_intro i c2 
+(CHead x1 (Bind Void) x2) x0 H6 H9)))))) H7)))))) H4)))))) H1)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/csubv/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/csubv/props.ma
new file mode 100644 (file)
index 0000000..126841b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/csubv/props.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubv/defs.ma".
+
+include "basic_1A/C/fwd.ma".
+
+include "basic_1A/T/props.ma".
+
+lemma csubv_bind_same:
+ \forall (c1: C).(\forall (c2: C).((csubv c1 c2) \to (\forall (b: B).(\forall 
+(v1: T).(\forall (v2: T).(csubv (CHead c1 (Bind b) v1) (CHead c2 (Bind b) 
+v2)))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubv c1 c2)).(\lambda (b: 
+B).(B_ind (\lambda (b0: B).(\forall (v1: T).(\forall (v2: T).(csubv (CHead c1 
+(Bind b0) v1) (CHead c2 (Bind b0) v2))))) (\lambda (v1: T).(\lambda (v2: 
+T).(csubv_bind c1 c2 H Abbr not_abbr_void Abbr v1 v2))) (\lambda (v1: 
+T).(\lambda (v2: T).(csubv_bind c1 c2 H Abst not_abst_void Abst v1 v2))) 
+(\lambda (v1: T).(\lambda (v2: T).(csubv_void c1 c2 H v1 v2))) b)))).
+
+lemma csubv_refl:
+ \forall (c: C).(csubv c c)
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(csubv c0 c0)) (\lambda (n: 
+nat).(csubv_sort n)) (\lambda (c0: C).(\lambda (H: (csubv c0 c0)).(\lambda 
+(k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(csubv (CHead c0 k0 t) (CHead 
+c0 k0 t)))) (\lambda (b: B).(\lambda (t: T).(csubv_bind_same c0 c0 H b t t))) 
+(\lambda (f: F).(\lambda (t: T).(csubv_flat c0 c0 H f f t t))) k)))) c).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/definitions.ma b/matita/matita/contribs/lambdadelta/basic_1A/definitions.ma
new file mode 100644 (file)
index 0000000..5d67783
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/definitions.ma
@@ -0,0 +1,68 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/tlt/defs.ma".
+
+include "basic_1A/iso/defs.ma".
+
+include "basic_1A/clen/defs.ma".
+
+include "basic_1A/flt/defs.ma".
+
+include "basic_1A/app/defs.ma".
+
+include "basic_1A/cnt/defs.ma".
+
+include "basic_1A/cimp/defs.ma".
+
+include "basic_1A/subst1/defs.ma".
+
+include "basic_1A/subst/defs.ma".
+
+include "basic_1A/csubst1/defs.ma".
+
+include "basic_1A/fsubst0/defs.ma".
+
+include "basic_1A/next_plus/defs.ma".
+
+include "basic_1A/sty1/defs.ma".
+
+include "basic_1A/llt/defs.ma".
+
+include "basic_1A/aprem/defs.ma".
+
+include "basic_1A/ex0/defs.ma".
+
+include "basic_1A/wcpr0/defs.ma".
+
+include "basic_1A/csubv/defs.ma".
+
+include "basic_1A/csuba/defs.ma".
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/ex2/defs.ma".
+
+include "basic_1A/csubc/defs.ma".
+
+include "basic_1A/pc1/defs.ma".
+
+include "basic_1A/ex1/defs.ma".
+
+include "basic_1A/csubt/defs.ma".
+
+include "basic_1A/wf3/defs.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop/defs.ma
new file mode 100644 (file)
index 0000000..84e6796
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+include "basic_1A/lift/defs.ma".
+
+include "basic_1A/r/defs.ma".
+
+inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
+| drop_refl: \forall (c: C).(drop O O c c)
+| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: 
+C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) 
+e))))))
+| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: 
+C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) 
+(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop/fwd.ma
new file mode 100644 (file)
index 0000000..eba1cbd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop/fwd.ma
@@ -0,0 +1,475 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop/defs.ma".
+
+include "basic_1A/lift/fwd.ma".
+
+include "basic_1A/r/props.ma".
+
+include "basic_1A/C/fwd.ma".
+
+implied rec lemma drop_ind (P: (nat \to (nat \to (C \to (C \to Prop))))) (f: 
+(\forall (c: C).(P O O c c))) (f0: (\forall (k: K).(\forall (h: nat).(\forall 
+(c: C).(\forall (e: C).((drop (r k h) O c e) \to ((P (r k h) O c e) \to 
+(\forall (u: T).(P (S h) O (CHead c k u) e))))))))) (f1: (\forall (k: 
+K).(\forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop 
+h (r k d) c e) \to ((P h (r k d) c e) \to (\forall (u: T).(P h (S d) (CHead c 
+k (lift h (r k d) u)) (CHead e k u))))))))))) (n: nat) (n0: nat) (c: C) (c0: 
+C) (d: drop n n0 c c0) on d: P n n0 c c0 \def match d with [(drop_refl c1) 
+\Rightarrow (f c1) | (drop_drop k h c1 e d0 u) \Rightarrow (f0 k h c1 e d0 
+((drop_ind P f f0 f1) (r k h) O c1 e d0) u) | (drop_skip k h d0 c1 e d1 u) 
+\Rightarrow (f1 k h d0 c1 e d1 ((drop_ind P f f0 f1) h (r k d0) c1 e d1) u)].
+
+lemma drop_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop 
+h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x: 
+C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c: 
+C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d 
+O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0: 
+nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) 
+\to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c: 
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C 
+c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq 
+nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal 
+nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda 
+(e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n)) 
+\to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C 
+(CHead c k u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
+(CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (and3 (eq C e 
+(CHead c k u)) (eq nat (S h0) O) (eq nat O O)) H4)))))))))) (\lambda (k: 
+K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c: C).(\lambda (e: 
+C).(\lambda (_: (drop h0 (r k d0) c e)).(\lambda (_: (((eq C c (CSort n)) \to 
+(and3 (eq C e c) (eq nat h0 O) (eq nat (r k d0) O))))).(\lambda (u: 
+T).(\lambda (H3: (eq C (CHead c k (lift h0 (r k d0) u)) (CSort n))).(let H4 
+\def (eq_ind C (CHead c k (lift h0 (r k d0) u)) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
+(CSort n) H3) in (False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r 
+k d0) u))) (eq nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) 
+H))))).
+
+lemma drop_gen_refl:
+ \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
+\def
+ \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O 
+(\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y: 
+nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y 
+n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0: 
+nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to 
+(eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq 
+nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c: 
+C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq 
+nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda 
+(_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S 
+h) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H5) in (False_ind (eq C (CHead c k u) e0) H6))))))))))) (\lambda 
+(k: K).(\lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e0: 
+C).(\lambda (H2: (drop h (r k d) c e0)).(\lambda (H3: (((eq nat (r k d) O) 
+\to ((eq nat h (r k d)) \to (eq C c e0))))).(\lambda (u: T).(\lambda (H4: (eq 
+nat (S d) O)).(\lambda (H5: (eq nat h (S d))).(let H6 \def (f_equal nat nat 
+(\lambda (e1: nat).e1) h (S d) H5) in (let H7 \def (eq_ind nat h (\lambda (n: 
+nat).((eq nat (r k d) O) \to ((eq nat n (r k d)) \to (eq C c e0)))) H3 (S d) 
+H6) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n (r k d) c e0)) H2 
+(S d) H6) in (eq_ind_r nat (S d) (\lambda (n: nat).(eq C (CHead c k (lift n 
+(r k d) u)) (CHead e0 k u))) (let H9 \def (eq_ind nat (S d) (\lambda (ee: 
+nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) 
+in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h 
+H6)))))))))))))) y y0 x e H1))) H0))) H))).
+
+lemma drop_gen_drop:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
+nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: 
+nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) 
+(\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x)) 
+(\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda 
+(n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to 
+(drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y 
+x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_: 
+nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x)))) 
+(\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) 
+\to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c 
+c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq 
+nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) 
+(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda 
+(ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I 
+(S h) H3) in (False_ind (drop (r k h) O c (CHead c k u)) H6)) c0 H5))))) 
+(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda 
+(H3: (drop (r k0 h0) O c0 e)).(\lambda (H4: (((eq nat (r k0 h0) (S h)) \to 
+((eq nat O O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c 
+e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S h))).(\lambda (_: 
+(eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H8 
+\def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | 
+(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let 
+H9 \def (f_equal C K (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
+k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c k u) H7) in 
+((let H10 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k 
+u) H7) in (\lambda (H11: (eq K k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 
+\def (eq_ind C c0 (\lambda (c1: C).((eq nat (r k0 h0) (S h)) \to ((eq nat O 
+O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h) O c e))))) H4 c H12) in 
+(let H14 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) H3 c 
+H12) in (let H15 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 h0) (S h)) 
+\to ((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) 
+H13 k H11) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c 
+e)) H14 k H11) in (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 
+with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in (let H18 
+\def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to ((eq nat O 
+O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h H17) in 
+(let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) H16 h 
+H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda (h0: 
+nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: (drop 
+h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0 d) O) 
+\to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e)))))).(\lambda 
+(u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d) 
+O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k 
+u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0 (lift n 
+(r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind nat h0 
+(\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq C c0 
+(CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let H10 
+\def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h) H5) in 
+(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 (lift (S h) (r 
+k0 d) u0)) (CHead c k u) H8) in ((let H12 \def (f_equal C K (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
+k1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 
+\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
+(lref_map (\lambda (x0: nat).(plus x0 (S h))) (r k0 d) u0) | (CHead _ _ t) 
+\Rightarrow t])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in 
+(\lambda (H14: (eq K k0 k)).(\lambda (H15: (eq C c0 c)).(let H16 \def (eq_ind 
+C c0 (\lambda (c1: C).((eq nat (S h) (S h)) \to ((eq nat (r k0 d) O) \to ((eq 
+C c1 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H9 c H15) in (let H17 
+\def (eq_ind C c0 (\lambda (c1: C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in 
+(let H18 \def (eq_ind K k0 (\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) 
+u)) H13 k H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) 
+(S h)) \to ((eq nat (r k1 d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) 
+(r k1 d) c e))))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: 
+K).(drop (S h) (r k1 d) c e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: 
+K).(drop (r k h) (S d) c (CHead e k1 u0))) (let H21 \def (eq_ind_r T u 
+(\lambda (t: T).((eq nat (S h) (S h)) \to ((eq nat (r k d) O) \to ((eq C c 
+(CHead c k t)) \to (drop (r k h) (r k d) c e))))) H19 (lift (S h) (r k d) u0) 
+H18) in (let H22 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r 
+k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11)))))))))))))))) 
+y1 y0 y x H2))) H1))) H0))) H)))))).
+
+lemma drop_gen_skip_r:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
+(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda 
+(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k 
+d) e c)))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k 
+u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0)) 
+(\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) 
+u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0: 
+(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y)) 
+(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x 
+(CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))) 
+(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d)) 
+\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k 
+(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda 
+(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k 
+u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C 
+c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))) 
+(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow 
+True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind (ex2 C (\lambda 
+(e: C).(eq C (CHead c k u) (CHead e k (lift O (r k d) u)))) (\lambda (e: 
+C).(drop O (r k d) e c))) H4)) c0 H3)))) (\lambda (k0: K).(\lambda (h0: 
+nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r k0 h0) O c0 
+e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e (CHead c k u)) \to (ex2 C 
+(\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 h0) (r k d) u)))) (\lambda 
+(e0: C).(drop (r k0 h0) (r k d) e0 c))))))).(\lambda (u0: T).(\lambda (H4: 
+(eq nat O (S d))).(\lambda (H5: (eq C e (CHead c k u))).(let H6 \def (eq_ind 
+C e (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to (ex2 
+C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0 h0) (r k d) u)))) 
+(\lambda (e0: C).(drop (r k0 h0) (r k d) e0 c)))))) H3 (CHead c k u) H5) in 
+(let H7 \def (eq_ind C e (\lambda (c1: C).(drop (r k0 h0) O c0 c1)) H2 (CHead 
+c k u) H5) in (let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
+\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2 
+C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d) 
+u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda 
+(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e: 
+C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0) 
+(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 
+(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 
+c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5: 
+(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead e k0 u0) (CHead c k u) H5) in ((let H7 \def (f_equal C K (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow 
+k1])) (CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u0 | (CHead _ _ t) 
+\Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda (H9: (eq K k0 
+k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t: T).(ex2 C (\lambda 
+(e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead e0 k (lift h0 (r k 
+d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let H11 \def (eq_ind C e 
+(\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u)) \to 
+(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda 
+(e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12 \def (eq_ind C e 
+(\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in (let H13 \def 
+(eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to ((eq C c (CHead c 
+k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) 
+(\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9) in (let H14 \def 
+(eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12 k H9) in 
+(eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 
+(lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: 
+C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0: 
+nat).(match e0 with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) 
+H4) in (let H16 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat (r k n) (S d)) 
+\to ((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k 
+(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H13 d 
+H15) in (let H17 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c0 
+c)) H14 d H15) in (eq_ind_r nat d (\lambda (n: nat).(ex2 C (\lambda (e0: 
+C).(eq C (CHead c0 k (lift h0 (r k n) u)) (CHead e0 k (lift h0 (r k d) u)))) 
+(\lambda (e0: C).(drop h0 (r k d) e0 c)))) (ex_intro2 C (\lambda (e0: C).(eq 
+C (CHead c0 k (lift h0 (r k d) u)) (CHead e0 k (lift h0 (r k d) u)))) 
+(\lambda (e0: C).(drop h0 (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift 
+h0 (r k d) u))) H17) d0 H15)))) k0 H9))))) u0 H8)))) H7)) H6)))))))))))) h y0 
+x y H1))) H0))) H))))))).
+
+lemma drop_gen_skip_l:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall 
+(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T 
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: 
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: 
+T).(drop h (r k d) c e))))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) 
+x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda 
+(_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0: 
+(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x)) 
+(\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e: 
+C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k 
+d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq 
+nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e: 
+C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k 
+d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda 
+(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1: 
+C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O 
+(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow 
+False])) I (S d) H2) in (False_ind (ex3_2 C T (\lambda (e: C).(\lambda (v: 
+T).(eq C (CHead c k u) (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T 
+u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k d) c 
+e)))) H4)) c0 H3)))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: 
+C).(\lambda (e: C).(\lambda (H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq 
+nat O (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: 
+C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T u (lift (r k0 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: 
+T).(drop (r k0 h0) (r k d) c e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq 
+nat O (S d))).(\lambda (H5: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H6 
+\def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | 
+(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let 
+H7 \def (f_equal C K (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow 
+k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0) (CHead c k u) H5) in 
+((let H8 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k 
+u) H5) in (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 
+\def (eq_ind C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k 
+u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v)))) 
+(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c 
+H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e)) 
+H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d)) 
+\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r 
+k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d) 
+c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r 
+k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) 
+H4) in (False_ind (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead 
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S h0) (r k d) v)))) 
+(\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c e0)))) H15))))))))) 
+H7)) H6))))))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (d0: 
+nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop h0 (r k0 d0) c0 
+e)).(\lambda (H3: (((eq nat (r k0 d0) (S d)) \to ((eq C c0 (CHead c k u)) \to 
+(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: 
+C).(\lambda (_: T).(drop h0 (r k d) c e0)))))))).(\lambda (u0: T).(\lambda 
+(H4: (eq nat (S d0) (S d))).(\lambda (H5: (eq C (CHead c0 k0 (lift h0 (r k0 
+d0) u0)) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0: C).(match e0 
+with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 
+k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in 
+((let H8 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow (lref_map (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead 
+_ _ t) \Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) 
+H5) in (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def 
+(eq_ind C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead 
+c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k 
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda 
+(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 
+\def (eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in 
+(let H13 \def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) 
+H8 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S 
+d)) \to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 
+(r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) 
+H11 k H9) in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c 
+e)) H12 k H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: 
+C).(\lambda (v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: 
+C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda 
+(_: T).(drop h0 (r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: 
+T).((eq nat (r k d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T 
+(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: 
+C).(\lambda (v: T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda 
+(_: T).(drop h0 (r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind 
+T (lift h0 (r k d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda 
+(v: T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 
+(r k d) c e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 
+with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H4) in (let H18 
+\def (eq_ind nat d0 (\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c 
+(CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T (\lambda (e0: C).(\lambda 
+(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift 
+h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop 
+h0 (r k d) c e0))))))) H16 d H17) in (let H19 \def (eq_ind nat d0 (\lambda 
+(n: nat).(drop h0 (r k n) c e)) H15 d H17) in (eq_ind_r nat d (\lambda (n: 
+nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead 
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift 
+h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))) 
+(ex3_2_intro C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead 
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h0 (r k d) u0) (lift 
+h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))) e 
+u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h0 (r k d) u0)) H19) d0 
+H17)))) u H13)) k0 H9))))))))) H7)) H6)))))))))))) h y0 y x H1))) H0))) 
+H))))))).
+
+lemma drop_S:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
+nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
+C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to 
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b) 
+u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O 
+O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort 
+n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O 
+(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead 
+e (Bind b) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
+(CHead _ _ _) \Rightarrow True])) I (CSort n) H0) in (False_ind (drop (S O) O 
+(CSort n) e) H3)) h H1)))) (drop_gen_sort n h O (CHead e (Bind b) u) H))))))) 
+(\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).(\forall (h: 
+nat).((drop h O c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 
+e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) 
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: 
+(drop O O (CHead c0 k t) (CHead e (Bind b) u))).(let H1 \def (f_equal C C 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead 
+c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) 
+(CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e 
+(Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 
+with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k 
+t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) 
+H0)) in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C 
+c0 (\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) 
+(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 
+c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: 
+(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 
+k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) 
+u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: 
+nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) 
+u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem drop_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h 
+d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 
+x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2: 
+C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq 
+nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort 
+n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C 
+x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 
+(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r 
+C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h 
+(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d 
+(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda 
+(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) 
+(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda 
+(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1 
+x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t) 
+x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2)))))) 
+(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1) 
+\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2))))) 
+(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1: 
+(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C 
+x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t))) 
+(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2 
+(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_: 
+(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t) 
+x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t) 
+x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O 
+(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n 
+H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n 
+(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq 
+C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t) 
+x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t) 
+x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0: 
+C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4: 
+(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0 
+x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: 
+C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: 
+(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r 
+C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 
+(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to 
+(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 
+(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 
+(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
+nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop 
+h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k 
+n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r 
+k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda 
+(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) 
+(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) 
+t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) 
+H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k 
+x3))) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (sym_eq C (CHead x4 k x3) 
+(CHead x0 k x3) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C 
+CHead x0 x4 k k x3 x3 (H x0 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal 
+T x3))))) x5 (lift_inj x5 x3 h (r k n) H11))))) x1 H6)) x2 H3)))))) 
+(drop_gen_skip_l c0 x1 t h n k H1))))))) (drop_gen_skip_l c0 x2 t h n k 
+H2)))))))) d))))))) c).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop/props.ma
new file mode 100644 (file)
index 0000000..2e17e17
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop/props.ma
@@ -0,0 +1,594 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop/fwd.ma".
+
+lemma drop_skip_bind:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
+d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) 
+(lift h d u)) (CHead e (Bind b) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
+(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) 
+d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e 
+(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
+
+lemma drop_skip_flat:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
+(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat 
+f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
+(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat 
+f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead 
+e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S 
+d))))))))).
+
+lemma drop_ctail:
+ \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
+h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) 
+(CTail k u c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: 
+T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda 
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) 
+c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h 
+O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: 
+(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) 
+(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u 
+(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O 
+(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 
+H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: 
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k 
+u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n 
+(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u 
+(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: 
+nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop 
+n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O 
+(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) 
+(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) 
+(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 
+H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to 
+(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail 
+k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: 
+K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O 
+(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: 
+nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to 
+(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail 
+k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) 
+c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda 
+(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) 
+(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: 
+(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k 
+t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 
+k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k 
+x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u 
+c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n 
+(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop 
+h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 
+(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: 
+T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) 
+(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) 
+x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) 
+c1).
+
+theorem drop_conf_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall 
+(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h (r k d) c0 e0)))))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: 
+T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: 
+T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k 
+u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop 
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S 
+(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in 
+(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) 
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda 
+(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) 
+x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k 
+x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T 
+(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda 
+(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda 
+(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k 
+d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 
+e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall 
+(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) 
+c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda 
+(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) 
+O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S 
+i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind 
+(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort 
+n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) 
+O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) H5))))) (drop_gen_sort n h 
+(S (plus (S i0) d)) e H1)))))))) (\lambda (c1: C).(\lambda (H0: (((drop (S 
+i0) O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (r k d) c0 e0))))))))))).(\lambda (k0: K).(K_ind (\lambda (k1: 
+K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 t) (CHead c0 k u)) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) 
+d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
+(CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 
+(Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e 
+(CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r 
+(Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S 
+i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 
+(Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) (plus (S i0) d)) 
+x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
+(CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
+(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
+d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) c1 (CHead c0 k 
+u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq 
+T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O x0 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead 
+e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) 
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: (eq T u (lift h (r k d) 
+x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k x2))).(\lambda (H9: (drop h (r 
+k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind 
+b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0))) x2 x3 H7 (drop_drop (Bind b) i0 x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) 
+e H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Bind b) H2))))))))) 
+(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c1 (Flat 
+f) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 (Flat f) t) 
+e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat 
+f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus (S 
+i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus (S 
+i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 (Flat f) 
+x1))).(\lambda (_: (eq T t (lift h (r (Flat f) (plus (S i0) d)) 
+x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
+(CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
+(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
+d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O x0 (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) (CHead e0 k v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (x2: 
+T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) x2))).(\lambda (H7: 
+(drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h (r k d) c0 
+x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
+x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) H8)))))) (H0 
+(drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e H3)))))) 
+(drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) k0)))) c)))))) 
+i)).
+
+theorem drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
+(plus d h) i) \to (drop (minus i h) O e a)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: 
+C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e 
+a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
+d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda 
+(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def 
+(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus 
+O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 
+\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def 
+(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O 
+(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: 
+C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h 
+H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: 
+nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
+(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: 
+C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d 
+h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: 
+nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: 
+(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d 
+O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda 
+(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) 
+(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: 
+(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O 
+O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 
+O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S 
+i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) 
+O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 
+a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) 
+c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop 
+(minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4)))))) 
+(drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1))))))))) 
+(\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e: 
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) 
+(S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind 
+(\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0 
+t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))) 
+(\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind 
+b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
+(drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S 
+i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le 
+(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h 
+O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind 
+(\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S 
+i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 
+(Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 
+(Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind 
+b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead 
+c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead 
+c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O 
+e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7: 
+(le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e 
+h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h 
+H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) 
+e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
+a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5: 
+(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t 
+(lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 
+x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) 
+h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead 
+x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 
+(drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) 
+x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus 
+d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d 
+H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
+(CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le 
+(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) 
+t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) 
+(\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O 
+h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) 
+\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda 
+(H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S 
+i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) 
+O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) 
+e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda 
+(_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to 
+(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 
+(Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop 
+(Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) 
+H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 
+(Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
+a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5: 
+(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t 
+(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) 
+h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: 
+nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) 
+H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n 
+(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop 
+n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) 
+(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 
+h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 
+H3))))))))) k)))) c))))) i).
+
+theorem drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
+(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: 
+C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
+\def
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: 
+C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) 
+\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 
+e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 
+e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let 
+H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl 
+e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: 
+C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda 
+(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: 
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O 
+c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind 
+(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: 
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: 
+nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda 
+(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 
+(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: 
+(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O 
+O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) 
+H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C 
+(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
+(CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2: 
+C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2: 
+C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k 
+t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 
+e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda 
+(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 
+k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let 
+H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C 
+(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
+(CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x 
+c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) 
+(CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 
+(lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2))))) 
+(\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def 
+(IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop 
+(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: 
+C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat 
+f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: 
+(drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) 
+(\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) 
+(lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat 
+f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))) k (drop_gen_drop k 
+e2 e3 t j0 H))))))))))) e1)))) j).
+
+theorem drop_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O 
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h 
+(minus d i) e1 e2)))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to 
+(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to 
+(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 
+e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: 
+nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O 
+c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 
+(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda 
+(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C 
+(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 
+(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: 
+nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: 
+C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
+C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda 
+(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind 
+(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: 
+C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O 
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
+C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) 
+O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h 
+O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat 
+(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C 
+(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S 
+i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: 
+(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
+\Rightarrow True | (S _) \Rightarrow False])) I (S x) H2) in (False_ind (ex2 
+C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O 
+(S i0)) e1 e2))) H4))))) (le_gen_S i0 O H))))))))) (\lambda (d0: 
+nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2: 
+C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O 
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
+C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S 
+d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
+nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to 
+(ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda 
+(H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq 
+nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
+(e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort 
+n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def 
+(eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in 
+(and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda 
+(e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) 
+(S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S 
+i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 
+C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h 
+(minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee: 
+nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) 
+in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
+(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) 
+(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) 
+(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h 
+(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda 
+(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) 
+e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
+T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to 
+(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S 
+i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: 
+C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda 
+(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: 
+T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O 
+(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 
+(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda 
+(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: 
+C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h 
+(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O 
+(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop 
+h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 
+(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S 
+d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda 
+(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) 
+O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop 
+h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r 
+(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 
+(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 
+t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) 
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T 
+(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: 
+C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C 
+c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) 
+x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 
+(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in 
+(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) 
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) 
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: 
+C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S 
+i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) 
+(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S 
+i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) 
+H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) 
+(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
+
+theorem drop_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 
+e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
+C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O 
+c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) 
+O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d 
+(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 
+H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall 
+(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall 
+(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 
+e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: 
+C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c 
+e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: 
+(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort 
+n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda 
+(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) 
+e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) 
+in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort 
+n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop 
+(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda 
+(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) 
+(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind 
+nat (S i0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
+\Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n) 
+(CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3)))) 
+(drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall 
+(c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall 
+(e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O 
+c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t) 
+c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop 
+(S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind 
+(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop 
+(S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
+e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda 
+(H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def 
+(eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t) 
+(drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n: 
+nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k 
+c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda 
+(_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 
+e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
+e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2: 
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind 
+nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2)) 
+(drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n))) 
+(\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n)) 
+(\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k 
+c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus 
+i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h)) 
+(\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k 
+t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to 
+(drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda 
+(H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop 
+(S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r 
+k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: 
+(eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 d0 (CHead c2 k 
+t) c) \to (\forall (e3: C).((drop (S i0) O c e3) \to ((le d0 (S i0)) \to 
+(drop (S (plus i0 h0)) O (CHead c2 k t) e3))))))) IHd (CHead x0 k x1) H2) in 
+(let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0 
+k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
+nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k x1)) \to (\forall (e3: 
+C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S i0)) \to (drop (S (plus 
+i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r 
+T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k 
+t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r 
+k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 
+e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda 
+(H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 
+(le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r 
+(Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r 
+(Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h 
+(r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) 
+d))))))) c1)))) i).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop1/defs.ma
new file mode 100644 (file)
index 0000000..629dba9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop1/defs.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop/defs.ma".
+
+include "basic_1A/lift1/defs.ma".
+
+inductive drop1: PList \to (C \to (C \to Prop)) \def
+| drop1_nil: \forall (c: C).(drop1 PNil c c)
+| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds 
+c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
+
+rec definition ptrans (hds: PList) on hds: nat \to PList \def \lambda (i: 
+nat).(match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow 
+(let j \def (trans hds0 i) in (let q \def (ptrans hds0 i) in (match (blt j d) 
+with [true \Rightarrow (PCons h (minus d (S j)) q) | false \Rightarrow 
+q])))]).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop1/fwd.ma
new file mode 100644 (file)
index 0000000..33ff698
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop1/fwd.ma
@@ -0,0 +1,81 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop1/defs.ma".
+
+implied rec lemma drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: 
+(\forall (c: C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: 
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: 
+C).(\forall (hds: PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons 
+h d hds) c1 c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P 
+p c c0 \def match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1 
+c2 h d0 d1 c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f 
+f0) hds c2 c3 d2))].
+
+lemma drop1_gen_pnil:
+ \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq 
+PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1 
+c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda 
+(p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c 
+c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c))) 
+(\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds: 
+PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to 
+(eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def 
+(eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee with [PNil 
+\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H4) in 
+(False_ind (eq C c3 c5) H5)))))))))))) y c1 c2 H0))) H))).
+
+lemma drop1_gen_pcons:
+ \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h: 
+nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda 
+(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
+\def
+ \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq 
+PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_: 
+PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds 
+c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind 
+(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d 
+hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 
+hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d 
+hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee with 
+[PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds) 
+H1) in (False_ind (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: 
+C).(drop1 hds c2 c))) H2)))) (\lambda (c2: C).(\lambda (c4: C).(\lambda (h0: 
+nat).(\lambda (d0: nat).(\lambda (H1: (drop h0 d0 c2 c4)).(\lambda (c5: 
+C).(\lambda (hds0: PList).(\lambda (H2: (drop1 hds0 c4 c5)).(\lambda (H3: 
+(((eq PList hds0 (PCons h d hds)) \to (ex2 C (\lambda (c6: C).(drop h d c4 
+c6)) (\lambda (c6: C).(drop1 hds c6 c5)))))).(\lambda (H4: (eq PList (PCons 
+h0 d0 hds0) (PCons h d hds))).(let H5 \def (f_equal PList nat (\lambda (e: 
+PList).(match e with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) 
+(PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat 
+(\lambda (e: PList).(match e with [PNil \Rightarrow d0 | (PCons _ n _) 
+\Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H7 \def 
+(f_equal PList PList (\lambda (e: PList).(match e with [PNil \Rightarrow hds0 
+| (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds) H4) in 
+(\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let H10 \def 
+(eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds)) \to (ex2 
+C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6 c5))))) 
+H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p: PList).(drop1 p 
+c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0 
+n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda (n: nat).(drop n 
+d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h d c2 c6)) 
+(\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6)) H5)))))))))))) y 
+c1 c3 H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop1/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop1/getl.ma
new file mode 100644 (file)
index 0000000..6b0ed89
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop1/getl.ma
@@ -0,0 +1,107 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop1/fwd.ma".
+
+include "basic_1A/getl/drop.ma".
+
+lemma drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) 
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl 
+i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i) 
+e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1 
+(ptrans hds i) v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: 
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to 
+(ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl 
+(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v)))))))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda 
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl 
+i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in 
+(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) 
+(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda 
+(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) 
+v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: 
+C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: 
+T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda 
+(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) 
+c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda 
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl 
+i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d 
+H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3)) 
+(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match 
+(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) 
+(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow 
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
+(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4: 
+(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: 
+bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons 
+h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans 
+hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow 
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 
+(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) 
+(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 
+i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow 
+(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow 
+(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true 
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 
+(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d 
+(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) 
+v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def 
+(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2: 
+C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x 
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: 
+C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) 
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h 
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0: 
+C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans 
+hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def 
+(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b 
+x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda 
+(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans 
+hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S 
+(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S 
+(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 
+i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans 
+hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2 
+(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 
+i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1 
+x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 
+i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead 
+e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) 
+v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0 
+i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) 
+false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in 
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: 
+C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) 
+(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl 
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) 
+(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: 
+(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let 
+H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b) 
+(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1 
+(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans 
+hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/drop1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/drop1/props.ma
new file mode 100644 (file)
index 0000000..b604d7d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/drop1/props.ma
@@ -0,0 +1,88 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop1/fwd.ma".
+
+include "basic_1A/drop/props.ma".
+
+include "basic_1A/getl/defs.ma".
+
+lemma drop1_skip_bind:
+ \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: 
+C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) 
+(lift1 hds u)) (CHead e (Bind b) u)))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: 
+PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) 
+(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H_y \def 
+(drop1_gen_pnil c e H) in (eq_ind_r C e (\lambda (c0: C).(drop1 PNil (CHead 
+c0 (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c 
+H_y))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead 
+c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H_x \def 
+(drop1_gen_pcons c e p n n0 H0) in (let H1 \def H_x in (ex2_ind C (\lambda 
+(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (drop1 (PCons n (S 
+n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)) 
+(\lambda (x: C).(\lambda (H2: (drop n n0 c x)).(\lambda (H3: (drop1 p x 
+e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead x (Bind b) 
+(lift1 p u)) n (S n0) (drop_skip_bind n n0 c x H2 b (lift1 p u)) (CHead e 
+(Bind b) u) (Ss p) (H x u H3))))) H1)))))))))) hds))).
+
+lemma drop1_cons_tail:
+ \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop 
+h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to 
+(drop1 (PConsTail hds h d) c1 c3))))))))
+\def
+ \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda 
+(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 
+c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def 
+(drop1_gen_pnil c1 c2 H0) in (eq_ind_r C c2 (\lambda (c: C).(drop1 (PCons h d 
+PNil) c c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 H_y)))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: 
+((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 
+c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H_x 
+\def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def H_x in (ex2_ind C 
+(\lambda (c4: C).(drop n n0 c1 c4)) (\lambda (c4: C).(drop1 p c4 c2)) (drop1 
+(PCons n n0 (PConsTail p h d)) c1 c3) (\lambda (x: C).(\lambda (H3: (drop n 
+n0 c1 x)).(\lambda (H4: (drop1 p x c2)).(drop1_cons c1 x n n0 H3 c3 
+(PConsTail p h d) (H0 x H4))))) H2))))))))) hds)))))).
+
+theorem drop1_trans:
+ \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0) 
+\to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1 
+(papp is1 is2) c1 c2)))))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (c1: 
+C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: 
+C).((drop1 is2 c0 c2) \to (drop1 (papp p is2) c1 c2)))))))) (\lambda (c1: 
+C).(\lambda (c0: C).(\lambda (H: (drop1 PNil c1 c0)).(\lambda (is2: 
+PList).(\lambda (c2: C).(\lambda (H0: (drop1 is2 c0 c2)).(let H_y \def 
+(drop1_gen_pnil c1 c0 H) in (let H1 \def (eq_ind_r C c0 (\lambda (c: 
+C).(drop1 is2 c c2)) H0 c1 H_y) in H1)))))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0: 
+C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 
+c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0: 
+C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda 
+(c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0 
+p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop n n0 c1 
+c3)) (\lambda (c3: C).(drop1 p c3 c0)) (drop1 (PCons n n0 (papp p is2)) c1 
+c2) (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda (H4: (drop1 p x 
+c0)).(drop1_cons c1 x n n0 H3 c2 (papp p is2) (H x c0 H4 is2 c2 H1))))) 
+H2))))))))))))) is1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/etc/performance.txt b/matita/matita/contribs/lambdadelta/basic_1A/etc/performance.txt
new file mode 100644 (file)
index 0000000..6c46086
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/etc/performance.txt
@@ -0,0 +1,17 @@
+full validation of lambdadelta_1
+
+command: time ../../matitac.opt basic_1
+
+- machine: "monica"
+  date   : ven  6 mar 2015, 20.31.46, CET
+
+  real    4m39.904s
+  user    3m58.580s
+  sys     0m11.473s
+
+- machine: "dev.helm"
+  date   : Sat Mar  7 16:41:46 CET 2015
+
+  real    30m36.357s
+  user    6m35.749s
+  sys     0m31.518s

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/etc/planes.txt b/matita/matita/contribs/lambdadelta/basic_1A/etc/planes.txt
new file mode 100644 (file)
index 0000000..d4b66b3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/etc/planes.txt
@@ -0,0 +1,13 @@
+T s tlist tlt iso
+C r flt app
+lift cnt drop clear getl clen cimp
+lift1 drop1
+subst0 subst1 subst csubst0 csubst1 fsubst0 
+G next_plus
+sty0 sty1
+A asucc aplus leq llt aprem ex0 
+pr0 wcpr0 pr1 pr2 pr3 
+csubv arity csuba
+nf2 sn3 sc3 csubc ex2
+pc1 pc3
+ty3 csubt wf3 ex1

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex0/defs.ma
new file mode 100644 (file)
index 0000000..43bdf85
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex0/defs.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/A/defs.ma".
+
+include "basic_1A/G/defs.ma".
+
+definition gz:
+ G
+\def
+ mk_G S lt_n_Sn.
+
+inductive leqz: A \to (A \to Prop) \def
+| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
+(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort 
+h2 n2))))))
+| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: 
+A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex0/fwd.ma
new file mode 100644 (file)
index 0000000..e0236a5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex0/fwd.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ex0/defs.ma".
+
+implied rec lemma leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: 
+nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus 
+h1 n2) (plus h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall 
+(a1: A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3: 
+A).(\forall (a4: A).((leqz a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead 
+a2 a4))))))))))) (a: A) (a0: A) (l: leqz a a0) on l: P a a0 \def match l with 
+[(leqz_sort h1 h2 n1 n2 e) \Rightarrow (f h1 h2 n1 n2 e) | (leqz_head a1 a2 
+l0 a3 a4 l1) \Rightarrow (f0 a1 a2 l0 ((leqz_ind P f f0) a1 a2 l0) a3 a4 l1 
+((leqz_ind P f f0) a3 a4 l1))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex0/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex0/props.ma
new file mode 100644 (file)
index 0000000..072b79e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex0/props.ma
@@ -0,0 +1,188 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ex0/fwd.ma".
+
+include "basic_1A/leq/fwd.ma".
+
+include "basic_1A/aplus/props.ma".
+
+lemma aplus_gz_le:
+ \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A 
+(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
+\def
+ \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: 
+nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) 
+n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y 
+\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) 
+(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: 
+nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A 
+(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: 
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A 
+(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O 
+\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: 
+nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) 
+k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O 
+(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 
+n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) 
+(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: 
+nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S 
+(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) 
+(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz 
+(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: 
+nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz 
+(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S 
+k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda 
+(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz 
+(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) 
+k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: 
+A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) 
+k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S 
+n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k).
+
+lemma aplus_gz_ge:
+ \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A 
+(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
+\def
+ \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: 
+nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) 
+n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: 
+nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) 
+(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 
+h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda 
+(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz 
+(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le 
+(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: 
+nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) 
+(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 
+x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
+\Rightarrow True | (S _) \Rightarrow False])) I (S x) H0) in (False_ind (eq A 
+(asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) H2))))) (le_gen_S k0 O H))) 
+(\lambda (n0: nat).(\lambda (_: (((le (S k0) n0) \to (eq A (asucc gz (aplus 
+gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n))))).(\lambda (H0: (le (S k0) 
+(S n0))).(let H_y \def (le_S_n k0 n0 H0) in (eq_ind A (aplus gz (ASort n0 n) 
+k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0) n) k0)) a)) 
+(eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) k0) (\lambda (a: A).(eq A a 
+(aplus gz (ASort n0 n) k0))) (refl_equal A (aplus gz (ASort n0 n) k0)) (asucc 
+gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S n0) n))) 
+(ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)).
+
+lemma next_plus_gz:
+ \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat 
+(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: 
+nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat 
+S (next_plus gz n n0) (plus n0 n) H))) h)).
+
+lemma leqz_leq:
+ \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz 
+(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda 
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda 
+(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k 
+h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 
+(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def 
+(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort 
+h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 
+(le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) (le_n_S (S k) h1 H1)))))) in (let 
+H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort 
+(minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n 
+k h2 (le_S_n (S k) (S h2) (le_S (S (S k)) (S h2) (le_n_S (S k) h2 H2)))))) in 
+(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n _) 
+\Rightarrow n | (AHead _ _) \Rightarrow (minus h1 k)])) (ASort (minus h1 k) 
+n1) (ASort (minus h2 k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: 
+A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) 
+(ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in (\lambda (H7: (eq nat 
+(minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: nat).(leqz (ASort h1 
+n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: nat).(leqz (ASort h1 n1) 
+(ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal nat (plus h1 n1))) h2 
+(minus_minus k h1 h2 (le_S_n k h1 (le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) 
+(le_n_S (S k) h1 H1)))) (le_S_n k h2 (le_S_n (S k) (S h2) (le_S (S (S k)) (S 
+h2) (le_n_S (S k) h2 H2)))) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 
+k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a 
+(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 
+(le_S_n k h1 (le_S_n (S k) (S h1) (le_S (S (S k)) (S h1) (le_n_S (S k) h1 
+H1)))))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: 
+A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus (minus k h2) n2)) 
+(aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat (minus h1 k) (\lambda 
+(n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) n2)))) H4 (S (minus 
+h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind A (ASort (S (minus 
+h1 (S k))) n1) (\lambda (ee: A).(match ee with [(ASort n _) \Rightarrow 
+(match n with [O \Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) 
+\Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in (False_ind 
+(leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 
+k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k 
+h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A 
+a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) 
+(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) 
+k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort 
+(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S_n (S k) (S h2) (le_S 
+(S (S k)) (S h2) (le_n_S (S k) h2 H2)))))) in (let H5 \def (sym_eq A (ASort O 
+(plus (minus k h1) n1)) (ASort (minus h2 k) n2) H4) in (let H6 \def (eq_ind 
+nat (minus h2 k) (\lambda (n: nat).(eq A (ASort n n2) (ASort O (plus (minus k 
+h1) n1)))) H5 (S (minus h2 (S k))) (minus_x_Sy h2 k H2)) in (let H7 \def 
+(eq_ind A (ASort (S (minus h2 (S k))) n2) (\lambda (ee: A).(match ee with 
+[(ASort n _) \Rightarrow (match n with [O \Rightarrow False | (S _) 
+\Rightarrow True]) | (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus 
+k h1) n1)) H6) in (False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H7))))))) 
+(\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) 
+(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus 
+k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz 
+(ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) 
+H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def 
+(f_equal A nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | 
+(AHead _ _) \Rightarrow (plus (minus k h1) n1)])) (ASort O (plus (minus k h1) 
+n1)) (ASort O (plus (minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 
+n1 n2 H1 H2 H5) in (leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: 
+A).(\lambda (a3: A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 
+a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda 
+(H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+
+lemma leq_leqz:
+ \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind 
+(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda 
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus 
+h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A 
+(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) 
+(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A 
+(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) 
+(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus 
+(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus 
+h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) 
+(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: 
+nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus 
+h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq 
+A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz 
+n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 
+h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: 
+nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus 
+h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal 
+nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 
+n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) 
+(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) 
+(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) 
+(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus 
+(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) 
+(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 
+h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: 
+A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 
+a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda 
+(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex1/defs.ma
new file mode 100644 (file)
index 0000000..bf7a591
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex1/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+definition ex1_c:
+ C
+\def
+ CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O).
+
+definition ex1_t:
+ T
+\def
+ THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex1/props.ma
new file mode 100644 (file)
index 0000000..aea8bcb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex1/props.ma
@@ -0,0 +1,516 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ex1/defs.ma".
+
+include "basic_1A/ty3/fwd.ma".
+
+include "basic_1A/pc3/fwd.ma".
+
+include "basic_1A/nf2/pr3.ma".
+
+include "basic_1A/nf2/props.ma".
+
+include "basic_1A/arity/defs.ma".
+
+include "basic_1A/leq/props.ma".
+
+fact ex1__leq_sort_SS:
+ \forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc 
+g (asucc g (ASort (S (S k)) n))))))
+\def
+ \lambda (g: G).(\lambda (k: nat).(\lambda (n: nat).(leq_refl g (asucc g 
+(asucc g (ASort (S (S k)) n)))))).
+
+lemma ex1_arity:
+ \forall (g: G).(arity g ex1_c ex1_t (ASort O O))
+\def
+ \lambda (g: G).(arity_appl g (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef O) (ASort (S 
+(S O)) O) (arity_abst g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) O (getl_refl Abst (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O)) 
+(ASort (S (S O)) O) (arity_abst g (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O) 
+O (getl_refl Abst (CHead (CSort O) (Bind Abst) (TSort O)) (TSort O)) (asucc g 
+(ASort (S (S O)) O)) (arity_repl g (CHead (CSort O) (Bind Abst) (TSort O)) 
+(TSort O) (ASort O O) (arity_sort g (CHead (CSort O) (Bind Abst) (TSort O)) 
+O) (asucc g (asucc g (ASort (S (S O)) O))) (ex1__leq_sort_SS g O O)))) (THead 
+(Bind Abst) (TLRef (S (S O))) (TSort O)) (ASort O O) (arity_head g (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (TLRef (S (S O))) (ASort (S (S O)) O) (arity_abst g (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (CSort O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (TLRef O) (clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) 
+(TSort O)) (S O) (getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort 
+O)) (CHead (CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort 
+O)) (TSort O))) (asucc g (ASort (S (S O)) O)) (arity_repl g (CSort O) (TSort 
+O) (ASort O O) (arity_sort g (CSort O) O) (asucc g (asucc g (ASort (S (S O)) 
+O))) (ex1__leq_sort_SS g O O))) (TSort O) (ASort O O) (arity_sort g (CHead 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) O))).
+
+lemma ex1_ty3:
+ \forall (g: G).(\forall (u: T).((ty3 g ex1_c ex1_t u) \to (\forall (P: 
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (u: T).(\lambda (H: (ty3 g (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort 
+O))) u)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u0: T).(\lambda (t: 
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind 
+Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) (THead (Bind Abst) 
+u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) 
+(TLRef O) u0))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (THead (Flat Appl) (TLRef O) (THead (Bind Abst) x0 x1)) 
+u)).(\lambda (H1: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef 
+(S (S O))) (TSort O)) (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (TLRef O) x0)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda 
+(_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O t) x0)))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
+t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
+t) x0)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O 
+x4) x0)).(\lambda (H5: (getl O (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind 
+Abbr) x3))).(\lambda (_: (ty3 g x2 x3 x4)).(ex3_2_ind T T (\lambda (t2: 
+T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind Abst) (TLRef (S (S 
+O))) t2) (THead (Bind Abst) x0 x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t))) (\lambda (t2: T).(\lambda (_: 
+T).(ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort 
+O) t2))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (pc3 (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 
+x1))).(\lambda (H8: (ty3 g (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
+x6)).(\lambda (_: (ty3 g (CHead (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind Abst) (TLRef 
+(S (S O)))) (TSort O) x5)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) 
+O u0) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
+T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x9) 
+x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
+(Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
+x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 
+(Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S 
+O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) x8))) P (\lambda (x: 
+C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abbr) 
+x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abbr) 
+x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) 
+H5))) in (False_ind P H17))))) H14)))))))) H10)) (\lambda (H10: (ex3_3 C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
+T).(\lambda (_: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x8) 
+x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
+(Bind Abst) x8))).(\lambda (_: (ty3 g x7 x8 x9)).(let H14 \def (getl_gen_all 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
+x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind Abst) (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x7 
+(Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: C).(drop (S 
+O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P (\lambda (x: 
+C).(\lambda (_: (drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) x)).(\lambda (_: (clear x (CHead x7 (Bind Abst) 
+x8))).(let H17 \def (eq_ind C (CHead x2 (Bind Abbr) x3) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abbr) 
+x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abbr) x3) 
+H5))) in (False_ind P H17))))) H14)))))))) H10)) (ty3_gen_lref g (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) x6 (S (S O)) H8))))))) (ty3_gen_bind g Abst (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (TLRef (S (S O))) (TSort O) (THead (Bind Abst) x0 x1) H1)))))))) 
+H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e 
+u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O u0) x0)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e 
+u0 t)))) P (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H4: 
+(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (Bind Abst) (TLRef O)) (lift (S O) O x3) x0)).(\lambda (H5: (getl O 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3))).(\lambda (H6: (ty3 g x2 x3 
+x4)).(ex3_2_ind T T (\lambda (t2: T).(\lambda (_: T).(pc3 (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (THead (Bind Abst) (TLRef (S (S O))) t2) (THead (Bind Abst) x0 
+x1)))) (\lambda (_: T).(\lambda (t: T).(ty3 g (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef 
+(S (S O))) t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) t2))) P (\lambda (x5: 
+T).(\lambda (x6: T).(\lambda (H7: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (THead (Bind 
+Abst) (TLRef (S (S O))) x5) (THead (Bind Abst) x0 x1))).(\lambda (H8: (ty3 g 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) x6)).(\lambda (_: (ty3 g (CHead 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (Bind Abst) (TLRef (S (S O)))) (TSort O) x5)).(or_ind 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind 
+Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t))))) P (\lambda (H10: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S 
+O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S 
+O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (lift (S (S (S O))) O t) x6)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda 
+(x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (_: (pc3 (CHead (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) 
+(TLRef O)) (lift (S (S (S O))) O x9) x6)).(\lambda (H12: (getl (S (S O)) 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (CHead x7 (Bind Abbr) x8))).(\lambda (_: (ty3 g x7 x8 
+x9)).(let H14 \def (getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (CHead x7 (Bind Abbr) x8) (r (Bind Abst) (S O)) 
+(getl_gen_S (Bind Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (CHead x7 (Bind Abbr) x8) (TLRef O) (S O) H12)) in (ex2_ind 
+C (\lambda (e: C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abbr) 
+x8))) P (\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x 
+(CHead x7 (Bind Abbr) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow x2 | (CHead c _ _) \Rightarrow c])) (CHead x2 
+(Bind Abst) x3) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) 
+x3) (TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) 
+H5))) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) 
+(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda 
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def 
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def 
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abbr) x8))) H16 (CHead 
+(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) 
+(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort 
+O)) x (TSort O) O H15))) in (let H24 \def (eq_ind C (CHead x7 (Bind Abbr) x8) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead (CSort O) (Bind Abst) (TSort O)) 
+(clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abbr) x8) (TSort O) H23)) in 
+(False_ind P H24)))))))) H17))))) H14)))))))) H10)) (\lambda (H10: (ex3_3 C T 
+T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (lift (S (S (S O))) O u0) x6)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead e (Bind Abst) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O u0) x6)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl (S (S O)) (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t)))) P (\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: 
+T).(\lambda (H11: (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) O x8) 
+x6)).(\lambda (H12: (getl (S (S O)) (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x7 
+(Bind Abst) x8))).(\lambda (H13: (ty3 g x7 x8 x9)).(let H14 \def 
+(getl_gen_all (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (CHead x7 (Bind Abst) x8) (r (Bind Abst) (S O)) (getl_gen_S (Bind 
+Abst) (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(CHead x7 (Bind Abst) x8) (TLRef O) (S O) H12)) in (ex2_ind C (\lambda (e: 
+C).(drop (S O) O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) e)) (\lambda (e: C).(clear e (CHead x7 (Bind Abst) x8))) P 
+(\lambda (x: C).(\lambda (H15: (drop (S O) O (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) x)).(\lambda (H16: (clear x (CHead x7 
+(Bind Abst) x8))).(let H17 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow x2 | (CHead c _ _) \Rightarrow c])) (CHead x2 (Bind 
+Abst) x3) (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) 
+(TLRef O) (getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) 
+in ((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow x3 | (CHead _ _ t) \Rightarrow t])) (CHead x2 (Bind Abst) x3) 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (clear_gen_bind Abst (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead x2 (Bind Abst) x3) (TLRef O) 
+(getl_gen_O (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (Bind Abst) (TLRef O)) (CHead x2 (Bind Abst) x3) H5))) in (\lambda 
+(H19: (eq C x2 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)))).(let H20 \def (eq_ind T x3 (\lambda (t: T).(ty3 g x2 t x4)) H6 
+(TLRef O) H18) in (let H21 \def (eq_ind T x3 (\lambda (t: T).(pc3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (lift (S O) O t) x0)) H4 (TLRef O) H18) in (let H22 \def 
+(eq_ind C x2 (\lambda (c: C).(ty3 g c (TLRef O) x4)) H20 (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) H19) in (let H23 \def 
+(eq_ind_r C x (\lambda (c: C).(clear c (CHead x7 (Bind Abst) x8))) H16 (CHead 
+(CSort O) (Bind Abst) (TSort O)) (drop_gen_refl (CHead (CSort O) (Bind Abst) 
+(TSort O)) x (drop_gen_drop (Bind Abst) (CHead (CSort O) (Bind Abst) (TSort 
+O)) x (TSort O) O H15))) in (let H24 \def (f_equal C C (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow x7 | (CHead c _ _) \Rightarrow c])) (CHead x7 
+(Bind Abst) x8) (CHead (CSort O) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
+(CSort O) (CHead x7 (Bind Abst) x8) (TSort O) H23)) in ((let H25 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow x8 | (CHead 
+_ _ t) \Rightarrow t])) (CHead x7 (Bind Abst) x8) (CHead (CSort O) (Bind 
+Abst) (TSort O)) (clear_gen_bind Abst (CSort O) (CHead x7 (Bind Abst) x8) 
+(TSort O) H23)) in (\lambda (H26: (eq C x7 (CSort O))).(let H27 \def (eq_ind 
+T x8 (\lambda (t: T).(ty3 g x7 t x9)) H13 (TSort O) H25) in (let H28 \def 
+(eq_ind T x8 (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S (S (S O))) 
+O t) x6)) H11 (TSort O) H25) in (let H29 \def (eq_ind C x7 (\lambda (c: 
+C).(ty3 g c (TSort O) x9)) H27 (CSort O) H26) in (or_ind (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O t) x4)))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) (ex3_3 C 
+T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (lift (S O) O u0) 
+x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl O (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead e (Bind Abst) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))) P 
+(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (lift (S O) O t) x4)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: 
+T).(\lambda (x12: T).(\lambda (_: (pc3 (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x12) x4)).(\lambda (H32: 
+(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(CHead x10 (Bind Abbr) x11))).(\lambda (_: (ty3 g x10 x11 x12)).(let H34 \def 
+(eq_ind C (CHead x10 (Bind Abbr) x11) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
+(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abbr) x11) (TSort O) 
+(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (CHead x10 (Bind Abbr) x11) H32))) in (False_ind P H34)))))))) H30)) 
+(\lambda (H30: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t: T).(ty3 g e u0 t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (lift (S O) O u0) x4)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (t: T).(ty3 g e u0 t)))) P (\lambda (x10: C).(\lambda (x11: 
+T).(\lambda (x12: T).(\lambda (H31: (pc3 (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (lift (S O) O x11) x4)).(\lambda (H32: 
+(getl O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(CHead x10 (Bind Abst) x11))).(\lambda (H33: (ty3 g x10 x11 x12)).(let H34 
+\def (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow x10 | 
+(CHead c _ _) \Rightarrow c])) (CHead x10 (Bind Abst) x11) (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst 
+(CHead (CSort O) (Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) 
+(getl_gen_O (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (CHead x10 (Bind Abst) x11) H32))) in ((let H35 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow x11 | (CHead _ _ t) 
+\Rightarrow t])) (CHead x10 (Bind Abst) x11) (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (clear_gen_bind Abst (CHead (CSort O) 
+(Bind Abst) (TSort O)) (CHead x10 (Bind Abst) x11) (TSort O) (getl_gen_O 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead 
+x10 (Bind Abst) x11) H32))) in (\lambda (H36: (eq C x10 (CHead (CSort O) 
+(Bind Abst) (TSort O)))).(let H37 \def (eq_ind T x11 (\lambda (t: T).(ty3 g 
+x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: 
+T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10 
+(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst) 
+(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
+x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind 
+b) u0) x5 x1))) P (\lambda (H40: (pc3 (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S 
+O))) x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (Bind b) u0) x5 x1))))).(let H42 \def (eq_ind T (lift (S O) 
+O (TLRef O)) (\lambda (t: T).(pc3 (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
+t)) (pc3_t x0 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind 
+Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) H40 (lift (S O) O 
+(TLRef O)) (ex2_sym T (pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort 
+O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (lift (S O) O (TLRef O))) 
+(pr3 (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort 
+O)) (Bind Abst) (TLRef O)) x0) H21)) (TLRef (plus O (S O))) (lift_lref_ge O 
+(S O) O (le_O_n O))) in (let H43 \def H42 in (ex2_ind T (\lambda (t: T).(pr3 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) t)) (\lambda (t: T).(pr3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (TLRef (S O)) t)) P (\lambda (x13: T).(\lambda (H44: (pr3 
+(CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) 
+(Bind Abst) (TLRef O)) (TLRef (S (S O))) x13)).(\lambda (H45: (pr3 (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) (TLRef (S O)) x13)).(let H46 \def (eq_ind_r T x13 (\lambda 
+(t: T).(eq T (TLRef (S (S O))) t)) (nf2_pr3_unfold (CHead (CHead (CHead 
+(CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef 
+O)) (TLRef (S (S O))) x13 H44 (nf2_lref_abst (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CSort 
+O) (TSort O) (S (S O)) (getl_clear_bind Abst (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead 
+(CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (TLRef O) 
+(clear_bind Abst (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) 
+(TSort O)) (TLRef O)) (CHead (CSort O) (Bind Abst) (TSort O)) (S O) 
+(getl_head (Bind Abst) O (CHead (CSort O) (Bind Abst) (TSort O)) (CHead 
+(CSort O) (Bind Abst) (TSort O)) (getl_refl Abst (CSort O) (TSort O)) (TSort 
+O))))) (TLRef (S O)) (nf2_pr3_unfold (CHead (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S O)) 
+x13 H45 (nf2_lref_abst (CHead (CHead (CHead (CSort O) (Bind Abst) (TSort O)) 
+(Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (CHead (CSort O) (Bind Abst) 
+(TSort O)) (TSort O) (S O) (getl_head (Bind Abst) O (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (CHead (CHead (CSort O) (Bind 
+Abst) (TSort O)) (Bind Abst) (TSort O)) (getl_refl Abst (CHead (CSort O) 
+(Bind Abst) (TSort O)) (TSort O)) (TLRef O))))) in (let H47 \def (eq_ind T 
+(TLRef (S (S O))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef n) \Rightarrow (match n with [O \Rightarrow False | (S n0) 
+\Rightarrow (match n0 with [O \Rightarrow False | (S _) \Rightarrow True])]) 
+| (THead _ _ _) \Rightarrow False])) I (TLRef (S O)) H46) in (False_ind P 
+H47)))))) H43))))) (pc3_gen_abst (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) x0 
+x5 x1 H7))))))) H34)))))))) H30)) (ty3_gen_lref g (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) x4 O H22))))))) H24)))))))) 
+H17))))) H14)))))))) H10)) (ty3_gen_lref g (CHead (CHead (CHead (CSort O) 
+(Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) x6 (S (S 
+O)) H8))))))) (ty3_gen_bind g Abst (CHead (CHead (CHead (CSort O) (Bind Abst) 
+(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O))) 
+(TSort O) (THead (Bind Abst) x0 x1) H1)))))))) H3)) (ty3_gen_lref g (CHead 
+(CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind 
+Abst) (TLRef O)) x0 O H2))))))) (ty3_gen_appl g (CHead (CHead (CHead (CSort 
+O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) 
+(TLRef O) (THead (Bind Abst) (TLRef (S (S O))) (TSort O)) u H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex2/defs.ma
new file mode 100644 (file)
index 0000000..29b444d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex2/defs.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+definition ex2_c:
+ C
+\def
+ CSort O.
+
+definition ex2_t:
+ T
+\def
+ THead (Flat Appl) (TSort O) (TSort O).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ex2/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ex2/props.ma
new file mode 100644 (file)
index 0000000..01c8eed
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ex2/props.ma
@@ -0,0 +1,152 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ex2/defs.ma".
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/pr2/fwd.ma".
+
+include "basic_1A/arity/fwd.ma".
+
+lemma ex2_nf2:
+ nf2 ex2_c ex2_t
+\def
+ \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O) 
+(TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2 
+H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
+O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
+(Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort 
+O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat 
+Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O) 
+(TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2 
+(THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O) 
+x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1 
+(\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O) 
+(pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t: 
+T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort 
+(CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O)) 
+(\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal 
+T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1: 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O) 
+(Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T 
+(THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead 
+(Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2 
+x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def 
+(eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O) 
+(pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O) 
+x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7 
+\def (eq_ind T (TSort O) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (THead (Bind Abst) x0 x1) H2) in (False_ind (eq T (THead (Flat 
+Appl) (TSort O) (TSort O)) (THead (Bind Abbr) (TSort O) x3)) H7)) t2 
+H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) 
+(Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O) 
+(Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2) 
+(\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq 
+T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0) 
+x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O) 
+(TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead 
+(CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in 
+(let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O 
+H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+(TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) 
+t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (THead (Bind x0) x1 x2) H3) in (False_ind (eq T (THead (Flat Appl) 
+(TSort O) (TSort O)) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+(TSort O)) x3))) H9)) t2 H8))))))))))))))) H1)) H0))).
+
+lemma ex2_arity:
+ \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P: 
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat 
+Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def 
+(arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda 
+(a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O) 
+(TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O) 
+(TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let 
+H_x \def (leq_gen_head1 g x a (ASort O O) (arity_gen_sort g (CSort O) O 
+(AHead x a) H2)) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g x a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a 
+a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O O) (AHead a3 a4)))) P 
+(\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g x x0)).(\lambda (_: 
+(leq g a x1)).(\lambda (H6: (eq A (ASort O O) (AHead x0 x1))).(let H7 \def 
+(eq_ind A (ASort O O) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
+(False_ind P H7))))))) H3)))))) H0))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/flt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/flt/defs.ma
new file mode 100644 (file)
index 0000000..58047be
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/flt/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/C/defs.ma".
+
+definition fweight:
+ C \to (T \to nat)
+\def
+ \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
+
+definition flt:
+ C \to (T \to (C \to (T \to Prop)))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt 
+(fweight c1 t1) (fweight c2 t2))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/flt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/flt/fwd.ma
new file mode 100644 (file)
index 0000000..77fc7fa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/flt/fwd.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/flt/defs.ma".
+
+fact flt_wf__q_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C 
+\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq 
+nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall 
+(t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall 
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda 
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c: 
+C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c: 
+C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
+
+lemma flt_wf_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: 
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) 
+\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall 
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda 
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2: 
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) 
+\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda 
+(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: 
+nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda 
+(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: 
+nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq 
+nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0 
+(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2 
+(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c 
+t))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/flt/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/flt/props.ma
new file mode 100644 (file)
index 0000000..e81e734
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/flt/props.ma
@@ -0,0 +1,105 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/flt/defs.ma".
+
+include "basic_1A/C/props.ma".
+
+lemma flt_thead_sx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c 
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l 
+(tweight u) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
+u) (plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t))))))).
+
+lemma flt_thead_dx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c 
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l 
+(tweight t) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight 
+t) (plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t))))))).
+
+lemma flt_shift:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c 
+k u) t c (THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat 
+(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt 
+(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus 
+(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus 
+(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight 
+c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight 
+t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S 
+(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u) 
+(tweight t))))))).
+
+lemma flt_arith0:
+ \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t 
+(CHead c k t) (TLRef i)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: 
+nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
+
+lemma flt_arith1:
+ \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle 
+(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: 
+nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
+\def
+ \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda 
+(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_: 
+K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) 
+(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H 
+(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n: 
+nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2) 
+(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2) 
+(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S 
+O))))))))))).
+
+lemma flt_arith2:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 
+t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt 
+c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda 
+(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda 
+(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1) 
+(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight 
+t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S 
+O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
+
+lemma cle_flt_trans:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (c3: C).(\forall 
+(u2: T).(\forall (u3: T).((flt c2 u2 c3 u3) \to (flt c1 u2 c3 u3)))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight 
+c2))).(\lambda (c3: C).(\lambda (u2: T).(\lambda (u3: T).(\lambda (H0: (lt 
+(plus (cweight c2) (tweight u2)) (plus (cweight c3) (tweight 
+u3)))).(le_lt_trans (plus (cweight c1) (tweight u2)) (plus (cweight c2) 
+(tweight u2)) (plus (cweight c3) (tweight u3)) (le_plus_plus (cweight c1) 
+(cweight c2) (tweight u2) (tweight u2) H (le_n (tweight u2))) H0))))))).
+
+theorem flt_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1 
+t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt 
+c1 t1 c3 t3))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3: 
+T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1 
+t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/defs.ma
new file mode 100644 (file)
index 0000000..d7b32c4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubst0/defs.ma".
+
+inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
+| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 
+t2))
+| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 
+t1))
+| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: 
+C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/fwd.ma
new file mode 100644 (file)
index 0000000..f477697
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/fsubst0/fwd.ma
@@ -0,0 +1,57 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/fsubst0/defs.ma".
+
+implied lemma fsubst0_ind:
+ \forall (i: nat).(\forall (v: T).(\forall (c1: C).(\forall (t1: T).(\forall 
+(P: ((C \to (T \to Prop)))).(((\forall (t2: T).((subst0 i v t1 t2) \to (P c1 
+t2)))) \to (((\forall (c2: C).((csubst0 i v c1 c2) \to (P c2 t1)))) \to 
+(((\forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: C).((csubst0 i v c1 
+c2) \to (P c2 t2)))))) \to (\forall (c: C).(\forall (t: T).((fsubst0 i v c1 
+t1 c t) \to (P c t)))))))))))
+\def
+ \lambda (i: nat).(\lambda (v: T).(\lambda (c1: C).(\lambda (t1: T).(\lambda 
+(P: ((C \to (T \to Prop)))).(\lambda (f: ((\forall (t2: T).((subst0 i v t1 
+t2) \to (P c1 t2))))).(\lambda (f0: ((\forall (c2: C).((csubst0 i v c1 c2) 
+\to (P c2 t1))))).(\lambda (f1: ((\forall (t2: T).((subst0 i v t1 t2) \to 
+(\forall (c2: C).((csubst0 i v c1 c2) \to (P c2 t2))))))).(\lambda (c: 
+C).(\lambda (t: T).(\lambda (f2: (fsubst0 i v c1 t1 c t)).(match f2 with 
+[(fsubst0_snd x x0) \Rightarrow (f x x0) | (fsubst0_fst x x0) \Rightarrow (f0 
+x x0) | (fsubst0_both x x0 x1 x2) \Rightarrow (f1 x x0 x1 x2)]))))))))))).
+
+lemma fsubst0_gen_base:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall 
+(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 
+c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 
+i v t1 t2) (csubst0 i v c1 c2)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(fsubst0_ind 
+i v c1 t1 (\lambda (c: C).(\lambda (t: T).(or3 (land (eq C c1 c) (subst0 i v 
+t1 t)) (land (eq T t1 t) (csubst0 i v c1 c)) (land (subst0 i v t1 t) (csubst0 
+i v c1 c))))) (\lambda (t0: T).(\lambda (H0: (subst0 i v t1 t0)).(or3_intro0 
+(land (eq C c1 c1) (subst0 i v t1 t0)) (land (eq T t1 t0) (csubst0 i v c1 
+c1)) (land (subst0 i v t1 t0) (csubst0 i v c1 c1)) (conj (eq C c1 c1) (subst0 
+i v t1 t0) (refl_equal C c1) H0)))) (\lambda (c0: C).(\lambda (H0: (csubst0 i 
+v c1 c0)).(or3_intro1 (land (eq C c1 c0) (subst0 i v t1 t1)) (land (eq T t1 
+t1) (csubst0 i v c1 c0)) (land (subst0 i v t1 t1) (csubst0 i v c1 c0)) (conj 
+(eq T t1 t1) (csubst0 i v c1 c0) (refl_equal T t1) H0)))) (\lambda (t0: 
+T).(\lambda (H0: (subst0 i v t1 t0)).(\lambda (c0: C).(\lambda (H1: (csubst0 
+i v c1 c0)).(or3_intro2 (land (eq C c1 c0) (subst0 i v t1 t0)) (land (eq T t1 
+t0) (csubst0 i v c1 c0)) (land (subst0 i v t1 t0) (csubst0 i v c1 c0)) (conj 
+(subst0 i v t1 t0) (csubst0 i v c1 c0) H0 H1)))))) c2 t2 H))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/clear.ma
new file mode 100644 (file)
index 0000000..8608116
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/clear.ma
@@ -0,0 +1,139 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/props.ma".
+
+include "basic_1A/clear/drop.ma".
+
+lemma clear_getl_trans:
+ \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to 
+(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: 
+C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 
+c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 
+c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 
+(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda 
+(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to 
+(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: 
+C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall 
+(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda 
+(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda 
+(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 
+c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) 
+\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) 
+c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).(K_ind (\lambda 
+(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to 
+(getl (S n) c1 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c 
+(Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def 
+(getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in 
+(ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) 
+(getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda 
+(H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n 
+H6) H7)))) H5))))) (\lambda (f: F).(\lambda (_: (getl (S n) (CHead c (Flat f) 
+t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 
+c t H4 (getl (S n) c1 c3))))) k H1 H2))))))))) c2)))) i).
+
+lemma getl_clear_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to 
+(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1 
+c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all 
+c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
+C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1 
+x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in 
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 
+(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6 
+\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5) 
+in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind 
+x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1 
+c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0 
+x1 c3 x2 H7)))))))) H4))))) H1))))))).
+
+lemma getl_clear_bind:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c 
+(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) 
+\to (getl (S n) c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: 
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: 
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda 
+(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) 
+(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: 
+(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) 
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: 
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: 
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda 
+(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear 
+(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: 
+nat).(\lambda (H1: (getl n e1 e2)).(K_ind (\lambda (k0: K).((clear (CHead c0 
+k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) (\lambda 
+(b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) 
+v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) (CHead e1 (Bind b) v) 
+(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in 
+((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow v | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) 
+(clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in (\lambda (H6: (eq B b 
+b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1: 
+C).(getl n c1 e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(getl (S n) 
+(CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) n c0 e2 H8 t) b0 H6))))) H4)) 
+H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 
+(Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 
+(Bind b) v) t H2) e2 n H1) f t))) k H0))))))))))) c)).
+
+lemma getl_clear_conf:
+ \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to 
+(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: 
+C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 
+c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 
+c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: 
+C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in 
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 
+(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 
+\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead 
+x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: 
+C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 
+(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall 
+(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) 
+\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall 
+(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) 
+c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) 
+(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) 
+c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: 
+C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: 
+C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: 
+T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda 
+(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).(K_ind (\lambda (k0: 
+K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl 
+(S n) c2 c3)))) (\lambda (b: B).(\lambda (H3: (getl (S n) (CHead c (Bind b) 
+t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c 
+(Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 
+(getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4))))) 
+(\lambda (f: F).(\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda 
+(H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n 
+H3) c2 (clear_gen_flat f c c2 t H4))))) k H1 H2))))))))) c1)))) i).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/dec.ma
new file mode 100644 (file)
index 0000000..1a4ebcb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/dec.ma
@@ -0,0 +1,97 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/props.ma".
+
+lemma getl_dec:
+ \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda 
+(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: 
+C).((getl i c d) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T 
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) 
+v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) 
+(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: 
+C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) 
+v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) 
+(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: 
+Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall 
+(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
+T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to 
+(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: 
+nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: 
+B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall 
+(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind 
+(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
+T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O 
+(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl 
+(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead 
+c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 
+(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: 
+C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e 
+(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def 
+(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: 
+B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: 
+C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: 
+C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e 
+(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to 
+(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda 
+(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T 
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) 
+v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl 
+O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O 
+(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: 
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind 
+x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: 
+T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: 
+C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro 
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat 
+f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) 
+O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to 
+(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: 
+B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) 
+(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) 
+(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: 
+Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t 
+(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n: 
+nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
+(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: 
+C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H 
+(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda 
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall 
+(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T 
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) 
+(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to 
+(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda 
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind 
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead 
+e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
+(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: 
+C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: 
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0 
+(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: 
+B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) 
+(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) 
+(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) 
+(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 
+(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n) 
+c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: 
+C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind 
+b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: 
+Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t) 
+d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1))))) 
+i)))))) c).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/defs.ma
new file mode 100644 (file)
index 0000000..e9efe10
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/drop/defs.ma".
+
+include "basic_1A/clear/defs.ma".
+
+inductive getl (h: nat) (c1: C) (c2: C): Prop \def
+| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h 
+c1 c2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/drop.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/drop.ma
new file mode 100644 (file)
index 0000000..3e1444c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/drop.ma
@@ -0,0 +1,483 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/props.ma".
+
+include "basic_1A/clear/drop.ma".
+
+lemma getl_drop:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
+nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
+C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to 
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) 
+u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) 
+e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: 
+T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 
+e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) 
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: 
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear 
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) 
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind 
+b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind b) u) (CHead 
+c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 
+\def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow b | 
+(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) \Rightarrow b1 | (Flat 
+_) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) 
+(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C 
+T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind 
+b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: 
+(eq C e c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind 
+b0) t) c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) 
+c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) 
+H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e 
+(Bind b) u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead 
+e (Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O 
+(drop_refl e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) 
+(\lambda (n: nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) 
+\to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k 
+t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) 
+(\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e 
+(Bind b) u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+lemma getl_drop_conf_lt:
+ \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c0 e0)))))))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: 
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) 
+c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i 
+(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i 
+(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda 
+(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) 
+(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def 
+(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C 
+(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 
+(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: 
+C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead 
+c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to 
+((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead 
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) 
+(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda 
+(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind 
+b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda 
+(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda 
+(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda 
+(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: 
+K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) 
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: 
+B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda 
+(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def 
+(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c1 | 
+(CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) 
+t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H10 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow b | 
+(CHead _ k1 _) \Rightarrow (match k1 with [(Bind b1) \Rightarrow b1 | (Flat 
+_) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) 
+(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def 
+(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
+(CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) 
+t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in (\lambda (H12: (eq 
+B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 \def (eq_ind_r T t0 (\lambda 
+(t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind b0) t1))) H7 u H11) in (let 
+H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i O (CHead c0 k t) (CHead x0 
+(Bind b1) u))) H14 b H12) in (let H16 \def (eq_ind_r C x0 (\lambda (c2: 
+C).((drop i O (CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in (let H17 \def 
+(eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead c2 (Bind b) 
+u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) 
+d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: 
+T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) d) 
+x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: 
+(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: 
+T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) 
+H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
+T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) 
+(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind 
+b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 
+e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O 
+(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat 
+f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n 
+d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) 
+\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) 
+e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda 
+(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow c0 | (CHead c2 _ _) 
+\Rightarrow c2])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead 
+c0 k t) (CHead x0 (Flat f) t0) H10)) in ((let H12 \def (f_equal C K (\lambda 
+(e0: C).(match e0 with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow 
+k1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) 
+(CHead x0 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow t | (CHead _ _ t1) \Rightarrow t1])) 
+(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 
+(Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda (H15: (eq C 
+c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: C).(clear c2 (CHead c1 
+(Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) c0 H15) in 
+(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O (CHead c0 k t) c2) 
+\to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) IHx0 c0 H15) in (let H18 \def (eq_ind K k (\lambda (k1: K).((drop O 
+O (CHead c0 k1 t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) H14) in (let H19 \def 
+(eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) (CHead c0 k1 t) e)) H9 
+(Flat f) H14) in (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e 
+(CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r 
+(Flat f) (plus O d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat 
+f) (plus O d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda 
+(H21: (eq T t (lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r 
+(Flat f) (plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) 
+t (lift h (r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e 
+(\lambda (c2: C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead 
+c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 
+(Flat f) x2) H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: 
+C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t 
+(\lambda (t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead 
+c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) H24 (lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O 
+c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O 
+(CHead x1 (Flat f) x2) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T 
+u (lift h d x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda 
+(H29: (drop h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O 
+(CHead c0 (Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) 
+t1)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 
+(lift h d x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 
+(CHead c1 (Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) 
+(\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h 
+d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) 
+x2) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))) x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) 
+O H28 f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h 
+(plus O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda 
+(IHi: (((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k 
+t) (CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear 
+x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq 
+T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus 
+(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) 
+(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) 
+\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) 
+e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0)))))))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 
+k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S i0) d)) 
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r 
+k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 
+x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S 
+i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S 
+(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 
+(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 
+(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let 
+H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to 
+((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda 
+(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T 
+t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead 
+c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind 
+T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k 
+x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 
+O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S 
+(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) 
+(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S 
+i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h 
+(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k 
+i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) 
+(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 
+(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d 
+H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: 
+C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 
+(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def 
+(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S 
+(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) 
+H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind 
+b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) 
+in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
+T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift 
+h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) 
+H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k 
+H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
+
+lemma getl_drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d 
+h) i) \to (getl (minus i h) e a)))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
+d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) 
+in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 
+a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c 
+x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i 
+x c H3 e h d H0 H1) H4)))) H2)))))))))).
+
+lemma getl_conf_ge_drop:
+ \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 
+c2) \to (drop i O c2 e))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda 
+(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) 
+(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e 
+u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S 
+i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in 
+H3)))))))).
+
+lemma getl_drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
+(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i 
+c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) 
+(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
+\def
+ \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 
+e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 
+H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
+
+lemma drop_getl_trans_lt:
+ \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: 
+C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda 
+(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda 
+(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
+c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i 
+c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) 
+v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: 
+C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead 
+e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d 
+(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: 
+(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 
+e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: 
+C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: 
+C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
+c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat 
+(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) 
+(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b 
+e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h 
+(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C 
+(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) 
+(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda 
+(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda 
+(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i 
+c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h 
+(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus 
+d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d 
+(le_S_n (S i) (S d) (le_S (S (S i)) (S d) (le_n_S (S i) d H)))) c1 c2 h H0 x 
+H3))))) H2)))))))))))).
+
+lemma drop_getl_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 
+e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) 
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: 
+C).(\lambda (e1: C).(clear e1 e2))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
+c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def 
+(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
+(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: 
+C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) 
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: 
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def 
+(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i 
+O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda 
+(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: 
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
+e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h 
+(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i 
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) 
+(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) 
+H2)))))))))).
+
+lemma drop_getl_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) 
+\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: 
+C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def 
+(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
+(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: 
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro 
+(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
+
+lemma getl_drop_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to 
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i 
+h)) O c1 e2)))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
+nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 
+e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 
+H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda 
+(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: 
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 
+e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall 
+(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: 
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead 
+c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) 
+t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop 
+(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead 
+c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S 
+i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) 
+H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 
+(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead 
+c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 
+t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: 
+(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: 
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) 
+e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S 
+i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop 
+(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 
+i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: 
+F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: 
+nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: 
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) 
+e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) 
+(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) 
+(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) 
+t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to 
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i 
+n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 
+(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) 
+O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S 
+(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/flt.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/flt.ma
new file mode 100644 (file)
index 0000000..a6161f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/flt.ma
@@ -0,0 +1,60 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/fwd.ma".
+
+include "basic_1A/flt/props.ma".
+
+lemma getl_flt:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to 
+(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) 
+u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef 
+i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: 
+T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef 
+i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) 
+(CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) (\lambda (H0: 
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear 
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef 
+O)))) (\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e 
+(Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 with 
+[(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind b) 
+u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) 
+in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 
+\def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) 
+(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b 
+b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 
+(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t 
+(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b (\lambda (b1: B).(flt c0 t 
+(CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) 
+u H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) 
+t) (CHead e (Bind b) u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e 
+(Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))) 
+k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: 
+nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u 
+(CHead c0 k t) (TLRef n))))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead 
+e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind 
+b) u) t n H1)) in (flt_arith2 e c0 u (r k n) H_y k t (S n)))))) i)))))))) c)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/fwd.ma
new file mode 100644 (file)
index 0000000..ef6d598
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/fwd.ma
@@ -0,0 +1,154 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/defs.ma".
+
+include "basic_1A/drop/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+implied lemma getl_ind:
+ \forall (h: nat).(\forall (c1: C).(\forall (c2: C).(\forall (P: 
+Prop).(((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to P)))) \to 
+((getl h c1 c2) \to P)))))
+\def
+ \lambda (h: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (P: 
+Prop).(\lambda (f: ((\forall (e: C).((drop h O c1 e) \to ((clear e c2) \to 
+P))))).(\lambda (g: (getl h c1 c2)).(match g with [(getl_intro x x0 x1) 
+\Rightarrow (f x x0 x1)])))))).
+
+lemma getl_gen_all:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 
+C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 
+c2)).(getl_ind i c1 c2 (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: 
+C).(clear e c2))) (\lambda (e: C).(\lambda (H0: (drop i O c1 e)).(\lambda 
+(H1: (clear e c2)).(ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda 
+(e0: C).(clear e0 c2)) e H0 H1)))) H)))).
+
+lemma getl_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to 
+(\forall (P: Prop).P))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h 
+(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H) 
+in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear 
+e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2: 
+(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P 
+(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: 
+(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2 
+(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0 
+H1))))) H0)))))).
+
+lemma getl_gen_O:
+ \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
+\def
+ \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def 
+(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0)) 
+(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1: 
+(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0 
+(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
+
+lemma getl_gen_S:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: 
+nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: 
+nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all 
+(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c 
+k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0: 
+C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0 
+x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
+
+lemma getl_gen_2:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex_3 
+B C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind 
+b) v)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1 
+c2)).(let H0 \def (getl_gen_all c1 c2 i H) in (ex2_ind C (\lambda (e: 
+C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)) (ex_3 B C T (\lambda (b: 
+B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind b) v)))))) 
+(\lambda (x: C).(\lambda (_: (drop i O c1 x)).(\lambda (H2: (clear x 
+c2)).(let H3 \def (clear_gen_all x c2 H2) in (ex_3_ind B C T (\lambda (b: 
+B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))) (ex_3 B 
+C T (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(eq C c2 (CHead c (Bind 
+b) v)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: 
+(eq C c2 (CHead x1 (Bind x0) x2))).(let H5 \def (eq_ind C c2 (\lambda (c: 
+C).(clear x c)) H2 (CHead x1 (Bind x0) x2) H4) in (eq_ind_r C (CHead x1 (Bind 
+x0) x2) (\lambda (c: C).(ex_3 B C T (\lambda (b: B).(\lambda (c0: C).(\lambda 
+(v: T).(eq C c (CHead c0 (Bind b) v))))))) (ex_3_intro B C T (\lambda (b: 
+B).(\lambda (c: C).(\lambda (v: T).(eq C (CHead x1 (Bind x0) x2) (CHead c 
+(Bind b) v))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0) x2))) c2 H4)))))) 
+H3))))) H0))))).
+
+lemma getl_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n 
+e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e 
+(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d 
+H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to 
+(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v) 
+d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
+
+lemma getl_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d 
+(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda 
+(j: nat).(getl j e d)))))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or 
+(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: 
+nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl 
+O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c: 
+C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda 
+(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl 
+(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat 
+(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e 
+(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind 
+b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d 
+(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n: 
+nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n 
+O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S 
+j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e 
+(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b) 
+v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl 
+j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: 
+nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n 
+H0)))))) i))))).
+
+theorem getl_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to 
+(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
+\def
+ \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c 
+x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all 
+c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: 
+C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c 
+x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in 
+(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq 
+C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear 
+x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0 
+(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0: 
+C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def 
+(eq_ind_r C x0 (\lambda (c0: C).(clear c0 x1)) H6 x (drop_mono c x O h H2 x0 
+H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/getl.ma
new file mode 100644 (file)
index 0000000..389ee52
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/getl.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/drop.ma".
+
+include "basic_1A/getl/clear.ma".
+
+lemma getl_conf_le:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
+(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e 
+a))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda 
+(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda 
+(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e 
+a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x 
+e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1) 
+e H4)))) H2))))))))).
+
+theorem getl_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl 
+h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 
+e2)))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: 
+nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 
+e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: 
+C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) 
+(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x 
+e2)).(nat_ind (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 
+e2))) (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: 
+C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O 
+h) c1 c2 H e2 H5))) (\lambda (i0: nat).(\lambda (_: (((drop i0 O c2 x) \to 
+(getl (plus i0 h) c1 e2)))).(\lambda (H4: (drop (S i0) O c2 x)).(let H_y \def 
+(getl_drop_trans c1 c2 h H x i0 H4) in (getl_intro (plus (S i0) h) c1 e2 x 
+H_y H3))))) i H2)))) H1)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/getl/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/getl/props.ma
new file mode 100644 (file)
index 0000000..5ae39bf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/getl/props.ma
@@ -0,0 +1,72 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/getl/fwd.ma".
+
+include "basic_1A/clear/props.ma".
+
+include "basic_1A/drop/props.ma".
+
+lemma getl_refl:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) 
+(CHead c (Bind b) u))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind 
+b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b) 
+u)) (clear_bind b c u)))).
+
+lemma getl_head:
+ \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k 
+h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
+\def
+ \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
+(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k 
+h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0: 
+C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1: 
+(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k 
+u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
+
+lemma getl_flat:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall 
+(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c 
+e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in 
+(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) 
+(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c 
+x)).(\lambda (H2: (clear x e)).(nat_ind (\lambda (n: nat).((drop n O c x) \to 
+(getl n (CHead c (Flat f) u) e))) (\lambda (H3: (drop O O c x)).(let H4 \def 
+(eq_ind_r C x (\lambda (c0: C).(clear c0 e)) H2 c (drop_gen_refl c x H3)) in 
+(getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c 
+(Flat f) u)) (clear_flat c e H4 f u)))) (\lambda (h0: nat).(\lambda (_: 
+(((drop h0 O c x) \to (getl h0 (CHead c (Flat f) u) e)))).(\lambda (H3: (drop 
+(S h0) O c x)).(getl_intro (S h0) (CHead c (Flat f) u) e x (drop_drop (Flat 
+f) h0 c x H3 u) H2)))) h H1)))) H0))))))).
+
+lemma getl_ctail:
+ \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: 
+T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda 
+(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C 
+(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b) 
+u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x: 
+C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b) 
+u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v 
+x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/iso/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/iso/defs.ma
new file mode 100644 (file)
index 0000000..46050e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/iso/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+inductive iso: T \to (T \to Prop) \def
+| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
+| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2)))
+| iso_head: \forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: 
+T).(\forall (k: K).(iso (THead k v1 t1) (THead k v2 t2)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/iso/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/iso/fwd.ma
new file mode 100644 (file)
index 0000000..9977071
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/iso/fwd.ma
@@ -0,0 +1,184 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/iso/defs.ma".
+
+include "basic_1A/tlist/defs.ma".
+
+implied lemma iso_ind:
+ \forall (P: ((T \to (T \to Prop)))).(((\forall (n1: nat).(\forall (n2: 
+nat).(P (TSort n1) (TSort n2))))) \to (((\forall (i1: nat).(\forall (i2: 
+nat).(P (TLRef i1) (TLRef i2))))) \to (((\forall (v1: T).(\forall (v2: 
+T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).(P (THead k v1 t1) 
+(THead k v2 t2)))))))) \to (\forall (t: T).(\forall (t0: T).((iso t t0) \to 
+(P t t0)))))))
+\def
+ \lambda (P: ((T \to (T \to Prop)))).(\lambda (f: ((\forall (n1: 
+nat).(\forall (n2: nat).(P (TSort n1) (TSort n2)))))).(\lambda (f0: ((\forall 
+(i1: nat).(\forall (i2: nat).(P (TLRef i1) (TLRef i2)))))).(\lambda (f1: 
+((\forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).(P (THead k v1 t1) (THead k v2 t2))))))))).(\lambda (t: T).(\lambda 
+(t0: T).(\lambda (i: (iso t t0)).(match i with [(iso_sort x x0) \Rightarrow 
+(f x x0) | (iso_lref x x0) \Rightarrow (f0 x x0) | (iso_head x x0 x1 x2 x3) 
+\Rightarrow (f1 x x0 x1 x2 x3)]))))))).
+
+lemma iso_gen_sort:
+ \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda 
+(n2: nat).(eq T u2 (TSort n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) 
+u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex 
+nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0: 
+(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1)) 
+\to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0: 
+nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort n) \Rightarrow n | 
+(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
+n1) H1) in (ex_intro nat (\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 
+(refl_equal T (TSort n2))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda 
+(H1: (eq T (TLRef i1) (TSort n1))).(let H2 \def (eq_ind T (TLRef i1) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex 
+nat (\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
+K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T 
+(THead k v1 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort 
+n1) H1) in (False_ind (ex nat (\lambda (n2: nat).(eq T (THead k v2 t2) (TSort 
+n2)))) H2)))))))) y u2 H0))) H))).
+
+lemma iso_gen_lref:
+ \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda 
+(n2: nat).(eq T u2 (TLRef n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) 
+u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex 
+nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0: 
+(iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1)) 
+\to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0: 
+nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2 
+\def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n3: nat).(eq T 
+(TSort n2) (TLRef n3)))) H2))))) (\lambda (i1: nat).(\lambda (i2: 
+nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2 \def (f_equal T nat 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow i1 | (TLRef n) 
+\Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in 
+(ex_intro nat (\lambda (n2: nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal 
+T (TLRef i2))))))) (\lambda (v1: T).(\lambda (v2: T).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H1: (eq T (THead k v1 t1) 
+(TLRef n1))).(let H2 \def (eq_ind T (THead k v1 t1) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ 
+_ _) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2: 
+nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
+
+lemma iso_gen_head:
+ \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso 
+(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2 
+(THead k v2 t2)))))))))
+\def
+ \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda 
+(H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t: 
+T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2: 
+T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y 
+u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to 
+(ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2)))))))) 
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k 
+v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda 
+(v2: T).(\lambda (t2: T).(eq T (TSort n2) (THead k v2 t2))))) H2))))) 
+(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (THead k 
+v1 t1))).(let H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda 
+(v2: T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) 
+(\lambda (v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(k0: K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | (TLRef 
+_) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 v0 t0) (THead 
+k v1 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) 
+\Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k 
+v1 t1) H1) in (\lambda (_: (eq T v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r 
+K k (\lambda (k1: K).(ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead 
+k1 v2 t2) (THead k v3 t3)))))) (ex_2_intro T T (\lambda (v3: T).(\lambda (t3: 
+T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 
+t2))) k0 H6)))) H3)) H2)))))))) y u2 H0))) H))))).
+
+lemma iso_flats_lref_bind_false:
+ \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall 
+(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind 
+b) v t)) \to (\forall (P: Prop).P)))))))
+\def
+ \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda 
+(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads 
+(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) 
+(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let 
+H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in 
+(ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P 
+(\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let 
+H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef x) H1) in (False_ind P H2)))) H0))))) (\lambda 
+(t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f) t1 (TLRef 
+i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso 
+(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) (THead (Bind b) v 
+t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f) t0 (THeads (Flat 
+f) t1 (TLRef i)) (THead (Bind b) v t) H0) in (let H1 \def H_x in (ex_2_ind T 
+T (\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Bind b) v t) (THead (Flat 
+f) v2 t2)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T (THead 
+(Bind b) v t) (THead (Flat f) x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) 
+v t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat f) x0 x1) 
+H2) in (False_ind P H3))))) H1)))))))) vs)))))).
+
+lemma iso_flats_flat_bind_false:
+ \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall 
+(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads 
+(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: 
+Prop).P)))))))))
+\def
+ \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda 
+(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind 
+(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) 
+(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead 
+(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def 
+(iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in 
+(ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t) 
+(THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: 
+(eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T 
+(THead (Bind b) v t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1: 
+TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) 
+(THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead 
+(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v 
+t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads 
+(Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1 
+\def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead 
+(Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3 
+\def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1)))))))) 
+vs)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/iso/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/iso/props.ma
new file mode 100644 (file)
index 0000000..8accf58
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/iso/props.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/fwd.ma".
+
+include "basic_1A/iso/fwd.ma".
+
+lemma iso_refl:
+ \forall (t: T).(iso t t)
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n: 
+nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_: 
+(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
+
+theorem iso_trans:
+ \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 
+t3) \to (iso t1 t3)))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) 
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso 
+(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in 
+(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3) 
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x) 
+(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1))))))) 
+(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso 
+(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in 
+(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3) 
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x) 
+(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1))))))) 
+(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def 
+(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda 
+(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0 
+x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t)) 
+(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1A/leq/asucc.ma
new file mode 100644 (file)
index 0000000..9453c41
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/leq/asucc.ma
@@ -0,0 +1,447 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/leq/props.ma".
+
+lemma asucc_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
+(asucc g a1) (asucc g a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g 
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: 
+nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g 
+(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) 
+(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O 
+(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow 
+(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq 
+A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: 
+nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g 
+(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S 
+h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) 
+(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind 
+A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O 
+(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq 
+A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) 
+(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) 
+(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) 
+H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g 
+(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: 
+nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) 
+\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next 
+g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g 
+(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) 
+n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g 
+(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: 
+A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) 
+n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) 
+k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort 
+O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) 
+(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) 
+(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
+(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g 
+n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort 
+O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A 
+(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda 
+(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to 
+((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g 
+(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow 
+(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) 
+\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O 
+\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) 
+(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) 
+k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) 
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
+\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 
+(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A 
+(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) 
+(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g 
+(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O 
+n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort 
+(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) 
+(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda 
+(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
+h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) 
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
+\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g 
+n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 
+with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h 
+n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
+(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
+(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next 
+g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 
+n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a 
+(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) 
+(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A 
+(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g 
+(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S 
+h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) 
+(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) 
+(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda 
+(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g 
+(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: 
+(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g 
+a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
+
+lemma asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc 
+g a2)) \to (leq g a1 a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g 
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda 
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) 
+(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort 
+n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) 
+(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 
+n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g 
+(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g 
+(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 
+g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind 
+nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort 
+h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
+x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) 
+(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
+[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g 
+n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match 
+e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with 
+[(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) 
+H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda 
+(n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) 
+x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A 
+(aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) 
+H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda 
+(a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) 
+(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
+(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) 
+a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in 
+(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: 
+nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) 
+\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g 
+(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O 
+(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda 
+(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g 
+(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
+x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
+\Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) 
+H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ 
+n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 
+x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 
+(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort 
+n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: 
+nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 
+n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) 
+(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) 
+(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
+(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 
+(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in 
+(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) 
+(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g 
+(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq 
+g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda 
+(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to 
+((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort 
+n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) 
+(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O 
+n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) 
+\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 
+n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g 
+(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus 
+g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
+\Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1 
+x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort 
+_ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _) 
+\Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda 
+(H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A 
+(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9 
+\def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2) 
+(aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A 
+(aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g 
+n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0 
+n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2) 
+(\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g 
+(ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O 
+n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_: 
+(((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g 
+(asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) 
+(ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda 
+(H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) 
+n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4) 
+n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def 
+(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind 
+nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
+(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 
+n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: 
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g 
+(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 
+n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
+[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2) 
+(ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e 
+with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 
+n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r 
+nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 
+x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A 
+(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 
+\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g 
+(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) 
+(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g 
+(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) 
+a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) 
+in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 
+H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n 
+n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda 
+(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) 
+a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a 
+a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g 
+a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) 
+(asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 
+n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) 
+(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O 
+n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq 
+g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g 
+(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc 
+g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g 
+n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g 
+(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda 
+(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g 
+(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 
+x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match 
+ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
+(ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8))))))) 
+H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) 
+(asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 
+n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort 
+n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a 
+a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to 
+(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) 
+n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g 
+(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def 
+(leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in 
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) 
+(ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0: 
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort 
+n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g 
+a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda 
+(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
+\Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1) 
+n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: 
+A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq 
+g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g 
+a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: 
+A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a 
+a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g 
+(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 
+n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O 
+n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) 
+H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) 
+(\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3 
+a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda 
+(H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A 
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
+(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: 
+nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) 
+\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g 
+(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a 
+(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda 
+(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq 
+g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) 
+(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: 
+A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g 
+a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def 
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
+(False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1)))) 
+(\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) 
+\to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc 
+g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: 
+(leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def 
+(leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def 
+H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5))) 
+(\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5: 
+A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g 
+(AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: 
+(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A 
+(AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda 
+(e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow 
+a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A 
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead 
+_ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda 
+(H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g 
+(asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0 
+(\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0 
+a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
+
+lemma leq_asucc:
+ \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
+a0)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: 
+A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro 
+A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) 
+(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda 
+(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A 
+(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A 
+(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g 
+(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc 
+g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) 
+(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) 
+a)).
+
+lemma leq_ahead_asucc_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 
+(asucc g a1)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead 
+(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: 
+nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O 
+(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g 
+(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 
+g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind 
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: 
+A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
+(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda 
+(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A 
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in 
+(False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead 
+(ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S 
+n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) 
+a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda 
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort 
+n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee: 
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
+False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) 
+(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g 
+a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall 
+(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: 
+Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) 
+(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g 
+(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind 
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: 
+A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A 
+(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 
+x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def 
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
+(AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in 
+((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) 
+\Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g 
+a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A 
+x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def 
+(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in 
+(leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
+
+lemma leq_asucc_false:
+ \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) 
+a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda 
+(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
+(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g 
+n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: 
+(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O 
+(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda 
+(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda 
+(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g 
+(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A 
+(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: 
+A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O])) 
+(ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e: 
+A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) 
+(ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def 
+(eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) 
+(aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 
+(\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O 
+n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g 
+n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g 
+(ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def 
+(aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 
+(\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) 
+H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow 
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to 
+P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def 
+(leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in 
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort 
+h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: 
+nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) 
+x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def 
+(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
+(AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in 
+((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2) 
+\Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1 
+x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1 
+(\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0) 
+x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A 
+(aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let 
+H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0 
+(aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2)) 
+(aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2 
+(ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le 
+n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda 
+(a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: 
+Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to 
+(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead 
+a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) 
+(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
+(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 
+x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) 
+(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1) 
+(AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) 
+(AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 
+(\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def 
+(eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) 
+H6))))))) H2))))))))) a)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/leq/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/leq/defs.ma
new file mode 100644 (file)
index 0000000..8c0605c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/leq/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/aplus/defs.ma".
+
+inductive leq (g: G): A \to (A \to Prop) \def
+| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall 
+(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
+h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
+| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: 
+A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 
+a4))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/leq/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/leq/fwd.ma
new file mode 100644 (file)
index 0000000..939990a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/leq/fwd.ma
@@ -0,0 +1,254 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/leq/defs.ma".
+
+implied rec lemma leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1: 
+nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k: 
+nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (P 
+(ASort h1 n1) (ASort h2 n2))))))))) (f0: (\forall (a1: A).(\forall (a2: 
+A).((leq g a1 a2) \to ((P a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leq 
+g a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead a2 a4))))))))))) (a: A) 
+(a0: A) (l: leq g a a0) on l: P a a0 \def match l with [(leq_sort h1 h2 n1 n2 
+k e) \Rightarrow (f h1 h2 n1 n2 k e) | (leq_head a1 a2 l0 a3 a4 l1) 
+\Rightarrow (f0 a1 a2 l0 ((leq_ind g P f f0) a1 a2 l0) a3 a4 l1 ((leq_ind g P 
+f f0) a3 a4 l1))].
+
+lemma leq_gen_sort1:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
+g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
+(ASort h2 n2))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda 
+(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort 
+h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A 
+a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g 
+(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat 
+nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a 
+k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: 
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
+k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal 
+A nat (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead _ _) 
+\Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H2) in ((let H4 \def (f_equal A 
+nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) 
+\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 
+h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 
+n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
+nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda 
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 
+n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort 
+n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) 
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) 
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
+(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 
+H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
+(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) 
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to 
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 
+n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def 
+(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H5) in 
+(False_ind (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(k: nat).(eq A (aplus g (AHead a1 a4) k) (aplus g (ASort h2 n2) k))))) 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a3 a5) 
+(ASort h2 n2)))))) H6))))))))))) y a2 H0))) H))))).
+
+lemma leq_gen_head1:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g 
+(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
+a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
+(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
+g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
+a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
+(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda 
+(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq 
+A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) 
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match 
+ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
+(AHead a1 a2) H2) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda 
+(a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) (AHead a3 a4))))) H3))))))))) 
+(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: 
+(((eq A a0 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: 
+A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda 
+(a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))))))).(\lambda (a4: 
+A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a4 
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a1 
+a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda (a6: 
+A).(\lambda (a7: A).(eq A a5 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead 
+a0 a4) (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e 
+with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 
+a4) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
+with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 
+a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 a1)).(let H9 \def (eq_ind A 
+a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: 
+A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 
+a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A a5 (AHead a7 a8))))))) H4 a2 
+H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: A).(leq g a6 a5)) H3 a2 H7) 
+in (let H11 \def (eq_ind A a0 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 \def (eq_ind A a0 (\lambda (a6: 
+A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A A (\lambda (a6: A).(\lambda (_: 
+A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda 
+(a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) a3 a5 H12 H10 
+(refl_equal A (AHead a3 a5))))))))) H6))))))))))) y a H0))) H))))).
+
+lemma leq_gen_sort2:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
+g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) 
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
+(ASort h2 n2))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
+A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda 
+(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) 
+(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq 
+A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind 
+g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat 
+nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus 
+g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: 
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
+k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal 
+A nat (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead _ _) 
+\Rightarrow h2])) (ASort h2 n2) (ASort h1 n1) H2) in ((let H4 \def (f_equal A 
+nat (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead _ _) 
+\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 
+h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 
+n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
+nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda 
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 
+n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort 
+h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) 
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) 
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 
+n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
+(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 
+H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
+(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to 
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 
+n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def 
+(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H5) in 
+(False_ind (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (AHead a3 a5) k))))) 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a1 a4) 
+(ASort h2 n2)))))) H6))))))))))) a2 y H0))) H))))).
+
+lemma leq_gen_head2:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a 
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 
+a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
+(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
+g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
+a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
+(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda 
+(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq 
+A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) 
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match 
+ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
+(AHead a1 a2) H2) in (False_ind (ex3_2 A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda 
+(a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) (AHead a3 a4))))) H3))))))))) 
+(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: 
+(((eq A a3 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: 
+A).(leq g a4 a1))) (\lambda (_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda 
+(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))))))).(\lambda (a4: 
+A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a5 
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a6 
+a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda (a6: 
+A).(\lambda (a7: A).(eq A a4 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead 
+a3 a5) (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e 
+with [(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 
+a5) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
+with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) (AHead a3 
+a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 a1)).(let H9 \def (eq_ind A 
+a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: 
+A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 
+a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A a4 (AHead a7 a8))))))) H4 a2 
+H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: A).(leq g a4 a6)) H3 a2 H7) 
+in (let H11 \def (eq_ind A a3 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 \def (eq_ind A a3 (\lambda (a6: 
+A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A A (\lambda (a6: A).(\lambda (_: 
+A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda 
+(a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) (AHead a6 a7)))) a0 a4 H12 H10 
+(refl_equal A (AHead a0 a4))))))))) H6))))))))))) a y H0))) H))))).
+
+lemma ahead_inj_snd:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
+(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda 
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def 
+(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A 
+(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda 
+(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) 
+(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda 
+(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead 
+a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e 
+with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a4) 
+(AHead x0 x1) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) 
+(AHead x0 x1) H3) in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 
+(\lambda (a: A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 
+(\lambda (a: A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/leq/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/leq/props.ma
new file mode 100644 (file)
index 0000000..624a3b8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/leq/props.ma
@@ -0,0 +1,188 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/leq/fwd.ma".
+
+include "basic_1A/aplus/props.ma".
+
+lemma leq_refl:
+ \forall (g: G).(\forall (a: A).(leq g a a))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) 
+(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A 
+(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 
+a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 
+H0))))) a)).
+
+lemma leq_eq:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 
+a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 
+a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
+
+lemma leq_sym:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
+a2 a1))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: 
+nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g 
+(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: 
+(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: 
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 
+H1 a6 a5 H3))))))))) a1 a2 H)))).
+
+theorem leq_trans:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
+(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g 
+(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 
+\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 
+(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus 
+g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def 
+(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 
+x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 
+n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g 
+(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r 
+nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) 
+(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) 
+x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus 
+g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 
+H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) 
+(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus 
+(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g 
+(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 
+x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g 
+(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: 
+A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: 
+A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: 
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) 
+\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) 
+a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in 
+(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: 
+(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 
+(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead 
+a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) 
+H5))))))))))))) a1 a2 H)))).
+
+lemma leq_ahead_false_1:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
+A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n 
+n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
+(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) 
+a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: 
+A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O 
+n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: 
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
+False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: 
+nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to 
+P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1) 
+n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort (S n1) n0) 
+H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 
+a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3 
+a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) 
+n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort (S n1) n0) 
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda (ee: 
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
+False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) 
+(\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to 
+(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: 
+A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: 
+A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: 
+Prop).(let H_x \def (leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let 
+H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead 
+a a0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: 
+A).(\lambda (x1: A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq 
+g a2 x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def 
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
+(AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7 
+\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | 
+(AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda 
+(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) 
+H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a 
+a0) a3)) H3 a H8) in (H a0 H10 P))))) H6))))))) H2)))))))))) a1)).
+
+lemma leq_ahead_false_2:
+ \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) 
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: 
+A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort 
+n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
+a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O 
+n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
+(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
+x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) 
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match 
+ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I 
+(AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: 
+nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to 
+P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) n0)) (ASort (S n1) 
+n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) (ASort (S n1) n0) 
+H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (ASort (S n1) n0) 
+a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) n0) (AHead a3 
+a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
+x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort (S 
+n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda 
+(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) 
+\Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n 
+H)))))) (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) 
+a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall 
+(a1: A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: 
+A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: 
+Prop).(let H_x \def (leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let 
+H2 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
+a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda 
+(a3: A).(\lambda (a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: 
+A).(\lambda (x1: A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead 
+a a0) x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def 
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
+(AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in ((let H7 
+\def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | 
+(AHead _ a3) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) in (\lambda 
+(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead 
+a a0) a3)) H4 a0 H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g 
+a1 a3)) H3 a H8) in (H0 a H9 P))))) H6))))))) H2)))))))))) a2)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift/defs.ma
new file mode 100644 (file)
index 0000000..4f50119
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift/defs.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/tlist/defs.ma".
+
+include "basic_1A/s/defs.ma".
+
+rec definition lref_map (f: (nat \to nat)) (d: nat) (t: T) on t: T \def match 
+t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match 
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u 
+t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))].
+
+definition lift:
+ nat \to (nat \to (T \to T))
+\def
+ \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: 
+nat).(plus x h)) i t))).
+
+rec definition lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def match ts 
+with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) 
+(lifts h d ts0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift/fwd.ma
new file mode 100644 (file)
index 0000000..88a0ecb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift/fwd.ma
@@ -0,0 +1,634 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/props.ma".
+
+lemma lift_gen_sort:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T 
+(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind 
+(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) 
+(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort 
+n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: 
+(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort 
+n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let 
+H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (lift h d (TLRef n0)) H) in (False_ind (lt n0 d) H1)))) in (let H2 
+\def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))) 
+(\lambda (_: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda 
+(t0: T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d (let 
+H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (lift h d (TLRef n0)) H) in (False_ind (le d n0) H1)))) in (let H2 
+\def (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef (plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) 
+H2))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) 
+(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq 
+T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T 
+(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d 
+(THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n) t2)) H1 (THead k (lift h d 
+t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T 
+(TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k 
+(lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k t0 t1) 
+(TSort n)) H3))))))))) t)))).
+
+lemma lift_gen_lref:
+ \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
+(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le 
+(plus d h) i) (eq T t (TLRef (minus i h)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h: 
+nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) 
+(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
+h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda 
+(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def 
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort 
+n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (TSort n) H0) in (False_ind (or (land 
+(lt i d) (eq T (TSort n) (TLRef i))) (land (le (plus d h) i) (eq T (TSort n) 
+(TLRef (minus i h))))) H1)))))))) (\lambda (n: nat).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H: (eq T (TLRef i) (lift h 
+d (TLRef n)))).(lt_le_e n d (or (land (lt i d) (eq T (TLRef n) (TLRef i))) 
+(land (le (plus d h) i) (eq T (TLRef n) (TLRef (minus i h))))) (\lambda (H0: 
+(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T 
+(TLRef i) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (f_equal 
+T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0) 
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H1) in 
+(eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef 
+n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) 
+(or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus d h) n) 
+(eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) (TLRef 
+n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d n)).(let H1 
+\def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H 
+(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal T nat 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow i | (TLRef n0) 
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef (plus n h)) 
+H1) in (eq_ind_r nat (plus n h) (\lambda (n0: nat).(or (land (lt n0 d) (eq T 
+(TLRef n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus 
+n0 h)))))) (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt (plus n h) d) (eq 
+T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h) (plus n h)) (eq T 
+(TLRef n) (TLRef n0))))) (or_intror (land (lt (plus n h) d) (eq T (TLRef n) 
+(TLRef (plus n h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef 
+n))) (conj (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n)) 
+(le_plus_plus d n h h H0 (le_n h)) (refl_equal T (TLRef n)))) (minus (plus n 
+h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda (k: K).(\lambda (t0: 
+T).(\lambda (_: ((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
+(TLRef i) (lift h d t0)) \to (or (land (lt i d) (eq T t0 (TLRef i))) (land 
+(le (plus d h) i) (eq T t0 (TLRef (minus i h))))))))))).(\lambda (t1: 
+T).(\lambda (_: ((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
+(TLRef i) (lift h d t1)) \to (or (land (lt i d) (eq T t1 (TLRef i))) (land 
+(le (plus d h) i) (eq T t1 (TLRef (minus i h))))))))))).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H1: (eq T (TLRef i) (lift 
+h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) 
+(\lambda (t2: T).(eq T (TLRef i) t2)) H1 (THead k (lift h d t0) (lift h (s k 
+d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (eq_ind T (TLRef i) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s 
+k d) t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef 
+i))) (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) 
+H3)))))))))))) t).
+
+lemma lift_gen_lref_lt:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall 
+(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n 
+d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x 
+\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) 
+(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq 
+T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind 
+(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n 
+d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: 
+(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d 
+h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus 
+d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef 
+(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq 
+T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S 
+n) d h H))) t H4))) H2)) H1)))))))).
+
+lemma lift_gen_lref_false:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
+(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall 
+(P: Prop).P)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T 
+(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d 
+h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) 
+(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land 
+(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P 
+(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H 
+H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n 
+h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda 
+(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false 
+(plus d h) n P H4 H0))) H3)) H2)))))))))).
+
+lemma lift_gen_lref_ge:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
+(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
+n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 
+t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in 
+(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus 
+d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) 
+(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind 
+(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda 
+(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r 
+T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq 
+T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d 
+(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) 
+(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
+h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
+h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda 
+(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n 
+h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus 
+(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
+
+lemma lift_gen_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (s k d) z)))))))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind 
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) 
+(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda 
+(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 
+\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
+T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u 
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) 
+z))))) H1))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d 
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: (lt n 
+d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead 
+k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (eq_ind T 
+(THead k u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
+(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))) 
+(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
+(t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d 
+H0)) in (let H2 \def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: 
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 
+t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: 
+T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
+(lift_head k0 t0 t1 h d)) in (let H3 \def (f_equal T K (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
+\Rightarrow k1])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
+H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) 
+(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | 
+(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead k u t) 
+(THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (\lambda (H6: (eq T u 
+(lift h d t0))).(\lambda (H7: (eq K k k0)).(let H8 \def (eq_ind_r K k0 
+(\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k H7) in (eq_ind K k 
+(\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k1 
+t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))) (let H9 
+\def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq 
+T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: 
+T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h0 
+d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h0 (s k d0) 
+z))))))))) H0 (lift h (s k d) t1) H8) in (let H10 \def (eq_ind T t (\lambda 
+(t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 
+d0 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t2 (lift h0 (s k d0) z))))))))) H (lift h (s k d) 
+t1) H8) in (eq_ind_r T (lift h (s k d) t1) (\lambda (t2: T).(ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t2 (lift h (s k d) z)))))) (let H11 \def (eq_ind T u 
+(\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k t2 
+(lift h (s k d) t1)) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda 
+(z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 
+(lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
+(lift h0 (s k d0) z))))))))) H10 (lift h d t0) H6) in (let H12 \def (eq_ind T 
+u (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: nat).((eq T (THead k t2 
+(lift h (s k d) t1)) (lift h0 d0 t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda 
+(z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 
+(lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
+(lift h0 (s k d0) z))))))))) H9 (lift h d t0) H6) in (eq_ind_r T (lift h d 
+t0) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+k t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k 
+d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 
+t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k d) t1) 
+(lift h (s k d) z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T 
+(lift h d t0)) (refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) 
+H4)) H3))))))))))) x)))).
+
+lemma lift_gen_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (S d) z)))))))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d 
+x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda 
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
+h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
+(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
+(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) 
+x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) 
+z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro 
+T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind 
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) 
+z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d 
+x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
+
+lemma lift_gen_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d 
+x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda 
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
+h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
+(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) 
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T 
+(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq 
+T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d 
+x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T 
+(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) 
+(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x 
+H1)))))) H0))))))))).
+
+lemma lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T 
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t 
+t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def 
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H 
+(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t 
+H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq 
+T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d 
+(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt 
+n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d 
+d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift 
+h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) 
+(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 
+t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
+T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) 
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) 
+\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: 
+nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t 
+t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
+(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
+(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
+(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z)))) 
+(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift 
+h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r 
+T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) 
+(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind 
+b) t t0) (THead (Bind b) x0 x1) (f_equal3 K T T T THead (Bind b) (Bind b) t 
+x0 t0 x1 (refl_equal K (Bind b)) (H x0 h d H4) (H0 x1 h (S d) H5)))) t1 
+H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d 
+H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to 
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall 
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: 
+(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T 
+(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 
+(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T 
+(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq 
+T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d 
+x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead 
+(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T 
+(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) 
+(THead (Flat f) x0 x1) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 x1 
+(refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)))) t1 H3)))))) 
+(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
+
+lemma lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: 
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1: 
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to 
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: 
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 
+t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda 
+(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1 
+d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1) 
+x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t 
+(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T 
+(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) 
+(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda 
+(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n) 
+(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T 
+(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1 
+d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T 
+(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2 
+(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda 
+(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda 
+(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2 
+h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) 
+(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n 
+d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t 
+(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in 
+(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift 
+h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T 
+(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
+(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: 
+T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n)) 
+(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef 
+n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 
+(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n 
+(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2)))) 
+(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) 
+(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1)) 
+(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x 
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) 
+(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2 
+T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) 
+(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1)) 
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef 
+n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) 
+t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n 
+h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) 
+(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x 
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2))) 
+(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq 
+T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) 
+(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus 
+n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2) 
+h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4) 
+(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda 
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 
+d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus 
+n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus 
+(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans 
+h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T 
+(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T 
+t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) 
+(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1 
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n 
+h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) 
+(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2) 
+h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T 
+(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2))) 
+(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2 
+n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2 
+h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2) 
+(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2))))) 
+(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t: 
+T).(eq T (TLRef (plus (minus n h2) h2)) t)) (sym_eq T (TLRef (plus (minus 
+(plus (minus n h2) h2) h2) h2)) (TLRef (plus (minus n h2) h2)) (f_equal nat T 
+TLRef (plus (minus (plus (minus n h2) h2) h2) h2) (plus (minus n h2) h2) 
+(f_equal2 nat nat nat plus (minus (plus (minus n h2) h2) h2) (minus n h2) h2 
+h2 (minus_plus_r (minus n h2) h2) (refl_equal nat h2)))) (lift h2 d2 (TLRef 
+(minus (plus (minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) 
+h2) h2) h2 d2 (le_minus d2 (plus (minus n h2) h2) h2 (le_plus_plus d2 (minus 
+n h2) h2 h2 (le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n 
+(le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 
+(plus d2 h1) (minus (plus n h1) h2) (arith0 h2 d2 n H4 h1) x 
+H5)))))))))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall 
+(x: T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: 
+nat).((le d1 d2) \to ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 
+T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift 
+h2 d2 t2))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: 
+T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: 
+nat).((le d1 d2) \to ((eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) x)) \to 
+(ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0 
+(lift h2 d2 t2))))))))))))).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2: 
+nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (H1: (le d1 d2)).(\lambda 
+(H2: (eq T (lift h1 d1 (THead k t t0)) (lift h2 (plus d2 h1) x))).(K_ind 
+(\lambda (k0: K).((eq T (lift h1 d1 (THead k0 t t0)) (lift h2 (plus d2 h1) 
+x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: 
+T).(eq T (THead k0 t t0) (lift h2 d2 t2)))))) (\lambda (b: B).(\lambda (H3: 
+(eq T (lift h1 d1 (THead (Bind b) t t0)) (lift h2 (plus d2 h1) x))).(let H4 
+\def (eq_ind T (lift h1 d1 (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 
+(lift h2 (plus d2 h1) x))) H3 (THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) 
+t0)) (lift_bind b t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: 
+T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
+(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) z)))) (ex2 T (\lambda 
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) t 
+t0) (lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
+x (THead (Bind b) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus 
+d2 h1) x0))).(\lambda (H7: (eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 
+h1)) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead 
+(Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift 
+h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2: 
+T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
+(THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T 
+x0 (lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T 
+(lift h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind 
+b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t t0) 
+(lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t3))) 
+(\lambda (t3: T).(eq T (THead (Bind b) t2 t0) (lift h2 d2 t3))))) (let H10 
+\def (refl_equal nat (plus (S d2) h1)) in (let H11 \def (eq_ind nat (S (plus 
+d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) t0) (lift h2 n x1))) H7 (plus 
+(S d2) h1) H10) in (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 (S d1) t2))) 
+(\lambda (t2: T).(eq T t0 (lift h2 (S d2) t2))) (ex2 T (\lambda (t2: T).(eq T 
+(THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
+(THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3: 
+T).(\lambda (H12: (eq T x1 (lift h1 (S d1) x3))).(\lambda (H13: (eq T t0 
+(lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S d1) x3) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) t2) (lift h1 d1 t3))) 
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2 
+t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: 
+T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 
+t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t2) (lift h2 d2 
+t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) 
+(lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) 
+(lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 t2))) (THead (Bind b) x2 x3) 
+(eq_ind_r T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (\lambda 
+(t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) t2)) 
+(refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3))) (lift h1 
+d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 d1)) (eq_ind_r T (THead 
+(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (\lambda (t2: T).(eq T (THead 
+(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) t2)) (refl_equal T (THead (Bind 
+b) (lift h2 d2 x2) (lift h2 (S d2) x3))) (lift h2 d2 (THead (Bind b) x2 x3)) 
+(lift_bind b x2 x3 h2 d2))) t0 H13) x1 H12)))) (H0 x1 h1 h2 (S d1) (S d2) 
+(le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x 
+H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S d1) t0) x h2 (plus d2 
+h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead (Flat f) t 
+t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead 
+(Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 
+(THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t t0 h1 d1)) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 
+h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0) (lift h2 
+(plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda 
+(t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda 
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T 
+(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0 
+x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) 
+(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T 
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1 
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) 
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T 
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3: 
+T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1 
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0) 
+(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2))) 
+(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T 
+(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T 
+(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3: 
+T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2 
+d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: 
+T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: 
+T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T 
+(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat 
+f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T 
+(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda 
+(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) 
+(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1 
+d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1 
+x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift 
+h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1)) 
+(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2: 
+T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T 
+(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f) 
+x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1 
+H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f 
+(lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2))))))))))))) 
+t1).
+
+lemma lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d: 
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts: 
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h 
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t: 
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts 
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_: 
+nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda 
+(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d: 
+nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t) 
+(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match 
+ee with [TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons 
+(lift h d t) (lifts h d t0)) H0) in (False_ind (eq TList TNil (TCons t t0)) 
+H1)))))))) ts)) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall 
+(ts: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t0) 
+(lifts h d ts)) \to (eq TList t0 ts))))))).(\lambda (ts: TList).(TList_ind 
+(\lambda (t1: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h 
+d (TCons t t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts 
+h d t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d 
+t0)) (\lambda (ee: TList).(match ee with [TNil \Rightarrow False | (TCons _ 
+_) \Rightarrow True])) I TNil H0) in (False_ind (eq TList (TCons t t0) TNil) 
+H1))))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: ((\forall (h: 
+nat).(\forall (d: nat).((eq TList (TCons (lift h d t) (lifts h d t0)) (lifts 
+h d t2)) \to (eq TList (TCons t t0) t2)))))).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H1: (eq TList (TCons (lift h d t) (lifts h d t0)) (TCons (lift 
+h d t1) (lifts h d t2)))).(let H2 \def (f_equal TList T (\lambda (e: 
+TList).(match e with [TNil \Rightarrow (lref_map (\lambda (x: nat).(plus x 
+h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) 
+(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList 
+TList (\lambda (e: TList).(match e with [TNil \Rightarrow (lifts h d t0) | 
+(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons 
+(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h 
+d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2))) 
+(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1 
+(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift/props.ma
new file mode 100644 (file)
index 0000000..7df8181
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift/props.ma
@@ -0,0 +1,323 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/defs.ma".
+
+include "basic_1A/s/props.ma".
+
+include "basic_1A/T/fwd.ma".
+
+lemma lift_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
+n)) (TSort n))))
+\def
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort 
+n)))).
+
+lemma lift_lref_lt:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T 
+(lift h d (TLRef n)) (TLRef n)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n 
+d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true 
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T 
+(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
+
+lemma lift_lref_ge:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
+(lift h d (TLRef n)) (TLRef (plus n h))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d 
+n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true 
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) 
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false 
+(le_bge d n H)))))).
+
+lemma lift_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) 
+t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
+
+lemma lift_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) 
+(lift h (S d) t)))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
+
+lemma lift_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) 
+(lift h d t)))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
+
+lemma thead_x_lift_y_y:
+ \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall 
+(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: 
+T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) 
+\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) 
+(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
+(TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H) in (False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TLRef n))) 
+(TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
+(TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H) in (False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: 
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
+h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: 
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
+h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 
+t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (lift h d (THead 
+k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead 
+_ t2 _) \Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 
+t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d 
+t0) (lref_map (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) 
+\Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d t0) (lref_map 
+(\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) \Rightarrow 
+t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in 
+(\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k 
+(\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall (d0: nat).((eq 
+T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: Prop).P0)))))) H0 k0 
+H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: 
+T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head 
+k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) H2)))))))))))) 
+t)).
+
+lemma lift_r:
+ \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) 
+t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda 
+(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) 
+(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef 
+n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) 
+(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T 
+t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) 
+(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) 
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) 
+t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) 
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) 
+(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) 
+(lift O (s k d) t1)) (THead k t0 t1) (f_equal3 K T T T THead k k (lift O d 
+t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))))) (lift O 
+d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
+
+lemma lift_lref_gt:
+ \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef 
+(pred n))) (TLRef n))))
+\def
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef 
+(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus 
+(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n 
+(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S 
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) 
+(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d 
+(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) 
+(S_pred n d H))))))).
+
+lemma lift_tle:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(tle t (lift h d t))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
+nat).(le (tweight t0) (tweight (lift h d t0)))))) (\lambda (_: nat).(\lambda 
+(_: nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (_: nat).(\lambda (_: 
+nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (k: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(le (tweight t0) 
+(tweight (lift h d t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: 
+nat).(\forall (d: nat).(le (tweight t1) (tweight (lift h d t1))))))).(\lambda 
+(h: nat).(\lambda (d: nat).(let H_y \def (H h d) in (let H_y0 \def (H0 h (s k 
+d)) in (le_n_S (plus (tweight t0) (tweight t1)) (plus (tweight (lref_map 
+(\lambda (x: nat).(plus x h)) d t0)) (tweight (lref_map (\lambda (x: 
+nat).(plus x h)) (s k d) t1))) (le_plus_plus (tweight t0) (tweight (lref_map 
+(\lambda (x: nat).(plus x h)) d t0)) (tweight t1) (tweight (lref_map (\lambda 
+(x: nat).(plus x h)) (s k d) t1)) H_y H_y0))))))))))) t).
+
+lemma lifts_tapp:
+ \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
+TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
+(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
+t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
+t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
+v)) H)))) vs)))).
+
+lemma lift_free:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
+(lift h d t)) (lift (plus k h) d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: 
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: 
+nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T 
+(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort 
+n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
+(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) 
+(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k 
+h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) 
+(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: 
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d 
+h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef 
+n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T 
+(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef 
+n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
+(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
+(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus 
+k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 
+H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d 
+n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift 
+(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda 
+(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n 
+(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal 
+nat T TLRef (plus (plus n h) k) (plus n (plus k h)) 
+(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) 
+(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 
+(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda 
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
+(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda 
+(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e 
+(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda 
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d 
+h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k 
+d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 
+t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift 
+h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 
+t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) 
+t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k 
+e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h 
+d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift 
+(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) 
+(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le 
+k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift 
+(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e 
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h 
+d))))))))))))) t).
+
+lemma lift_free_sym:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
+(lift h d t)) (lift (plus h k) d t))))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
+nat).(\lambda (e: nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d 
+e)).(eq_ind_r nat (plus k h) (\lambda (n: nat).(eq T (lift k e (lift h d t)) 
+(lift n d t))) (lift_free t h k d e H H0) (plus h k) (plus_sym h k)))))))).
+
+lemma lift_d:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) 
+(lift k e (lift h d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k 
+d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda 
+(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: 
+(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) 
+(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
+T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: 
+T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
+T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k 
+e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) 
+(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) 
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
+nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h 
+(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda 
+(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef 
+n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d 
+(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k 
+e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
+(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift 
+h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) 
+(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k 
+d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e 
+n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) 
+t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: 
+nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) 
+(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d 
+(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda 
+(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef 
+(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T 
+(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d 
+(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) 
+(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: 
+(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 
+(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda 
+(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef 
+(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) 
+(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) 
+(plus_permute_2_in_3 n k h)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
+(plus n h) k e (le_plus_trans e n h H0))) (lift h d (TLRef n)) (lift_lref_ge 
+n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge (plus n k) h 
+(plus d k) (le_plus_plus d n k k H1 (le_n k)))))) (plus k d) (plus_sym k d)) 
+(lift k e (TLRef n)) (lift_lref_ge n k e H0)))))))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 
+d) (lift k0 e t0)) (lift k0 e (lift h d t0)))))))))).(\lambda (t1: 
+T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e 
+t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda (k0: 
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(eq_ind_r T 
+(THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h 
+(plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) (eq_ind_r T (THead k 
+(lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
+e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d (THead k t0 t1))))) 
+(eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) (\lambda (t2: T).(eq 
+T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) 
+(lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead k (lift k0 e (lift 
+h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda (t2: T).(eq T (THead 
+k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
+e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda (n: nat).(eq T (THead 
+k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift k0 (s k e) t1))) (THead 
+k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))))) 
+(f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e 
+(lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) t1)) (lift k0 (s k 
+e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) (H0 h k0 (s k d) (s 
+k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 d)) (lift k0 e 
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h d)) 
+(lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 (s k e) t1))) (lift_head 
+k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 d))) (lift k0 e (THead k t0 
+t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift/tlt.ma
new file mode 100644 (file)
index 0000000..bc2ad0a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift/tlt.ma
@@ -0,0 +1,284 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/props.ma".
+
+include "basic_1A/tlt/props.ma".
+
+lemma lift_weight_map:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to 
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat 
+(weight_map f (lift h d t)) (weight_map f t))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0))))))) 
+(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
+O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
+O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f 
+(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat 
+(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0))) 
+(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq 
+nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda 
+(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_plus_trans d n h H0)) 
+(f n) (H n H0)) (lift h d (TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda 
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: 
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f 
+t0)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: 
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat 
+(f m) O)))) \to (eq nat (weight_map f (lift h d t1)) (weight_map f 
+t1)))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (H1: ((\forall (m: nat).((le d m) \to (eq nat (f m) 
+O))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 
+t1))) (weight_map f (THead k0 t0 t1)))) (\lambda (b: B).(eq_ind_r T (THead 
+(Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) (\lambda (t2: T).(eq nat 
+(weight_map f t2) (weight_map f (THead (Bind b) t0 t1)))) (B_ind (\lambda 
+(b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus (weight_map f (lift 
+h d t0)) (weight_map (wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) 
+t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map 
+(wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S (plus (weight_map f 
+(lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))))]) (match b0 with 
+[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S 
+(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0) 
+(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0) 
+(weight_map (wadd f O) t1)))]))) (eq_ind_r nat (weight_map f t0) (\lambda (n: 
+nat).(eq nat (S (plus n (weight_map (wadd f (S n)) (lift h (S d) t1)))) (S 
+(plus (weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1))))) 
+(eq_ind_r nat (weight_map (wadd f (S (weight_map f t0))) t1) (\lambda (n: 
+nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus (weight_map f t0) 
+(weight_map (wadd f (S (weight_map f t0))) t1))))) (refl_equal nat (S (plus 
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))) 
+(weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1)) (H0 h (S d) 
+(wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2: (le (S d) 
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
+n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x: nat).(\lambda 
+(H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda 
+(n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m H3)))) 
+(le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1)) (eq_ind_r 
+nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map 
+f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) 
+t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map 
+(wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 
+nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map 
+(wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat 
+(weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h 
+(S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat 
+(\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd 
+f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le 
+d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x 
+H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat (weight_map (wadd f O) t1) 
+(\lambda (n: nat).(eq nat (S (plus (weight_map f (lift h d t0)) n)) (S (plus 
+(weight_map f t0) (weight_map (wadd f O) t1))))) (f_equal nat nat S (plus 
+(weight_map f (lift h d t0)) (weight_map (wadd f O) t1)) (plus (weight_map f 
+t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat nat plus (weight_map f 
+(lift h d t0)) (weight_map f t0) (weight_map (wadd f O) t1) (weight_map (wadd 
+f O) t1) (H h d f H1) (refl_equal nat (weight_map (wadd f O) t1)))) 
+(weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: 
+nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S 
+n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O) (\lambda (x: 
+nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d 
+m H2)))))) b) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h 
+d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) (lift h (s 
+(Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f 
+(THead (Flat f0) t0 t1)))) (f_equal nat nat S (plus (weight_map f (lift h d 
+t0)) (weight_map f (lift h d t1))) (plus (weight_map f t0) (weight_map f t1)) 
+(f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f t0) 
+(weight_map f (lift h d t1)) (weight_map f t1) (H h d f H1) (H0 h d f H1))) 
+(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
+k)))))))))) t).
+
+lemma lift_weight:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d 
+t)) (weight t))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d 
+(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat 
+O)))))).
+
+lemma lift_weight_add:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: 
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to 
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat 
+(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: 
+nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat 
+(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) 
+\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d 
+t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: 
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) 
+w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f 
+m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n)))))))))))) 
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m 
+d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: 
+((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d 
+(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d 
+(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) 
+(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef 
+n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d 
+H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: 
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) 
+(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f 
+(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda 
+(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f 
+(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) 
+(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift 
+h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda 
+(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to 
+(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d 
+m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0)) 
+(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0: 
+((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f 
+m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g 
+(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift 
+(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m: 
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d) 
+w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f 
+m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0 
+t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b: 
+B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1)) 
+(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead 
+(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h) 
+(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b) 
+(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind 
+(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus 
+(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d 
+t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h 
+d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S 
+(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) 
+t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h) 
+d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h) 
+(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0)) 
+(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) 
+t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map 
+(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift 
+(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map 
+f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S 
+(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S 
+(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2 
+H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S 
+(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S 
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) 
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d 
+t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m 
+O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift 
+(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat 
+nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq 
+nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g 
+H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S 
+m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat 
+m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g 
+(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda 
+(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r 
+nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d 
+t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) 
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
+n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: 
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) 
+n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus 
+(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) 
+t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g 
+(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map 
+(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) 
+(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) 
+(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) 
+(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat 
+O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m 
+H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda 
+(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) 
+(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: 
+nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) 
+H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) 
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d 
+n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S 
+x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g 
+n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat 
+S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) 
+t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S 
+h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) 
+(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) 
+(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S 
+d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S 
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) 
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda 
+(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) 
+(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: 
+nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda 
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O 
+m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda 
+(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) 
+(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: 
+nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S 
+n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: 
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S 
+x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) 
+(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head 
+(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind 
+b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) 
+(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) 
+(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead 
+(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: 
+T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) 
+d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d 
+t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) 
+(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f 
+(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) 
+(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) 
+(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) 
+(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) 
+k))))))))))))) t)).
+
+lemma lift_weight_add_O:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to 
+nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) 
+O t))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to 
+nat))).(lift_weight_add (minus (wadd f w O) O) t h O f (wadd f w) (\lambda 
+(m: nat).(\lambda (H: (lt m O)).(lt_x_O m H (eq nat (wadd f w m) (f m))))) 
+(minus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal 
+nat (f m)))))))).
+
+lemma lift_tlt_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(tlt t (THead k u (lift h d t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight 
+(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t) 
+(lift_weight t h d)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift1/defs.ma
new file mode 100644 (file)
index 0000000..c65acca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift1/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/defs.ma".
+
+rec definition trans (hds: PList) on hds: nat \to nat \def \lambda (i: 
+nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let 
+j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false 
+\Rightarrow (plus j h)]))]).
+
+rec definition lift1 (hds: PList) on hds: T \to T \def \lambda (t: T).(match 
+hds with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 
+hds0 t))]).
+
+rec definition lifts1 (hds: PList) (ts: TList) on ts: TList \def match ts 
+with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) 
+(lifts1 hds ts0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift1/drop1.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift1/drop1.ma
new file mode 100644 (file)
index 0000000..d15754d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift1/drop1.ma
@@ -0,0 +1,127 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/props.ma".
+
+include "basic_1A/drop1/defs.ma".
+
+lemma lift1_lift1:
+ \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
+(lift1 is2 t)) (lift1 (papp is1 is2) t))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
+PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) 
+t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
+((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 
+(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat 
+T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal 
+nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
+
+lemma lift1_xhg:
+ \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
+(lift (S O) O (lift1 hds t))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T 
+(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: 
+T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) 
+(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T 
+(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S 
+O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: 
+nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d 
+(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda 
+(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift 
+(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 
+p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S 
+d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
+
+lemma lifts1_xhg:
+ \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts 
+(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+\def
+ \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq 
+TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) 
+(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq 
+TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds 
+t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList 
+(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 
+hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O 
+(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 
+hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds 
+t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O 
+(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) 
+(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
+
+lemma lift1_free:
+ \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
+(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: 
+nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O 
+(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T 
+(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
+PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 
+(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) 
+t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 
+i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift 
+(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | 
+false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 
+i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans 
+hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt 
+(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) 
+O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans 
+hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with 
+[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
+false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind 
+(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d 
+(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with 
+[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
+h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) 
+(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S 
+(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h 
+n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
+(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
+(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 
+i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
+(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans 
+hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S 
+(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) 
+(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans 
+hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S 
+(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d 
+(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: 
+(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans 
+hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S 
+(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus 
+h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) 
+t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) 
+(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
+(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans 
+hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 
+(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) 
+(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S 
+(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 
+i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) 
+(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda 
+(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d 
+(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) 
+(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans 
+hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) 
+hds).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/lift1/props.ma
new file mode 100644 (file)
index 0000000..8ff7f4d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/lift1/props.ma
@@ -0,0 +1,140 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift1/defs.ma".
+
+include "basic_1A/lift/props.ma".
+
+lemma lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
+\def
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T 
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: 
+nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p 
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
+
+lemma lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
+(trans hds i))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T 
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T 
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p 
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq 
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow 
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T 
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false 
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
+
+lemma lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss 
+hds) t))))))
+\def
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) 
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal 
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda 
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) 
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p 
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n 
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) 
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) 
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u 
+t)) (H u t)))))))) hds)).
+
+lemma lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds 
+t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) 
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T 
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: 
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) 
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) 
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p 
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift 
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) 
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) 
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+lemma lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq 
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: 
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) 
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d 
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d 
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p 
+h d) t) H))))) hds)))).
+
+lemma lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: 
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds 
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: 
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds 
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds 
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) 
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads 
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) 
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) 
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat 
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) 
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 
+(THeads (Flat f) t1 t)))))) ts)))).
+
+lemma lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) 
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: 
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq 
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 
+PNil t0) H)))) ts).
+
+lemma lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: 
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: 
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) 
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda 
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d 
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: 
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d 
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) 
+H)))) ts)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/llt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/llt/defs.ma
new file mode 100644 (file)
index 0000000..819ba1b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/llt/defs.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/A/defs.ma".
+
+rec definition lweight (a: A) on a: nat \def match a with [(ASort _ _) 
+\Rightarrow O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight 
+a2)))].
+
+definition llt:
+ A \to (A \to Prop)
+\def
+ \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/llt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/llt/fwd.ma
new file mode 100644 (file)
index 0000000..65e029b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/llt/fwd.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/llt/defs.ma".
+
+fact llt_wf__q_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to 
+Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 
+a))))) P n))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: 
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to 
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) 
+n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight 
+a)))))).
+
+lemma llt_wf_ind:
+ \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 
+a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: 
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to 
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) 
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind 
+(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: 
+A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
+\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat 
+(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
+(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P 
+a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt 
+(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight 
+a1))))))))))))) a)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/llt/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/llt/props.ma
new file mode 100644 (file)
index 0000000..4e7b48e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/llt/props.ma
@@ -0,0 +1,65 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/llt/defs.ma".
+
+include "basic_1A/leq/fwd.ma".
+
+lemma lweight_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat 
+(lweight a1) (lweight a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight 
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: 
+nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g 
+(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: 
+A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight 
+a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda 
+(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight 
+a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus 
+(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 
+H)))).
+
+lemma llt_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall 
+(a3: A).((llt a1 a3) \to (llt a2 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 
+\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 
+(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
+
+theorem llt_trans:
+ \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 
+a3) \to (llt a1 a3)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight 
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans 
+(lweight a1) (lweight a2) (lweight a3) H H0))))).
+
+lemma llt_head_sx:
+ \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) 
+(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
+
+lemma llt_head_dx:
+ \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) 
+(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/next_plus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/next_plus/defs.ma
new file mode 100644 (file)
index 0000000..ee23d9f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/next_plus/defs.ma
@@ -0,0 +1,21 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/G/defs.ma".
+
+rec definition next_plus (g: G) (n: nat) (i: nat) on i: nat \def match i with 
+[O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/next_plus/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/next_plus/props.ma
new file mode 100644 (file)
index 0000000..52d0785
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/next_plus/props.ma
@@ -0,0 +1,59 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/next_plus/defs.ma".
+
+lemma next_plus_assoc:
+ \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq 
+nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h1: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n n0) h2) 
+(next_plus g n (plus n0 h2))))) (\lambda (h2: nat).(refl_equal nat (next_plus 
+g n h2))) (\lambda (n0: nat).(\lambda (_: ((\forall (h2: nat).(eq nat 
+(next_plus g (next_plus g n n0) h2) (next_plus g n (plus n0 h2)))))).(\lambda 
+(h2: nat).(nat_ind (\lambda (n1: nat).(eq nat (next_plus g (next g (next_plus 
+g n n0)) n1) (next g (next_plus g n (plus n0 n1))))) (eq_ind nat n0 (\lambda 
+(n1: nat).(eq nat (next g (next_plus g n n0)) (next g (next_plus g n n1)))) 
+(refl_equal nat (next g (next_plus g n n0))) (plus n0 O) (plus_n_O n0)) 
+(\lambda (n1: nat).(\lambda (H0: (eq nat (next_plus g (next g (next_plus g n 
+n0)) n1) (next g (next_plus g n (plus n0 n1))))).(eq_ind nat (S (plus n0 n1)) 
+(\lambda (n2: nat).(eq nat (next g (next_plus g (next g (next_plus g n n0)) 
+n1)) (next g (next_plus g n n2)))) (f_equal nat nat (next g) (next_plus g 
+(next g (next_plus g n n0)) n1) (next g (next_plus g n (plus n0 n1))) H0) 
+(plus n0 (S n1)) (plus_n_Sm n0 n1)))) h2)))) h1))).
+
+lemma next_plus_next:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g 
+(next g n) h) (next g (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(eq_ind_r nat (next_plus 
+g n (plus (S O) h)) (\lambda (n0: nat).(eq nat n0 (next g (next_plus g n 
+h)))) (refl_equal nat (next g (next_plus g n h))) (next_plus g (next_plus g n 
+(S O)) h) (next_plus_assoc g n (S O) h)))).
+
+lemma next_plus_lt:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next 
+g n) h))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: 
+nat).(lt n0 (next_plus g (next g n0) n)))) (\lambda (n: nat).(next_lt g n)) 
+(\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(lt n0 (next_plus g (next 
+g n0) n))))).(\lambda (n0: nat).(eq_ind nat (next_plus g (next g (next g n0)) 
+n) (\lambda (n1: nat).(lt n0 n1)) (lt_trans n0 (next g n0) (next_plus g (next 
+g (next g n0)) n) (next_lt g n0) (H (next g n0))) (next g (next_plus g (next 
+g n0) n)) (next_plus_next g (next g n0) n))))) h)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/arity.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/arity.ma
new file mode 100644 (file)
index 0000000..dd9b38e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/arity.ma
@@ -0,0 +1,486 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/fwd.ma".
+
+include "basic_1A/arity/subst0.ma".
+
+lemma arity_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
+A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda 
+(n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort 
+n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort 
+n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n 
+(refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: 
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: 
+nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef 
+i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads 
+(Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 
+ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
+i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u 
+(THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef 
+i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda 
+(w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T 
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda 
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
+i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T 
+(TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda 
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef 
+i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda 
+(H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 
+(Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 
+(Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind 
+b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) 
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 
+(CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead 
+c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u 
+t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0 
+(Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws 
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda 
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq 
+B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 
+a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 
+H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) 
+u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 
+w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u 
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g 
+(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u 
+t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False with []) in 
+H9)))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 
+(Bind Void) u) t0 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let 
+H9 \def (arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O 
+(getl_refl Void c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O 
+v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind 
+Void) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u 
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (x: T).(\lambda (H10: (eq T t0 (lift (S O) O x))).(let H11 
+\def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Bind Void) u t1))) H8 
+(lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t1) 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T (THead (Bind Void) u t1) (TSort n)))) (ex3_2 TList 
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u t1) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) (nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Bind Void) u (lift (S O) O x)) (TSort n)))) (ex3_2 TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u (lift (S O) O x)) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) t0 H10)))) H9))))) b H0 H3 H5))))))))))))) (\lambda (c0: C).(\lambda 
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda 
+(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda 
+(a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: 
+(((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w) 
+u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind Abst) u) 
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind Abst) u) 
+(TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u t0))).(let H5 
+\def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 (CHead c0 (Bind 
+Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
+(Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort 
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead 
+(Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 
+(CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind 
+Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind 
+Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda 
+(H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def 
+(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList 
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def 
+(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9: 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq 
+T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
+t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq 
+T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: 
+(eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: 
+(nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: 
+T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in 
+(let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 
+(THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) 
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+(THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind 
+Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind 
+Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat 
+Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))))))) 
+H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind 
+nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: 
+nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda 
+(t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def 
+(eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x) 
+H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def 
+(leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12)) 
+in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq 
+g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead 
+(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda 
+(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda 
+(n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
+x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0 
+x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee with 
+[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 
+x1) H16) in (False_ind (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+(THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
+(TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) H17))))))) H13))) t0 H10))))) 
+H9)) (\lambda (H9: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: 
+TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0 
+(TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef 
+x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) 
+u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T 
+t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0 
+(TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda 
+(t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
+Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u 
+t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead 
+(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u 
+(THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads 
+(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda 
+(ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u 
+(THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11) 
+H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda 
+(u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda 
+(_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda 
+(_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 
+u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
+Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u 
+t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0 
+t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind 
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: 
+nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda 
+(u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) 
+u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind 
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind 
+Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind 
+Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w 
+u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead 
+(Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) 
+u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) 
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind 
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8)) 
+t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T 
+(TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: 
+T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) 
+(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: 
+T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort 
+x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda 
+(_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x 
+(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) 
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda 
+(x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0 
+(TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef 
+x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w 
+u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads 
+(Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) 
+x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws 
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda 
+(_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef 
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef 
+i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0 
+H6)))))) H5)) H4))))))))))) c t a H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/dec.ma
new file mode 100644 (file)
index 0000000..6f100f9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/dec.ma
@@ -0,0 +1,193 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/pr2/clen.ma".
+
+include "basic_1A/pr0/dec.ma".
+
+include "basic_1A/C/props.ma".
+
+lemma nf2_dec:
+ \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq 
+T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
+\def
+ \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall 
+(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda 
+(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in 
+(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) 
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to 
+(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 
+(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 
+H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) 
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to 
+(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: 
+T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T 
+t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) 
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x 
+H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or 
+(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 
+t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H 
+t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T 
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) 
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: 
+((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0: 
+K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 
+T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0: 
+B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 
+t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def 
+(dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: 
+T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) 
+(clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) 
+\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda 
+(x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq 
+T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) 
+(clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 
+(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
+(Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) 
+(clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def 
+H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t 
+c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind 
+Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) 
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
+t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t 
+c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 
+(CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
+(Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 
+t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) 
+x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 
+(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in 
+(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) 
+(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt 
+(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym 
+(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t 
+(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) 
+(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 
+(\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1 
+(lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda 
+(t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T 
+t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall 
+(t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T 
+(lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) 
+(clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail 
+(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda 
+(H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 
+\def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let 
+H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda 
+(_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) 
+(clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift 
+(S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) 
+t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) 
+(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda 
+(t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) 
+(\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: 
+(pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 
+t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) 
+(\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: 
+T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x 
+x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) 
+H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) 
+(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) 
+(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen 
+c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) 
+t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) 
+H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) 
+\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda 
+(t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def 
+(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) 
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
+(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
+K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: 
+K).(match ee with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow 
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) 
+H4)) H3)))))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 
+t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda 
+(t2: T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def 
+(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind 
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) 
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) 
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T 
+(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) 
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq 
+K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: 
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: 
+K).(match ee with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow 
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) 
+\Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) 
+H4)) H3)))))) b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail 
+(Flat f) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) 
+t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 
+t2)).(let H_x0 \def (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def 
+H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 
+t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: 
+(ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 
+t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: 
+T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: 
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: 
+(eq K (Flat f) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 
+(clen c0) t x0 t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match 
+ee with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind 
+Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))) k)) (\lambda 
+(H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) 
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) (or (\forall 
+(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: 
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t 
+c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) \to (\forall (P: 
+Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall (t2: T).((pr2 
+(CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2))) 
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 x H3 k 
+t)))))) H1)) H0)))))))) c).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/defs.ma
new file mode 100644 (file)
index 0000000..de14669
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/defs.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/defs.ma".
+
+definition nf2:
+ C \to (T \to Prop)
+\def
+ \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 
+t2)))).
+
+rec definition nfs2 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/fwd.ma
new file mode 100644 (file)
index 0000000..6173e86
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/fwd.ma
@@ -0,0 +1,173 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/pr2/clen.ma".
+
+include "basic_1A/subst0/dec.ma".
+
+include "basic_1A/T/props.ma".
+
+lemma nf2_gen_lref:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2 
+c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P: 
+Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0 
+(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef 
+i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
+
+lemma nf2_gen_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u 
+t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t) 
+t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2: 
+T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2: 
+T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ t0 _) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) t2 t) (H 
+(THead (Bind Abst) t2 t) (pr2_head_1 c u t2 H0 (Bind Abst) t))) in (let H2 
+\def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u t0)) H0 u H1) in (eq_ind T u 
+(\lambda (t0: T).(eq T u t0)) (refl_equal T u) t2 H1))))) (\lambda (t2: 
+T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t t2)).(let H1 \def (f_equal T 
+T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
+\Rightarrow t | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) u t) 
+(THead (Bind Abst) u t2) (H (THead (Bind Abst) u t2) (let H_y \def 
+(pr2_gen_cbind Abst c u t t2 H0) in H_y))) in (let H2 \def (eq_ind_r T t2 
+(\lambda (t0: T).(pr2 (CHead c (Bind Abst) u) t t0)) H0 t H1) in (eq_ind T t 
+(\lambda (t0: T).(eq T t t0)) (refl_equal T t) t2 H1))))))))).
+
+lemma nf2_gen_cast:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u 
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead 
+(Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t 
+(pr2_free c (THead (Flat Cast) u t) t (pr0_tau t t (pr0_refl t) u))) P))))).
+
+lemma nf2_gen_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c 
+(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) 
+\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: 
+Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) 
+(H (THead (Bind Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind 
+Abst) v t)) (THead (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t 
+(pr0_refl t))))) in (False_ind P H0))))))).
+
+lemma nf2_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c 
+(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f) 
+u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall 
+(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c 
+u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat f) u t) (THead (Flat f) t2 t) (H (THead (Flat f) t2 t) 
+(pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) (\lambda (t2: T).(\lambda (H0: 
+(pr2 c t t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) 
+(pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
+
+fact nf2_gen__nf2_gen_aux:
+ \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T 
+(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: 
+T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
+(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort 
+n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
+d (TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H) in (False_ind P H0))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TLRef n))) (TLRef 
+n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
+d (TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H) in (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: 
+((\forall (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) 
+t) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to 
+(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: 
+(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
+t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | 
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u (lift (S O) d (THead k t 
+t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) 
+\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
+t0) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t) 
+(lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (TLRef _) 
+\Rightarrow (THead k (lref_map (\lambda (x0: nat).(plus x0 (S O))) d t) 
+(lref_map (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) 
+\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
+t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) 
+H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t 
+t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift 
+(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 
+P)))))) H3)) H2))))))))))) x)).
+
+lemma nf2_gen_abbr:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u 
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) 
+t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x 
+in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t 
+(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift 
+(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O 
+x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O 
+x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (H (THead 
+(Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind Abbr) u t) (THead 
+(Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) t t (pr0_refl t) 
+(lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda (t0: T).(subst0 
+O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in (subst0_refl u (lift (S 
+O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) O x))).(let H3 \def 
+(eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c (THead (Bind Abbr) u 
+t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H (lift (S O) O x) H2) in 
+(nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c (THead (Bind Abbr) u (lift 
+(S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) u))) P))) H1))) 
+H0))))))).
+
+lemma nf2_gen_void:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u 
+(lift (S O) O t))) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind 
+Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux 
+Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t 
+(pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/iso.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/iso.ma
new file mode 100644 (file)
index 0000000..84c0293
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/iso.ma
@@ -0,0 +1,125 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/pr3.ma".
+
+include "basic_1A/iso/props.ma".
+
+lemma nf2_iso_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
+TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso 
+(THeads (Flat Appl) vs (TLRef i)) u))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads 
+(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u)))) 
+(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def 
+(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda 
+(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda 
+(t: T).(iso (TLRef i) t)) (iso_refl (TLRef i)) u H_y))))) (\lambda (t: 
+T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) 
+u))))).(\lambda (u: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (TLRef i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat 
+Appl) t0 (TLRef i)) u H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T u (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
+t0 (TLRef i)) t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
+i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u 
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat 
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0 
+x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 
+(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head t x0 
+(THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) u H4)))))) H3)) (\lambda 
+(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1 
+t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda 
+(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 
+x1))).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in 
+(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: 
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
+u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift 
+(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1 
+x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0 
+(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 
+H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
+u))))))))))))))) H3)) H2))))))) vs)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/lift1.ma
new file mode 100644 (file)
index 0000000..99d2654
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/lift1.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/props.ma".
+
+include "basic_1A/drop1/fwd.ma".
+
+lemma nf2_lift1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 
+hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
+t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c 
+e)).(\lambda (H0: (nf2 e t)).(let H_y \def (drop1_gen_pnil c e H) in 
+(eq_ind_r C e (\lambda (c0: C).(nf2 c0 t)) H0 c H_y)))))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: 
+C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p 
+t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) 
+c e)).(\lambda (H1: (nf2 e t)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) 
+in (let H2 \def H_x in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda 
+(c2: C).(drop1 p c2 e)) (nf2 c (lift n n0 (lift1 p t))) (\lambda (x: 
+C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x e)).(nf2_lift x 
+(lift1 p t) (H x t H4 H1) c n n0 H3)))) H2))))))))))) hds)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/pr3.ma
new file mode 100644 (file)
index 0000000..962d57b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/pr3.ma
@@ -0,0 +1,50 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/pr3/pr3.ma".
+
+lemma nf2_pr3_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
+t1) \to (eq T t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
+t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t 
+(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) 
+\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def 
+(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def 
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T 
+t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
+
+theorem nf2_pr3_confluence:
+ \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) 
+\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: 
+T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t 
+t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) 
+(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: 
+(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 
+x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 
+H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) 
+in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 
+(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: 
+T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 
+t1 H1))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/nf2/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/nf2/props.ma
new file mode 100644 (file)
index 0000000..61c1e8e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/nf2/props.ma
@@ -0,0 +1,309 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/nf2/defs.ma".
+
+include "basic_1A/pr2/fwd.ma".
+
+lemma nf2_sort:
+ \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
+\def
+ \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort 
+n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal 
+T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
+
+lemma nf2_csort_lref:
+ \forall (n: nat).(\forall (i: nat).(nf2 (CSort n) (TLRef i)))
+\def
+ \lambda (n: nat).(\lambda (i: nat).(\lambda (t2: T).(\lambda (H: (pr2 (CSort 
+n) (TLRef i) t2)).(let H0 \def (pr2_gen_lref (CSort n) t2 i H) in (or_ind (eq 
+T t2 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort n) 
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
+i) O u))))) (eq T (TLRef i) t2) (\lambda (H1: (eq T t2 (TLRef i))).(eq_ind_r 
+T (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
+H1)) (\lambda (H1: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
+n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
+(S i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i (CSort 
+n) (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift 
+(S i) O u)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H2: (getl i (CSort n) (CHead x0 (Bind Abbr) x1))).(\lambda (H3: (eq T t2 
+(lift (S i) O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T 
+(TLRef i) t)) (getl_gen_sort n i (CHead x0 (Bind Abbr) x1) H2 (eq T (TLRef i) 
+(lift (S i) O x1))) t2 H3))))) H1)) H0))))).
+
+theorem nf2_abst:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: 
+T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind 
+Abst) u t))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
+\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda 
+(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t 
+t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t) 
+t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead 
+(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2 
+(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5: 
+((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t 
+x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead 
+(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t 
+x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3)))))) 
+H2)))))))))).
+
+theorem nf2_abst_shift:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (t: T).((nf2 (CHead c 
+(Bind Abst) u) t) \to (nf2 c (THead (Bind Abst) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2) 
+\to (eq T u t2))))).(\lambda (t: T).(\lambda (H0: ((\forall (t2: T).((pr2 
+(CHead c (Bind Abst) u) t t2) \to (eq T t t2))))).(\lambda (t2: T).(\lambda 
+(H1: (pr2 c (THead (Bind Abst) u t) t2)).(let H2 \def (pr2_gen_abst c u t t2 
+H1) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+u0) t t3))))) (eq T (THead (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H3: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 
+c u x0)).(\lambda (H5: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind 
+b) u0) t x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T 
+(THead (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) 
+u x0 t x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 Abst u))) t2 
+H3)))))) H2)))))))).
+
+lemma nfs2_tapp:
+ \forall (c: C).(\forall (t: T).(\forall (ts: TList).((nfs2 c (TApp ts t)) 
+\to (land (nfs2 c ts) (nf2 c t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0: 
+TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H: 
+(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True 
+(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I 
+H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c 
+(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c 
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c 
+(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2: 
+(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let 
+H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c 
+t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj 
+(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5) 
+H6))) H4))))) H1)))))) ts))).
+
+lemma nf2_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: 
+TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(vs: TList).(TList_ind (\lambda (t: TList).((nfs2 c t) \to (nf2 c (THeads 
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda 
+(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0 
+(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in 
+(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat 
+Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c 
+t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def 
+(pr2_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) t2 H5) in (or3_ind (ex3_2 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t3)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 
+t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (H7: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THeads (Flat Appl) t0 (TLRef i)) t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THeads (Flat Appl) t0 (TLRef i)) t3))) (eq T (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H8: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H9: (pr2 c t 
+x0)).(\lambda (H10: (pr2 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (TLRef i))) t1)) (let H11 \def (eq_ind_r T x1 (\lambda (t1: 
+T).(pr2 c (THeads (Flat Appl) t0 (TLRef i)) t1)) H10 (THeads (Flat Appl) t0 
+(TLRef i)) (H_y x1 H10)) in (eq_ind T (THeads (Flat Appl) t0 (TLRef i)) 
+(\lambda (t1: T).(eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef 
+i))) (THead (Flat Appl) x0 t1))) (let H12 \def (eq_ind_r T x0 (\lambda (t1: 
+T).(pr2 c t t1)) H9 t (H3 x0 H9)) in (eq_ind T t (\lambda (t1: T).(eq T 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) (THead (Flat Appl) t1 
+(THeads (Flat Appl) t0 (TLRef i))))) (refl_equal T (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i)))) x0 (H3 x0 H9))) x1 (H_y x1 H10))) t2 
+H8)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t3))))))) (eq T (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) 
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H8: (eq T (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) 
+x0 x1))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 
+c t x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) 
+u) x1 x3))))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t1: T).(eq T 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind 
+(\lambda (t1: TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T 
+(THeads (Flat Appl) t1 (TLRef i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead 
+(Flat Appl) t (THeads (Flat Appl) t1 (TLRef i))) (THead (Bind Abbr) x2 
+x3))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil (TLRef i)))).(\lambda 
+(H13: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead (Bind Abst) x0 
+x1))).(let H14 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Bind Abst) x0 x1) H13) in (False_ind (eq T 
+(THead (Flat Appl) t (THeads (Flat Appl) TNil (TLRef i))) (THead (Bind Abbr) 
+x2 x3)) H14)))) (\lambda (t1: T).(\lambda (t3: TList).(\lambda (_: (((nf2 c 
+(THeads (Flat Appl) t3 (TLRef i))) \to ((eq T (THeads (Flat Appl) t3 (TLRef 
+i)) (THead (Bind Abst) x0 x1)) \to (eq T (THead (Flat Appl) t (THeads (Flat 
+Appl) t3 (TLRef i))) (THead (Bind Abbr) x2 x3)))))).(\lambda (_: (nf2 c 
+(THeads (Flat Appl) (TCons t1 t3) (TLRef i)))).(\lambda (H13: (eq T (THeads 
+(Flat Appl) (TCons t1 t3) (TLRef i)) (THead (Bind Abst) x0 x1))).(let H14 
+\def (eq_ind T (THead (Flat Appl) t1 (THeads (Flat Appl) t3 (TLRef i))) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x0 
+x1) H13) in (False_ind (eq T (THead (Flat Appl) t (THeads (Flat Appl) (TCons 
+t1 t3) (TLRef i))) (THead (Bind Abbr) x2 x3)) H14))))))) t0 H_y H8) t2 
+H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) 
+t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2) (\lambda (x0: 
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H9: (eq T 
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (H10: 
+(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)))).(\lambda (_: (pr2 c t x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)) (\lambda (t1: T).(eq T (THead (Flat Appl) 
+t (THeads (Flat Appl) t0 (TLRef i))) t1)) (TList_ind (\lambda (t1: 
+TList).((nf2 c (THeads (Flat Appl) t1 (TLRef i))) \to ((eq T (THeads (Flat 
+Appl) t1 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead (Flat Appl) t 
+(THeads (Flat Appl) t1 (TLRef i))) (THead (Bind x0) x5 (THead (Flat Appl) 
+(lift (S O) O x4) x3)))))) (\lambda (_: (nf2 c (THeads (Flat Appl) TNil 
+(TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) TNil (TLRef i)) (THead 
+(Bind x0) x1 x2))).(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ 
+_ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T 
+(THead (Flat Appl) t (THeads (Flat Appl) TNil (TLRef i))) (THead (Bind x0) x5 
+(THead (Flat Appl) (lift (S O) O x4) x3))) H16)))) (\lambda (t1: T).(\lambda 
+(t3: TList).(\lambda (_: (((nf2 c (THeads (Flat Appl) t3 (TLRef i))) \to ((eq 
+T (THeads (Flat Appl) t3 (TLRef i)) (THead (Bind x0) x1 x2)) \to (eq T (THead 
+(Flat Appl) t (THeads (Flat Appl) t3 (TLRef i))) (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3))))))).(\lambda (_: (nf2 c (THeads (Flat 
+Appl) (TCons t1 t3) (TLRef i)))).(\lambda (H15: (eq T (THeads (Flat Appl) 
+(TCons t1 t3) (TLRef i)) (THead (Bind x0) x1 x2))).(let H16 \def (eq_ind T 
+(THead (Flat Appl) t1 (THeads (Flat Appl) t3 (TLRef i))) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind x0) x1 x2) H15) in (False_ind (eq T 
+(THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t3) (TLRef i))) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H16))))))) t0 H_y H9) 
+t2 H10))))))))))))) H7)) H6))))))) H2)))))) vs)))).
+
+theorem nf2_appl_lref:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c 
+(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (nf2 c u)).(\lambda (i: 
+nat).(\lambda (H0: (nf2 c (TLRef i))).(let H_y \def (nf2_appls_lref c i H0 
+(TCons u TNil)) in (H_y (conj (nf2 c u) True H I))))))).
+
+lemma nf2_lref_abst:
+ \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c 
+(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2 
+(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d 
+(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O 
+u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T 
+(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2 
+H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
+(S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c 
+(CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift 
+(S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) 
+O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) 
+(let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c 
+c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H 
+(CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) 
+u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k 
+_) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e 
+(Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) H3)) in (False_ind (eq T (TLRef 
+i) (lift (S i) O x1)) H6))) t2 H4))))) H2)) H1)))))))).
+
+lemma nf2_lift:
+ \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2) 
+\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i: 
+nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c 
+(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind 
+T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3)) 
+(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i 
+x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq 
+T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x 
+(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq 
+T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3)))) 
+H2)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc1/defs.ma
new file mode 100644 (file)
index 0000000..264293d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc1/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr1/defs.ma".
+
+definition pc1:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda 
+(t: T).(pr1 t2 t)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc1/props.ma
new file mode 100644 (file)
index 0000000..57a6050
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc1/props.ma
@@ -0,0 +1,116 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc1/defs.ma".
+
+include "basic_1A/pr1/pr1.ma".
+
+lemma pc1_pr0_r:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) 
+(pr1_refl t2)))).
+
+lemma pc1_pr0_x:
+ \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) 
+(pr1_pr0 t2 t1 H)))).
+
+lemma pc1_refl:
+ \forall (t: T).(pc1 t t)
+\def
+ \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0: 
+T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
+
+lemma pc1_pr0_u:
+ \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: 
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda 
+(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) 
+H3)))) H1)))))).
+
+lemma pc1_s:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in 
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 
+t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 
+x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 
+H1)))) H0)))).
+
+lemma pc1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall 
+(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: 
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) 
+(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda 
+(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T 
+(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 
+t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) 
+H0)))))).
+
+lemma pc1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall 
+(k: K).(pc1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: 
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) 
+(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: 
+T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda 
+(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead 
+k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
+
+theorem pc1_t:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: 
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in 
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 
+t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 
+x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 
+t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x 
+x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 
+(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x 
+H2))))) H4))))) H1)))))).
+
+lemma pc1_pr0_u2:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 
+t2) \to (pc1 t1 t2)))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: 
+T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
+
+theorem pc1_head:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall 
+(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 
+t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead 
+k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) 
+(pc1_head_2 t1 t2 H0 u2 k)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/dec.ma
new file mode 100644 (file)
index 0000000..55ed35f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/dec.ma
@@ -0,0 +1,146 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/arity_props.ma".
+
+include "basic_1A/nf2/fwd.ma".
+
+theorem pc3_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c 
+u1 u2) ((pc3 c u1 u2) \to False)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2 
+t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T 
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2) 
+((pc3 c u1 u2) \to False)) (\lambda (x: T).(\lambda (H2: (pr3 c u1 
+x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let H4 
+\def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c 
+u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (x0: T).(\lambda 
+(H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def (term_dec x x0) 
+in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: 
+Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to False)) (\lambda (H8: (eq T x 
+x0)).(let H9 \def (eq_ind_r T x0 (\lambda (t: T).(nf2 c t)) H6 x H8) in (let 
+H10 \def (eq_ind_r T x0 (\lambda (t: T).(pr3 c u2 t)) H5 x H8) in (or_introl 
+(pc3 c u1 u2) ((pc3 c u1 u2) \to False) (pc3_pr3_t c u1 x H2 u2 H10))))) 
+(\lambda (H8: (((eq T x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 
+u2) ((pc3 c u1 u2) \to False) (\lambda (H9: (pc3 c u1 u2)).(let H10 \def H9 
+in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) 
+False (\lambda (x1: T).(\lambda (H11: (pr3 c u1 x1)).(\lambda (H12: (pr3 c u2 
+x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5 x1 H12) in (let H13 \def H_x2 
+in (ex2_ind T (\lambda (t: T).(pr3 c x0 t)) (\lambda (t: T).(pr3 c x1 t)) 
+False (\lambda (x2: T).(\lambda (H14: (pr3 c x0 x2)).(\lambda (H15: (pr3 c x1 
+x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14 H6) in (let H16 \def 
+(eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0 H_y1) in (let H17 \def 
+(nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17 (pr3_t x1 u1 c H11 x0 
+H16)) False))))))) H13)))))) H10))))) H7)))))) H4)))))) H1)))))))))))).
+
+theorem pc3_abst_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to False))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2 
+H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T 
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T 
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to False))) (\lambda (x: T).(\lambda (H4: (pr3 c u1 x)).(\lambda (H5: (nf2 c 
+x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def H_x0 in (ex2_ind T 
+(\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T 
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to False))) (\lambda (x0: T).(\lambda (H7: (pr3 c u2 x0)).(\lambda (H8: (nf2 
+c x0)).(let H_x1 \def (abst_dec x x0) in (let H9 \def H_x1 in (or_ind (ex T 
+(\lambda (t: T).(eq T x (THead (Bind Abst) x0 t)))) (\forall (t: T).((eq T x 
+(THead (Bind Abst) x0 t)) \to (\forall (P: Prop).P))) (or (ex4_2 T T (\lambda 
+(u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: 
+T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: 
+T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c 
+v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to False))) 
+(\lambda (H10: (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 
+t))))).(ex_ind T (\lambda (t: T).(eq T x (THead (Bind Abst) x0 t))) (or 
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) 
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: 
+T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind 
+Abst) u2 u)) \to False))) (\lambda (x1: T).(\lambda (H11: (eq T x (THead 
+(Bind Abst) x0 x1))).(let H12 \def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 
+(THead (Bind Abst) x0 x1) H11) in (let H13 \def (eq_ind T x (\lambda (t: 
+T).(pr3 c u1 t)) H4 (THead (Bind Abst) x0 x1) H11) in (let H_y \def 
+(ty3_sred_pr3 c u1 (THead (Bind Abst) x0 x1) H13 g t1 H) in (or_introl (ex4_2 
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) 
+\to False)) (ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead 
+(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind 
+Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda 
+(_: T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) 
+x0 x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 
+x1 (pr3_refl (CHead c (Bind Abst) x0) x1))) H_y H7 H8))))))) H10)) (\lambda 
+(H10: ((\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P: 
+Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 
+(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead 
+(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) 
+(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 
+(THead (Bind Abst) u2 u)) \to False)) (\lambda (u: T).(\lambda (H11: (pc3 c 
+u1 (THead (Bind Abst) u2 u))).(let H12 \def H11 in (ex2_ind T (\lambda (t: 
+T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 u) t)) False 
+(\lambda (x1: T).(\lambda (H13: (pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead 
+(Bind Abst) u2 u) x1)).(ex2_ind T (\lambda (t: T).(pr3 c x1 t)) (\lambda (t: 
+T).(pr3 c x t)) False (\lambda (x2: T).(\lambda (H15: (pr3 c x1 x2)).(\lambda 
+(H16: (pr3 c x x2)).(let H_y \def (nf2_pr3_unfold c x x2 H16 H5) in (let H17 
+\def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def 
+(pr3_gen_abst c u2 u x1 H14) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T x1 (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b) u0) u t3))))) False (\lambda (x3: T).(\lambda 
+(x4: T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x3 x4))).(\lambda (H20: 
+(pr3 c u2 x3)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c 
+(Bind b) u0) u x4))))).(let H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t 
+x)) H17 (THead (Bind Abst) x3 x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 
+x H22) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead 
+c (Bind b) u0) x4 t3))))) False (\lambda (x5: T).(\lambda (x6: T).(\lambda 
+(H24: (eq T x (THead (Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 
+x5)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: 
+T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10 
+(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t: 
+T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def 
+(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) 
+x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind 
+Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in 
+(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) 
+(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K 
+(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) False)))) 
+H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4))))) 
+H12)))))) H9)))))) H6)))))) H3)))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/defs.ma
new file mode 100644 (file)
index 0000000..2267960
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/defs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/defs.ma".
+
+definition pc3:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 
+c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
+
+inductive pc3_left (c: C): T \to (T \to Prop) \def
+| pc3_left_r: \forall (t: T).(pc3_left c t t)
+| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
+| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/fsubst0.ma
new file mode 100644 (file)
index 0000000..f8f6eb3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/fsubst0.ma
@@ -0,0 +1,697 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/left.ma".
+
+include "basic_1A/fsubst0/fwd.ma".
+
+include "basic_1A/csubst0/getl.ma".
+
+lemma pc3_pr2_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1 
+t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2 
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
+t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
+T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda 
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
+\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
+u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
+T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c 
+t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4 
+w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
+t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x: 
+T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x 
+t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3 
+(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl 
+u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: 
+C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 
+(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2 
+t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e: 
+C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2 
+T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0 
+t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3 
+H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: 
+T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda 
+(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0 
+t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x 
+H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c 
+c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5)) 
+(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1)))))))))) 
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
+u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0 
+H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0 
+t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
+(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 
+(S i)) u0 t5 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: 
+C).(getl i c0 (CHead c3 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in 
+(ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S 
+(plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda 
+(H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) 
+i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
+(\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in 
+(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 
+x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead 
+e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 
+t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda 
+(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
+b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B 
+Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda 
+(t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) H10 u H14) in (let H18 \def 
+(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H11 d 
+H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
+(Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 
+t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i)) 
+u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
+_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) 
+\Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i0 u0 t2 t5)).(\lambda (c0: C).(\lambda (H5: 
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t5 t0) (\lambda (H7: (lt i i0)).(let H8 
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
+(pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u2 
+c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 
+(CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 
+t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9: (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
+(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow 
+c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
+((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: 
+(eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 
+(S i)) u0 t6 x3)) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: 
+C).(getl i c0 (CHead c3 (Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in 
+(ex2_ind T (\lambda (t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S 
+(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda 
+(H21: (subst0 i x3 t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) 
+i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
+(\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in 
+(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c 
+c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 
+(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 
+x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
+e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back 
+t3 t0 u i H2 x3 u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) 
+(\lambda (H9: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C 
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) 
+u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 
+(CHead x2 (Bind x0) x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 
+x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) 
+\Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) H11 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 
+(pr2_delta c0 x2 u i H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) 
+(\lambda (H9: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 
+e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x4))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: 
+(csubst0 (minus i0 (S i)) u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
+((let H16 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H10) in (\lambda (H17: (eq B Abbr x0)).(\lambda (H18: 
+(eq C d x1)).(let H19 \def (eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 
+(S i)) u0 t6 x4)) H12 u H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: 
+C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in 
+(ex2_ind T (\lambda (t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S 
+(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda 
+(H22: (subst0 i x4 t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) 
+i)) u0 t0 x)).(let H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
+(\lambda (n: nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in 
+(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c 
+c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 
+(pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x c0 
+x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
+e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24)))))))) (subst0_subst0_back 
+t3 t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) 
+H8))) (\lambda (H7: (le i0 i)).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 
+(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 
+t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
+(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
+H2))))))))))) c2 t4 H3)))))))))))))))) c1 t1 t H)))).
+
+lemma pc3_pr2_fsubst0_back:
+ \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t t2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t 
+t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2 
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0 
+t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: 
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0: 
+T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda 
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) 
+\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3 
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr) 
+u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4 
+(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0: 
+C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c 
+(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0)))))) 
+(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda 
+(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e 
+(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0 
+t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e 
+(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1)))))))))) 
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3 
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr) 
+u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3 
+H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3 
+(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0 
+(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c 
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr) 
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def 
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8: 
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i 
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) 
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) 
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
+(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t5: T).(subst0 (minus i0 
+(S i)) u0 t5 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c3: 
+C).(getl i c0 (CHead c3 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18 Abbr H15) in 
+(ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5: T).(subst0 (S 
+(plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda 
+(H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus (minus i0 (S i)) 
+i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
+(\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0 H6)) in 
+(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0 (pc3_pr2_x c0 
+x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead 
+e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22))))))) (subst0_subst0_back t3 
+t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13)) H12))))))))) H8)) (\lambda 
+(H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H11: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
+b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) \Rightarrow t5])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B 
+Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda 
+(t5: T).(getl i c0 (CHead x2 (Bind x0) t5))) H10 u H14) in (let H18 \def 
+(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H11 d 
+H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
+(Bind b) u))) H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 
+t2 t3 H1 t0 H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i)) 
+u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
+_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t5) 
+\Rightarrow t5])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t5: T).(subst0 (minus i0 (S i)) u0 t5 x4)) H11 u 
+H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H10 Abbr H16) in (ex2_ind T (\lambda 
+(t5: T).(subst0 i x4 t3 t5)) (\lambda (t5: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x4 
+t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H6)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta 
+c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H4 (CHead e (Bind Abbr) 
+u0) H5) t0 t0 (pr0_refl t0) x H23))))))) (subst0_subst0_back t3 t0 u i H2 x4 
+u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: 
+(le i0 i)).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c 
+c0 u0 H4 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i0 u0 t0 t5)).(\lambda (c0: C).(\lambda (H5: 
+(csubst0 i0 u0 c c0)).(\lambda (e: C).(\lambda (H6: (getl i0 c (CHead e (Bind 
+Abbr) u0))).(lt_le_e i i0 (pc3 c0 t2 t5) (\lambda (H7: (lt i i0)).(let H8 
+\def (csubst0_getl_lt i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) in 
+(or4_ind (getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: 
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) 
+(pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_u 
+c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5 (pr3_sing c0 t0 t3 
+(pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5 (pr3_pr2 c0 t0 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda (H9: (ex3_4 B C 
+T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i0 (S i)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) 
+u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda 
+(H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def (f_equal C C (\lambda 
+(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c3 _ _) \Rightarrow 
+c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in 
+((let H15 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H10) in (\lambda (H16: (eq B Abbr x0)).(\lambda (H17: 
+(eq C d x1)).(let H18 \def (eq_ind_r T x2 (\lambda (t6: T).(subst0 (minus i0 
+(S i)) u0 t6 x3)) H12 u H15) in (let H19 \def (eq_ind_r C x1 (\lambda (c3: 
+C).(getl i c0 (CHead c3 (Bind x0) x3))) H11 d H17) in (let H20 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H19 Abbr H16) in 
+(ex2_ind T (\lambda (t6: T).(subst0 i x3 t3 t6)) (\lambda (t6: T).(subst0 (S 
+(plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda 
+(H21: (subst0 i x3 t3 x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) 
+i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) 
+(\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in 
+(pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 
+t0 x (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead 
+e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 
+t0 u i H2 x3 u0 (minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda 
+(H9: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S 
+i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead 
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) 
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def 
+(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
+(CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3) H10) in ((let H14 \def (f_equal C B (\lambda (e0: C).(match e0 with 
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind 
+b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) \Rightarrow t6])) 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in (\lambda (H16: (eq B 
+Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda 
+(t6: T).(getl i c0 (CHead x2 (Bind x0) t6))) H11 u H15) in (let H19 \def 
+(eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S i)) u0 c3 x2)) H12 d 
+H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 
+(Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 (pr2_delta c0 x2 u i H20 
+t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge 
+i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) 
+t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t5) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H12: 
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: (csubst0 (minus i0 (S i)) 
+u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort 
+_) \Rightarrow d | (CHead c3 _ _) \Rightarrow c3])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H16 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t6) 
+\Rightarrow t6])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in 
+(\lambda (H17: (eq B Abbr x0)).(\lambda (H18: (eq C d x1)).(let H19 \def 
+(eq_ind_r T x3 (\lambda (t6: T).(subst0 (minus i0 (S i)) u0 t6 x4)) H12 u 
+H16) in (let H20 \def (eq_ind_r C x1 (\lambda (c3: C).(csubst0 (minus i0 (S 
+i)) u0 c3 x2)) H13 d H18) in (let H21 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl i c0 (CHead x2 (Bind b) x4))) H11 Abbr H17) in (ex2_ind T (\lambda 
+(t6: T).(subst0 i x4 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S 
+i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H22: (subst0 i x4 
+t3 x)).(\lambda (H23: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let 
+H24 \def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: 
+nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x 
+t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5 (pc3_pr2_u2 c0 t0 x 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5 (pc3_pr2_r c0 t0 t5 
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e 
+(Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 
+t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15)) H14))))))))))) H9)) H8))) 
+(\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2 (pr2_delta c0 d u i 
+(csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr) u) H0) t2 t3 H1 t0 
+H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n 
+i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 
+H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
+
+lemma pc3_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1 
+t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2 
+t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3 
+t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: 
+C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u 
+c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e 
+(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0 
+i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3 
+H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e: 
+C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0))))) 
+(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda 
+(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i 
+(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3 
+H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0: 
+(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2: 
+((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: 
+T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
+Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2 
+t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e: 
+C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5: 
+T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 
+(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1 
+t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda 
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0 
+(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2 
+c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda 
+(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6: 
+(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0 
+t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2 
+(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda 
+(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u: 
+T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c1 t0 c2 t4) \to (\forall 
+(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t4 
+t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda 
+(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) 
+\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2 
+t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1 
+t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda 
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e 
+(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0 
+t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0 
+(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: 
+(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1 
+c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr) 
+u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0 
+t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst 
+i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/fwd.ma
new file mode 100644 (file)
index 0000000..775f6e0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/fwd.ma
@@ -0,0 +1,304 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/props.ma".
+
+lemma pc3_gen_sort:
+ \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort 
+n)) \to (eq nat m n))))
+\def
+ \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c 
+(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c 
+(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x: 
+T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let 
+H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n 
+H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat 
+(\lambda (e: T).(match e with [(TSort n0) \Rightarrow n0 | (TLRef _) 
+\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H3) in 
+H4))))) H0))))).
+
+lemma pc3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall 
+(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to 
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
+t1 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 
+t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst) 
+u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c 
+u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2)))) 
+(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda 
+(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2 
+x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: 
+T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2 
+x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3))))) 
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) 
+t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead 
+(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def 
+(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead 
+(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ t _) 
+\Rightarrow t])) (THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in 
+((let H13 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x3 | (TLRef _) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) 
+(THead (Bind Abst) x2 x3) (THead (Bind Abst) x0 x1) H11) in (\lambda (H14: 
+(eq T x2 x0)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t)))) H10 x1 H13) in (let H16 
+\def (eq_ind T x2 (\lambda (t: T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 
+u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))) 
+(pc3_pr3_t c u1 x0 H16 u2 H5) (\lambda (b: B).(\lambda (u: T).(pc3_pr3_t 
+(CHead c (Bind b) u) t1 x1 (H15 b u) t2 (H6 b u))))))))) H12)))))))) 
+H7))))))) H3))))) H0))))))).
+
+lemma pc3_gen_abst_shift:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pc3 c 
+(THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (pc3 (CHead c (Bind 
+Abst) u) t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def 
+(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u) 
+(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3 
+(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2: 
+((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2 
+Abst u))) H0))))))).
+
+lemma pc3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall 
+(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d 
+c e) \to (pc3 e t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e: 
+C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t: 
+T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e 
+t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3: 
+(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in 
+(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e 
+t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d 
+x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e 
+H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: 
+T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift 
+h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: 
+T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1 
+(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e 
+t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
+
+lemma pc3_gen_not_abst:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) 
+u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S 
+O) O (THead (Bind Abst) u2 t2))))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: 
+T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead 
+c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda 
+(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr) 
+u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind 
+Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind 
+Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1) 
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
+(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead 
+c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5: 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))))).(ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda 
+(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1 
+x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def 
+(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
+c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 
+x2)).(\lambda (_: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind 
+Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T 
+(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow (match b0 with [Abbr \Rightarrow False | 
+Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind 
+Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) H14)))))))) H9))))))) 
+H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))).(let 
+H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c 
+(Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 
+x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t2 x1))))).(let H10 \def (eq_ind T x 
+(\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O t))) H5 (THead 
+(Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) t1 (lift (S O) 
+O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind Abst) u2 t2)) 
+(pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind Abbr) O c c 
+(drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 x1) 
+(pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) H4))))) 
+H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: C).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pc3 
+c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def (match 
+(H (refl_equal B Abst)) in False with []) in H1)))))))) (\lambda (_: (not (eq 
+B Void Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Void) u1 t1) 
+(THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr3 c (THead (Bind Void) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind 
+Abst) u2 t2) t)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind 
+Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Void) u1 t1) 
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def 
+(pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 t3)))))) (pr3 (CHead c (Bind Void) u1) 
+t1 (lift (S O) O x)) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead 
+(Bind Abst) u2 t2))) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 
+c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t1 t3))))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t3))))) (pc3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H6: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_: ((\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(let H9 \def (pr3_gen_abst c u2 t2 x 
+H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind 
+Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind 
+Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq T x 
+(THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2 x2)).(\lambda (_: 
+((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 x3))))).(let 
+H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Void) x0 x1))) H6 
+(THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T (THead (Bind Abst) 
+x2 x3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow (match b0 with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
+Void) x0 x1) H13) in (False_ind (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) 
+O (THead (Bind Abst) u2 t2))) H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 
+(CHead c (Bind Void) u1) t1 (lift (S O) O x))).(let H6 \def (pr3_gen_abst c 
+u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O 
+(THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: 
+(eq T x (THead (Bind Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda 
+(H9: ((\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2 
+x1))))).(let H10 \def (eq_ind T x (\lambda (t: T).(pr3 (CHead c (Bind Void) 
+u1) t1 (lift (S O) O t))) H5 (THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t 
+(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) x0 x1)) H10 
+(lift (S O) O (THead (Bind Abst) u2 t2)) (pr3_lift (CHead c (Bind Void) u1) c 
+(S O) O (drop_drop (Bind Void) O c c (drop_refl c) u1) (THead (Bind Abst) u2 
+t2) (THead (Bind Abst) x0 x1) (pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 
+Abst x0)))))))))) H6))) H4))))) H1))))))))) b).
+
+lemma pc3_gen_lift_abst:
+ \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall 
+(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) 
+\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda 
+(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
+t1)))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind 
+Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H 
+in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3 
+c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: 
+T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: 
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) 
+(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c 
+(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0) 
+in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 
+e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind 
+Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) 
+(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T 
+x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2 
+t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead 
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e 
+t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 
+(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1 
+x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x 
+(\lambda (t0: T).(eq T t0 (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1: 
+T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: 
+T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) 
+t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead 
+(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14: 
+(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t0)))) H10 
+(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t0: T).(pr3 c 
+u2 t0)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0: 
+T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T 
+(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) 
+(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: 
+T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1 
+x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
+
+lemma pc3_gen_sort_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c 
+(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda 
+(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 
+\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: 
+T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c 
+(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def 
+(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u 
+x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) 
+(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind 
+T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H7) in (False_ind P H8)))))))) H3))))) H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/left.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/left.ma
new file mode 100644 (file)
index 0000000..a4a1aa0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/left.ma
@@ -0,0 +1,117 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/props.ma".
+
+implied rec lemma pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall 
+(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to 
+(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: 
+(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: 
+pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow 
+(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1 
+((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1) 
+\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
+
+fact pc3_ind_left__pc3_left_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda 
+(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 
+c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: 
+(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
+
+fact pc3_ind_left__pc3_left_trans:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: 
+T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda 
+(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: 
+(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda 
+(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+fact pc3_ind_left__pc3_left_sym:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(pc3_left c t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) 
+(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 
+H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 
+t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 
+(pc3_left_r c t3))))))))) t1 t2 H)))).
+
+fact pc3_ind_left__pc3_left_pc3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to 
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x 
+(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x 
+(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
+
+fact pc3_ind_left__pc3_pc3_left:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
+(pc3 c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda 
+(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c 
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 
+c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 
+(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
+
+lemma pc3_ind_left:
+ \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 
+t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: 
+T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to 
+((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c 
+t t0) \to (P t t0))))))))
+\def
+ \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall 
+(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 
+t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 
+t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
+\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 
+t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t 
+t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: 
+T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left 
+c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 
+(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/nf2.ma
new file mode 100644 (file)
index 0000000..cc79e08
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/nf2.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/defs.ma".
+
+include "basic_1A/nf2/pr3.ma".
+
+lemma pc3_nf2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
+t1) \to ((nf2 c t2) \to (eq T t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (H0: (nf2 c t1)).(\lambda (H1: (nf2 c t2)).(let H2 \def H in 
+(ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (eq T 
+t1 t2) (\lambda (x: T).(\lambda (H3: (pr3 c t1 x)).(\lambda (H4: (pr3 c t2 
+x)).(let H_y \def (nf2_pr3_unfold c t1 x H3 H0) in (let H5 \def (eq_ind_r T x 
+(\lambda (t: T).(pr3 c t2 t)) H4 t1 H_y) in (let H6 \def (eq_ind_r T x 
+(\lambda (t: T).(pr3 c t1 t)) H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c 
+t2 t1 H5 H1) in (let H7 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t1)) H5 t1 
+H_y0) in (eq_ind_r T t1 (\lambda (t: T).(eq T t1 t)) (refl_equal T t1) t2 
+H_y0))))))))) H2))))))).
+
+lemma pc3_nf2_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c 
+t2) \to (pr3 c t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (H0: (nf2 c t2)).(let H1 \def H in (ex2_ind T (\lambda (t: 
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pr3 c t1 t2) (\lambda (x: 
+T).(\lambda (H2: (pr3 c t1 x)).(\lambda (H3: (pr3 c t2 x)).(let H_y \def 
+(nf2_pr3_unfold c t2 x H3 H0) in (let H4 \def (eq_ind_r T x (\lambda (t: 
+T).(pr3 c t1 t)) H2 t2 H_y) in H4))))) H1)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/pc1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/pc1.ma
new file mode 100644 (file)
index 0000000..75648cf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/pc1.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/defs.ma".
+
+include "basic_1A/pc1/defs.ma".
+
+include "basic_1A/pr3/pr1.ma".
+
+lemma pc3_pc1:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c: 
+C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: 
+T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1 
+x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) 
+(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c))))) 
+H0))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/props.ma
new file mode 100644 (file)
index 0000000..af2a301
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/props.ma
@@ -0,0 +1,410 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/defs.ma".
+
+include "basic_1A/pr3/pr3.ma".
+
+lemma clear_pc3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to 
+(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1 
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind 
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1 
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2 
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1 
+H0))))) H1))))))).
+
+lemma pc3_pr2_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
+
+lemma pc3_pr2_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2 
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
+
+lemma pc3_pr3_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t2 H (pr3_refl c t2))))).
+
+lemma pc3_pr3_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2 
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) 
+t1 (pr3_refl c t1) H)))).
+
+lemma pc3_pr3_t:
+ \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall 
+(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1 
+t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t: 
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
+
+lemma pc3_refl:
+ \forall (c: C).(\forall (t: T).(pc3 c t t))
+\def
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0)) 
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
+
+lemma pc3_s:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c 
+t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1 
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t)) 
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
+
+lemma pc3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
+(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda 
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u 
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead 
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead 
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f))))) 
+H0))))))).
+
+lemma pc3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda 
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t) 
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2: 
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0)) 
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x 
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl 
+(CHead c k x) t)))))) H0))))))).
+
+lemma pc3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T 
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u) 
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1: 
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2 
+T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u 
+t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1) 
+(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
+
+lemma pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall 
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1 
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t)) 
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
+theorem pc3_t:
+ \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall 
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in 
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c 
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3 
+x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1 
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t)) 
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7: 
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c 
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2))))) 
+H4))))) H1))))))).
+
+lemma pc3_pr2_u2:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
+(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x 
+c t1 t0 H) t2 H0)))))).
+
+lemma pc3_pr3_conf:
+ \forall (c: C).(\forall (t: T).(\forall (t1: T).((pc3 c t t1) \to (\forall 
+(t2: T).((pr3 c t t2) \to (pc3 c t2 t1))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pc3 c t 
+t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(pc3_t t c t2 (pc3_pr3_x c 
+t2 t H0) t1 H)))))).
+
+theorem pc3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c 
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pc3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c 
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
+
+lemma pc3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
+t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda 
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: 
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0 
+(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k 
+u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k 
+u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in 
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) 
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k 
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1) 
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr) 
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead 
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda 
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
+u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
+((let H12 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
+((let H13 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in 
+(\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H16 \def (eq_ind 
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13) in (eq_ind B Abbr 
+(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0)) (pc3 (CHead c (Bind 
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O u1 t4 x)).(\lambda 
+(H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x (pr3_pr2 (CHead c 
+(Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl 
+Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr) u1) t x (pr2_free 
+(CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2 t4 t O H16 u1 H)) b 
+H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10: (clear (CHead c (Flat 
+f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t3 
+t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t (pr2_delta (CHead d (Bind Abbr) u) 
+d u O (getl_refl Abbr d u) t3 t4 H3 t H9)) (CHead c (Flat f) u1) (clear_flat 
+c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H10) 
+f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H8)))) (\lambda 
+(i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pc3 (CHead c k u1) t3 t))))).(\lambda (H8: (getl 
+(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) 
+u t4 t)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind 
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c k0 u1) t3 t)))) \to 
+((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) 
+(\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind 
+Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Bind b) u1) t3 
+t))))).(\lambda (H10: (getl (r (Bind b) i0) c (CHead d (Bind Abbr) 
+u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d 
+u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H10 u1) t3 t4 H3 t 
+H9))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead 
+d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3 (CHead c (Flat f) u1) t3 
+t))))).(\lambda (H10: (getl (r (Flat f) i0) c (CHead d (Bind Abbr) 
+u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u 
+(r (Flat f) i0) H10 t3 t4 H3 t H9) f u1))))) k IHi (getl_gen_S k c (CHead d 
+(Bind Abbr) u) u2 i0 H8)))))) i H7 H4)))))))))))))) y t1 t2 H1))) H0)))))))).
+
+lemma pc3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3 
+(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
+K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0 
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
+(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1: 
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y: 
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4 
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4 
+t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
+c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n: 
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3 
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 
+(CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14 \def (f_equal C B (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
+C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
+H11 t1 H15) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 
+t6 t7)) (pc3 (CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: 
+(subst0 O t2 t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: 
+T).(subst0 O t t5 t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t 
+t5 x0)).(\lambda (H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal 
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S 
+i) H23) in (pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 
+(Bind Abbr) t) c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x 
+(CHead c0 (Bind Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) 
+(getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 
+H24)))))))) (subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd 
+t1 t5 t6 O H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda 
+(H12: (clear (CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) 
+u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 
+(Bind Abbr) u0) t4 t6 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl 
+Abbr d0 u0) t4 t5 H6 t6 H11)) (CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 
+(Bind Abbr) u0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f 
+t)))) k (getl_gen_O (CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) 
+(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind 
+Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pc3 (CHead c0 k t) t4 
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t1) (CHead d0 (Bind Abbr) 
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl 
+(r k0 i1) c0 (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c0 k0 t) t4 t6))) 
+(\lambda (b: B).(\lambda (H12: (getl (r (Bind b) i1) c0 (CHead d0 (Bind Abbr) 
+u0))).(pc3_pr2_r (CHead c0 (Bind b) t) t4 t6 (pr2_delta (CHead c0 (Bind b) t) 
+d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind Abbr) u0) H12 t) t4 t5 
+H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (r (Flat f) i1) c0 (CHead 
+d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 
+t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) H12 t4 t5 H6 t6 H11) f t)))) k 
+(getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1 i1 H10)))))) i0 H9 
+H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1 H)))).
+
+lemma pc3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to 
+(pc3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3 
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
+u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1) 
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2 
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
+u1 H3)))))))))) t1 t2 H)))))).
+
+lemma pc3_pr3_pc3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2 
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1 
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
+K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pc3 
+(CHead c k t2) t4 t5) \to (pc3 (CHead c k t3) t4 t5))))))).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0 
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c 
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2) 
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6: 
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0 
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2 
+H0)))))) H4))))))))))))) u2 u1 H)))).
+
+lemma pc3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift 
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1 
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t: 
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda 
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1) 
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H 
+t2 x H3))))) H1))))))))).
+
+lemma pc3_eta:
+ \forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: T).((pc3 c t 
+(THead (Bind Abst) w u)) \to (\forall (v: T).((pc3 c v w) \to (pc3 c (THead 
+(Bind Abst) v (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t)))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: T).(\lambda (H: 
+(pc3 c t (THead (Bind Abst) w u))).(\lambda (v: T).(\lambda (H0: (pc3 c v 
+w)).(pc3_t (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O 
+(THead (Bind Abst) w u)))) c (THead (Bind Abst) v (THead (Flat Appl) (TLRef 
+O) (lift (S O) O t))) (pc3_head_21 c v w H0 (Bind Abst) (THead (Flat Appl) 
+(TLRef O) (lift (S O) O t)) (THead (Flat Appl) (TLRef O) (lift (S O) O (THead 
+(Bind Abst) w u))) (pc3_thin_dx (CHead c (Bind Abst) v) (lift (S O) O t) 
+(lift (S O) O (THead (Bind Abst) w u)) (pc3_lift (CHead c (Bind Abst) v) c (S 
+O) O (drop_drop (Bind Abst) O c c (drop_refl c) v) t (THead (Bind Abst) w u) 
+H) (TLRef O) Appl)) t (pc3_t (THead (Bind Abst) w u) c (THead (Bind Abst) w 
+(THead (Flat Appl) (TLRef O) (lift (S O) O (THead (Bind Abst) w u)))) 
+(pc3_pr3_r c (THead (Bind Abst) w (THead (Flat Appl) (TLRef O) (lift (S O) O 
+(THead (Bind Abst) w u)))) (THead (Bind Abst) w u) (pr3_eta c w u w (pr3_refl 
+c w))) t (pc3_s c (THead (Bind Abst) w u) t H))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/subst1.ma
new file mode 100644 (file)
index 0000000..dc3ec30
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/subst1.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/props.ma".
+
+include "basic_1A/pr3/subst1.ma".
+
+lemma pc3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall 
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall 
+(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
+t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d 
+c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c 
+a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1: 
+T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda 
+(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda 
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x: 
+T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T 
+(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2 
+x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d 
+x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x 
+(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda 
+(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3 
+a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0 
+(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2 
+H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3))))) 
+(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pc3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1A/pc3/wcpr0.ma
new file mode 100644 (file)
index 0000000..91d464f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pc3/wcpr0.ma
@@ -0,0 +1,87 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/props.ma".
+
+include "basic_1A/wcpr0/getl.ma".
+
+fact pc3_wcpr0__pc3_wcpr0_t_aux:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall 
+(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 
+(CHead c2 k u) t1 t2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: 
+K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: 
+T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4 
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3: 
+(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead 
+c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4 
+t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead 
+c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda 
+(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2 
+k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d 
+(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6 
+t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c 
+(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead 
+d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: 
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda 
+(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7: 
+T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6 
+t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t 
+x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6 
+t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x 
+H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k 
+u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) 
+H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
+
+lemma pc3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 
+t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0: 
+C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: 
+T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def 
+(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T 
+(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k 
+u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3 
+(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x 
+(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2 
+(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x 
+H6)))))) H4))))))))))))) c1 c2 H))).
+
+lemma pc3_wcpr0:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: 
+T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind 
+T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2 
+x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2 
+(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/dec.ma
new file mode 100644 (file)
index 0000000..b5ccaf4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/dec.ma
@@ -0,0 +1,520 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/props.ma".
+
+include "basic_1A/subst0/dec.ma".
+
+include "basic_1A/T/dec.ma".
+
+include "basic_1A/T/props.ma".
+
+lemma nf0_dec:
+ \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t1 t2))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to 
+(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl 
+(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) 
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T 
+(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl 
+(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) 
+t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T 
+(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: 
+T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 
+t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(K_ind (\lambda (k0: K).(or 
+(\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) 
+(ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) (\lambda (b: 
+B).(B_ind (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) 
+t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Bind b0) t t0) t2))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind 
+Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: 
+T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in 
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) 
+O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t 
+t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 
+(lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S 
+O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) 
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) 
+t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind 
+Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let 
+H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind 
+Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)) H4) in (let H6 \def 
+(eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S O) O x))) H3 (lift (S 
+O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 P))))) (pr0_delta t t 
+(pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) (\lambda (H3: (eq T 
+t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) t3)))) (ex_intro2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) O x)) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t (lift (S 
+O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t (lift (S O) O x)) 
+x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S O) O H4 P))) 
+(pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) H1)))) (let 
+H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to 
+(eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 t t2) 
+\to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) 
+\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead 
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda 
+(H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
+(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) 
+(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 
+t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def 
+(H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H8 t0 
+H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) x0 t3))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(pr0 t t3)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t H_y0) in (eq_ind_r T (THead 
+(Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) 
+(refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 
+t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: 
+T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) 
+(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead 
+(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t 
+t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) 
+t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: 
+Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 
+(THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind 
+Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t 
+x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) t x) H7) in 
+(let H9 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H6 t0 H8) in (let 
+H10 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H5 t0 H8) in (H10 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: 
+T).(\lambda (H3: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr0 
+t x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq 
+T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind 
+Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind 
+Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind Abst) t t0) t2)) (THead (Bind Abst) x t0) (\lambda (H5: (eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) x t0))).(\lambda (P: Prop).(let H6 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
+_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) 
+(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: 
+T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T 
+t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) 
+(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) (let H_x 
+\def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or 
+(subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall 
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) 
+(\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 
+(lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift 
+(S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T 
+(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind 
+Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let 
+H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to 
+(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) 
+\to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) 
+\to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead 
+(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda 
+(H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall 
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) 
+(\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t 
+t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 
+t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t 
+x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def 
+(H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 
+t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead 
+(Bind Void) x0 t3))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda 
+(t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(eq T t2 (THead (Bind Void) t3 t0))) H14 t H_y0) in (eq_ind_r T (THead 
+(Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) 
+(refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: 
+(pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let 
+H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 
+(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq 
+T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) 
+(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead 
+(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 
+H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
+t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall 
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) 
+(\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: 
+Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 
+(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T 
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind 
+Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) 
+t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Bind Void) t t0) (THead (Bind Void) t x0) H10) in 
+(let H12 \def (eq_ind_r T x0 (\lambda (t2: T).(pr0 t0 t2)) H9 t0 H11) in (let 
+H13 \def (eq_ind_r T x0 (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H8 t0 H11) in (H13 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6))) (\lambda (H5: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
+(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda 
+(x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: Prop).P)))).(\lambda 
+(H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) 
+t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) x0 t0) (\lambda (H8: 
+(eq T (THead (Bind Void) t t0) (THead (Bind Void) x0 t0))).(\lambda (P: 
+Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) 
+(THead (Bind Void) t t0) (THead (Bind Void) x0 t0) H8) in (let H10 \def 
+(eq_ind_r T x0 (\lambda (t2: T).(pr0 t t2)) H7 t H9) in (let H11 \def 
+(eq_ind_r T x0 (\lambda (t2: T).((eq T t t2) \to (\forall (P0: Prop).P0))) H6 
+t H9) in (H11 (refl_equal T t) P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) 
+(Bind Void))))))) H5)) H4))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(let 
+H4 \def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to 
+(eq T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 t2 t3))))) H0 (lift (S O) O x) H3) in 
+(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or (\forall (t3: T).((pr0 
+(THead (Bind Void) t t2) t3) \to (eq T (THead (Bind Void) t t2) t3))) (ex2 T 
+(\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2) t3))))) (or_intror 
+(\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O x)) t2) \to (eq T 
+(THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T (\lambda (t2: T).((eq T 
+(THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2))) (ex_intro2 
+T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S 
+O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) 
+x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Void) x t (S O) O H5 P))) 
+(pr0_zeta Void not_void_abst x x (pr0_refl x) t))) t0 H3))) H2))) H1))) b)) 
+(\lambda (f: F).(F_ind (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead 
+(Flat f0) t t0) t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda 
+(t2: T).((eq T (THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) 
+(\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2))))) (let H_x \def 
+(binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: 
+B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) 
+(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat 
+Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: 
+T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T 
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w 
+u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq 
+T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
+T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 
+\def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq 
+T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) 
+(\lambda (t3: T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r 
+T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead 
+(Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T 
+(\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: 
+Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) (B_ind 
+(\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to 
+(eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+(THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 
+x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
+b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
+Appl) t (THead (Bind b) x1 x2)) t2)))))) (\lambda (_: (or (\forall (t2: 
+T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1 x2) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2) 
+t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind 
+Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) 
+x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat 
+Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T 
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 
+x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead 
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
+I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
+(False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 
+(pr0_refl x1) x2 x2 (pr0_refl x2))))) (\lambda (_: (or (\forall (t2: T).((pr0 
+(THead (Bind Abst) x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 
+T (\lambda (t2: T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror 
+(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
+\to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) 
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead 
+(Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda 
+(t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead 
+(Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) 
+x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T 
+(THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ 
+_) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) 
+(pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) (\lambda (_: (or 
+(\forall (t2: T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind 
+Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) 
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 
+x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead 
+(Bind Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 
+x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind 
+Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: 
+T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) 
+x1 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead 
+(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: 
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) 
+I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in 
+(False_ind P H7)))) (pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 
+(pr0_refl x1) x2 x2 (pr0_refl x2))))) x0 H4) t0 H3)))))) H2)) (\lambda (H2: 
+((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w 
+u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: 
+T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: 
+T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
+(\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def 
+H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T 
+(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to 
+(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead 
+(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 
+(THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) 
+\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t 
+t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq 
+T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 
+(THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) 
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 
+t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t 
+x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def 
+(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 
+t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3: 
+T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead 
+(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) 
+(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead 
+(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead 
+(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall 
+(t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in 
+(let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
+T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
+Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind 
+Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl 
+(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 
+x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: 
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T 
+t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda 
+(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat 
+Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4: 
+T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let 
+H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: 
+T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: 
+Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) 
+x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 
+(THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind 
+x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t 
+(THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
+x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) 
+(\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T 
+t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall 
+(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) 
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to 
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) 
+(\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: 
+Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 
+(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat 
+Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t 
+x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in 
+(let H11 \def (eq_ind_r T x (\lambda (t2: T).(pr0 t0 t2)) H8 t0 H10) in (let 
+H12 \def (eq_ind_r T x (\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: 
+Prop).P0))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t 
+(pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T 
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead 
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T 
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: 
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: 
+T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 
+t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq 
+T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead 
+(Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef 
+_) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) 
+(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: 
+T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq 
+T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) 
+P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) 
+H1))) (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq 
+T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat 
+Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat 
+Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead 
+(Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) 
+t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_tau t0 t0 
+(pr0_refl t0) t))) f)) k)))))) t1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/defs.ma
new file mode 100644 (file)
index 0000000..bcbc781
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/defs.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/defs.ma".
+
+inductive pr0: T \to (T \to Prop) \def
+| pr0_refl: \forall (t: T).(pr0 t t)
+| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) 
+(THead k u2 t2))))))))
+| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to 
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
+| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: 
+T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 
+u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t2)))))))))))))
+| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to 
+(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall 
+(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O 
+t1)) t2))))))
+| pr0_tau: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: 
+T).(pr0 (THead (Flat Cast) u t1) t2)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/fwd.ma
new file mode 100644 (file)
index 0000000..bff7d0b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/fwd.ma
@@ -0,0 +1,1561 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/defs.ma".
+
+include "basic_1A/subst0/fwd.ma".
+
+implied rec lemma pr0_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
+t))) (f0: (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to 
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall 
+(k: K).(P (THead k u1 t1) (THead k u2 t2)))))))))))) (f1: (\forall (u: 
+T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall 
+(t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (P (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))))))) (f2: 
+(\forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
+T).((pr0 v1 v2) \to ((P v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 
+u2) \to ((P u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to ((P 
+t1 t2) \to (P (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))))))))))))))) (f3: (\forall 
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to ((P u1 u2) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (w: T).((subst0 
+O u2 t2 w) \to (P (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 
+w))))))))))))) (f4: (\forall (b: B).((not (eq B b Abst)) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead 
+(Bind b) u (lift (S O) O t1)) t2))))))))) (f5: (\forall (t1: T).(\forall (t2: 
+T).((pr0 t1 t2) \to ((P t1 t2) \to (\forall (u: T).(P (THead (Flat Cast) u 
+t1) t2))))))) (t: T) (t0: T) (p: pr0 t t0) on p: P t t0 \def match p with 
+[(pr0_refl t1) \Rightarrow (f t1) | (pr0_comp u1 u2 p0 t1 t2 p1 k) 
+\Rightarrow (f0 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 p1 
+((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) k) | (pr0_beta u v1 v2 p0 t1 t2 
+p1) \Rightarrow (f1 u v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) v1 v2 p0) t1 
+t2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1)) | (pr0_upsilon b n v1 v2 p0 
+u1 u2 p1 t1 t2 p2) \Rightarrow (f2 b n v1 v2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
+f5) v1 v2 p0) u1 u2 p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p1) t1 t2 p2 
+((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p2)) | (pr0_delta u1 u2 p0 t1 t2 p1 w 
+s0) \Rightarrow (f3 u1 u2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) u1 u2 p0) t1 t2 
+p1 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p1) w s0) | (pr0_zeta b n t1 t2 p0 
+u) \Rightarrow (f4 b n t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 f5) t1 t2 p0) u) 
+| (pr0_tau t1 t2 p0 u) \Rightarrow (f5 t1 t2 p0 ((pr0_ind P f f0 f1 f2 f3 f4 
+f5) t1 t2 p0) u)].
+
+lemma pr0_gen_sort:
+ \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq 
+T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in 
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TSort n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t1)) (TSort n))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H8) in 
+(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda 
+(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
+Abbr) u1 t1) (TSort n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in 
+(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
+O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TSort n))).(let H4 \def 
+(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
+H4)))))))) y x H0))) H))).
+
+lemma pr0_gen_lref:
+ \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq 
+T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H5) in (False_ind (eq T (THead k u2 t2) (THead k u1 t1)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t1)) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 t2) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in 
+(False_ind (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t1))) H9))))))))))))))))) (\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 
+(TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda 
+(w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind 
+Abbr) u1 t1) (TLRef n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in 
+(False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H4) in (False_ind (eq T t2 (THead (Bind b) u (lift (S O) 
+O t1))) H5)))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) (TLRef n))).(let H4 \def 
+(eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) 
+H4)))))))) y x H0))) H))).
+
+lemma pr0_gen_abst:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 
+t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda 
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def 
+(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r 
+T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind 
+Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 
+(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) in ((let H7 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H5) 
+in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Abst))).(eq_ind_r 
+K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind 
+Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 
+u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10)))) 
+H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda 
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H6) in (False_ind 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H7))))))))))))) (\lambda (b: 
+B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H4: (eq T 
+(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(let H5 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abst) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
+Abst) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(TLRef _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
+(Bind Abst) u1 t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b 
+Abst)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 
+Abst H9) in (let H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead 
+(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in 
+(eq_ind T (lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t 
+t3))))) (let H12 \def (match (H10 (refl_equal B Abst)) in False with []) in 
+H12) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead (Flat Cast) u t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T 
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) H4)))))))) y x H0))) H)))).
+
+lemma pr0_gen_appl:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 
+t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) 
+v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2))))))))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T 
+t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead 
+(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T 
+t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl 
+t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 
+u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
+T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat 
+Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
+\Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
+(THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: 
+(eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Appl) 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+(THead k0 u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))) (let H11 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let H12 \def (eq_ind 
+T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 
+(\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda 
+(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead 
+(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 
+(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ 
+_ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead 
+(Flat Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T 
+v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead 
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead 
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 
+\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in 
+(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda 
+(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) 
+(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead 
+(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind 
+Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 
+y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
+v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 
+(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 
+t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not 
+(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 
+v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead 
+(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 
+(THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H8) in ((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ 
+t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead 
+(Flat Appl) u1 t1) H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind 
+T v1 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead 
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead 
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def 
+(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: 
+T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in 
+(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 
+t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead 
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in 
+(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda 
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat 
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat 
+Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat 
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 
+Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) 
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0)) 
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) 
+H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
+(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 
+t1) H6) in (False_ind (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Bind Abbr) u2 w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
+u2 w) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Bind 
+Abbr) u2 w) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))) H7))))))))))))) 
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) 
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq 
+T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 
+\def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
+(Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: 
+T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 
+t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
+True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
+
+lemma pr0_gen_cast:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) 
+(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda 
+(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat 
+Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat 
+Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: 
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat 
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) 
+u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T 
+(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda 
+(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda 
+(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) in ((let H7 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Flat Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H5) 
+in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat Cast))).(eq_ind_r 
+K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 
+u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 
+t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) u1 t1) H5) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Bind Abbr) v2 t2))) H6)))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 v2))))).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) u1 t1) H8) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead 
+(Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H6) in (False_ind (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) u2 
+w))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u 
+(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(let H5 \def (eq_ind T (THead 
+(Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
+(THead (Flat Cast) u1 t1) H4) in (False_ind (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(let H4 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) 
+(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 
+t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) 
+H8))))) H4)))))))) y x H0))) H)))).
+
+lemma pr0_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 
+(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t 
+x)) (\lambda (_: T).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(pr0 t1 t2)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat 
+d (\lambda (n: nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T 
+x (lift h n t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: 
+T).(\forall (x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq 
+T x (lift h x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 
+x0)) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: 
+T).(pr0 x0 t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H1: (eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq 
+T t (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) 
+(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: 
+((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T 
+(\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 
+t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
+t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (k: K).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H5: (eq T (THead k u1 t2) (lift h x1 x0))).(K_ind (\lambda 
+(k0: K).((eq T (THead k0 u1 t2) (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T (THead k0 u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))))) 
+(\lambda (b: B).(\lambda (H6: (eq T (THead (Bind b) u1 t2) (lift h x1 
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
+b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
+(Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T 
+t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t: 
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h 
+(S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) 
+x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) 
+x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) x4) (\lambda (t: 
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T (\lambda (t4: 
+T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda 
+(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T 
+(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) 
+t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) 
+x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
+x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind 
+b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h x1 (THead (Bind b) x5 
+x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift_bind b x5 x4 h 
+x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 H_x0)))) (H2 x2 x1 H8)) t3 
+H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind b u1 t2 x0 h x1 H6)))) 
+(\lambda (f: F).(\lambda (H6: (eq T (THead (Flat f) u1 t2) (lift h x1 
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
+f) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda 
+(t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
+(Flat f) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: (eq T 
+t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T 
+(\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) 
+u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) 
+(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H10: (pr0 
+x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T (THead (Flat f) u2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat f) x2 x3) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) 
+u2 (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 
+x3) t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda 
+(H11: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Flat f) t (lift h x1 x4)) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq 
+T (THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat f) x2 x3) t4)) (THead (Flat f) x5 x4) (sym_eq T 
+(lift h x1 (THead (Flat f) x5 x4)) (THead (Flat f) (lift h x1 x5) (lift h x1 
+x4)) (lift_flat f x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 
+H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat 
+f u1 t2 x0 h x1 H6)))) k H5))))))))))))) (\lambda (u: T).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: 
+T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
+T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda 
+(x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda 
+(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: 
+T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead 
+(Bind Abst) u t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
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+T).(\lambda (x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 
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+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift 
+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (z: T).(eq T x3 (THead (Bind Abst) y0 z)))) (\lambda (y0: 
+T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) 
+t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H9: (eq T x3 (THead (Bind 
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+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda 
+(t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T 
+(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda 
+(x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H12: (pr0 x5 
+x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex2_ind T (\lambda 
+(t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T 
+(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1) x6)) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) 
+t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h x1 x7))).(\lambda 
+(H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) 
+(ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h 
+(S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 
+(THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h 
+x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S 
+x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2 
+H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3 H9)))))) 
+(lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat Appl v1 
+(THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b: B).(\lambda 
+(H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 (lift h 
+x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) (\lambda (t2: 
+T).(pr0 x0 t2)))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (H5: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 
+x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: 
+T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
+t3)).(\lambda (H7: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T 
+(\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z)))) 
+(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h x1 z)))) (ex2 T 
+(\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl) x2 
+x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead 
+(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
+(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0 
+z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda 
+(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1) 
+x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T 
+(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4)) 
+(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 
+(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift 
+h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq 
+T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S 
+x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead 
+(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 
+x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: 
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift 
+(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: 
+T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O 
+v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T 
+v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) 
+(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) 
+(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T 
+(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t 
+(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat 
+Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat 
+Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat 
+Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead 
+(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S 
+x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat 
+Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 
+x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1) 
+x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1 
+x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 
+H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) 
+(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1 
+(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: 
+T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
+T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (w: 
+T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1 
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
+Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead 
+(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9: 
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda 
+(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 
+(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: 
+T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3 
+(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T 
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0: 
+(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 
+x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let 
+H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11 
+(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in 
+(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5) 
+(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq 
+T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: 
+T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17: 
+(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t: 
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T 
+(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O 
+x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) 
+t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5 
+x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) 
+(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17)) 
+w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1 
+H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1 
+H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: 
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u 
+(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda 
+(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq 
+T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) 
+(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1 
+x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T 
+(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift 
+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S 
+O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (le_antisym (S x1) (plus (S O) 
+x1) (le_n (plus (S O) x1)) (le_n (S x1))) (plus x1 (S O)) (plus_sym x1 (S 
+O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: nat).(eq T (lift (S O) O 
+t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in (ex2_ind T (\lambda (t4: T).(eq 
+T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq T t2 (lift h x1 t4))) (ex2 T 
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind 
+b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H10: (eq T x3 (lift (S O) O 
+x4))).(\lambda (H11: (eq T t2 (lift h x1 x4))).(eq_ind_r T (lift (S O) O x4) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Bind b) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T 
+t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq 
+T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
+x4)) t4))) (\lambda (x5: T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda 
+(H12: (pr0 x4 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 
+(lift (S O) O x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) 
+t4)) x5 (refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 
+H_x)))) (H3 x4 x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n 
+x1) H9)))) x0 H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 
+H4)))))))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
+t3)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H3: (eq T (THead (Flat Cast) u t2) (lift h x1 x0))).(ex3_2_ind 
+T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Cast) y0 z)))) 
+(\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T t3 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) x2 x3))).(\lambda (_: (eq T 
+u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead 
+(Flat Cast) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T 
+t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4))) 
+(\lambda (x4: T).(\lambda (H_x: (eq T t3 (lift h x1 x4))).(\lambda (H7: (pr0 
+x3 x4)).(eq_ind_r T (lift h x1 x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) 
+t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x4) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)) x4 (refl_equal T (lift h 
+x1 x4)) (pr0_tau x3 x4 H7 x2)) t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) 
+(lift_gen_flat Cast u t2 x0 h x1 H3)))))))))) y x H0))))) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/pr0.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/pr0.ma
new file mode 100644 (file)
index 0000000..9687b04
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/pr0.ma
@@ -0,0 +1,2303 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/subst0.ma".
+
+include "basic_1A/lift/tlt.ma".
+
+include "basic_1A/tlt/fwd.ma".
+
+fact pr0_confluence__pr0_cong_upsilon_refl:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
+T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to 
+(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) 
+\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) 
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t5)) t)))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda 
+(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda 
+(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda 
+(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 
+(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 
+t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) 
+(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S 
+O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind 
+b))))))))))))))).
+
+fact pr0_confluence__pr0_cong_upsilon_cong:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: 
+T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall 
+(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: 
+T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T 
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) 
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
+t5)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda 
+(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 
+x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 
+x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda 
+(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T 
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) 
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) 
+t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) 
+(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat 
+Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp 
+(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat 
+Appl)) (Bind b))))))))))))))))))).
+
+fact pr0_confluence__pr0_cong_upsilon_delta:
+ (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: 
+T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: 
+T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 
+x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to 
+((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))))
+\def
+ \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: 
+T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: 
+T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: 
+(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 
+x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda 
+(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T 
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T 
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead 
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H 
+u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O 
+v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) 
+(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind 
+Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: 
+T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda 
+(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) 
+u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 
+(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: 
+(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: 
+T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) 
+(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon 
+Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift 
+(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) 
+O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) 
+(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 
+(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 
+H5))))))))))))))))))).
+
+fact pr0_confluence__pr0_cong_upsilon_zeta:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: 
+T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 
+u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: 
+T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat 
+Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v2) (lift (S O) O x))) t)))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda 
+(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda 
+(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: 
+T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: 
+(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: 
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: 
+T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead 
+(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O 
+(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 
+t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat 
+Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) 
+(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) 
+O)))))))))))))))).
+
+fact pr0_confluence__pr0_cong_delta:
+ \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to 
+(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall 
+(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind 
+Abbr) u3 w) t))))))))))))))
+\def
+ \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 
+t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda 
+(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 
+x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: 
+T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: 
+T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
+u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) 
+(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 
+x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) 
+(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w 
+w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) 
+(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 
+x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta 
+u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) 
+(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
+
+fact pr0_confluence__pr0_upsilon_upsilon:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
+T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: 
+T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to 
+(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 
+x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) 
+(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
+(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 
+x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 
+x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T 
+(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) 
+t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) 
+x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat 
+Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) 
+(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 
+H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O 
+x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S 
+O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
+
+fact pr0_confluence__pr0_delta_delta:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
+(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to 
+(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) 
+\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))))))))))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 
+t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: 
+(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: 
+(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 
+x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: 
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 
+x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: 
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w 
+x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp 
+u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) 
+(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O 
+x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 
+O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: 
+(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: 
+T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) 
+u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) 
+(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x 
+H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: 
+T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda 
+(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 
+w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: 
+T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 
+w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 
+w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
+T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 
+H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda 
+(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda 
+(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: 
+T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) 
+(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 
+x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in 
+(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: 
+T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x 
+H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda 
+(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x 
+x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: 
+T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) 
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: 
+T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 
+x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta 
+u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) 
+(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) 
+(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 
+w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead 
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 
+(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 
+x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) 
+(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
+
+fact pr0_confluence__pr0_delta_tau:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to 
+(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T 
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 
+t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 
+t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda 
+(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda 
+(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) 
+(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S 
+O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: 
+T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w 
+(S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) 
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda 
+(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) 
+(pr0_gen_lift t4 t3 (S O) O H0)))))))).
+
+theorem pr0_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 
+t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
+\def
+ \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to 
+(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) 
+(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall 
+(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 
+v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 
+t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: 
+T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 with [(pr0_refl t3) 
+\Rightarrow (\lambda (H2: (eq T t3 t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T 
+t (\lambda (t4: T).((eq T t4 t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) 
+(\lambda (t5: T).(pr0 t2 t5))))) (\lambda (H4: (eq T t t1)).(eq_ind T t1 
+(\lambda (_: T).(ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 
+t5)))) (let H5 \def (match H1 with [(pr0_refl t4) \Rightarrow (\lambda (H5: 
+(eq T t4 t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T 
+t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 
+t6))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T 
+(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def 
+(eq_ind T t (\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind 
+T t (\lambda (t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t 
+(\lambda (t5: T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t 
+(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
+\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 
+(\lambda (t5: T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) 
+\to (\forall (t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda 
+(t5: T).(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) 
+(let H13 \def (eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in 
+(ex_intro2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 
+(pr0_refl t1) (pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T 
+t4 t H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq 
+T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T 
+(THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) 
+\to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T 
+(THead k u2 t5) (\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda 
+(H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k 
+u1 t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: 
+T).(pr0 (THead k u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: 
+T).(eq T t3 t6)) H2 (THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T 
+(\lambda (t6: T).(pr0 (THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 
+t5) t6)) (THead k u2 t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k 
+u2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 
+H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead 
+(Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T 
+(THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda 
+(t6: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda 
+(H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) 
+H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda 
+(t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) 
+(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t4)) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 
+t5) t6)) (THead (Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl 
+(THead (Bind Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | 
+(pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to 
+((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: 
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) 
+\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) 
+(\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda 
+(H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 
+t4)) H9) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda 
+(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def 
+(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall 
+(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 
+t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in 
+(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 
+(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta 
+u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 
+t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead 
+(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to 
+((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T 
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
+(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda 
+(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 
+w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead 
+(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: 
+T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) 
+H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: 
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
+T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T 
+(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 
+(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 
+t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) 
+t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: 
+(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 
+t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 
+t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 
+t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to 
+(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) 
+(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def 
+(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) 
+O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: 
+T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
+H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u 
+(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: 
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
+T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in 
+(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) 
+(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl 
+t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_tau t4 t5 
+H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) t)).(\lambda 
+(H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T 
+t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T t2 (\lambda 
+(t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) in (eq_ind T 
+(THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 
+t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t (\lambda 
+(t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 \def 
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u 
+t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) 
+(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_tau t4 t2 H9 u) (pr0_refl t2)))) t1 
+H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) 
+(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | 
+(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 
+t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) 
+(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) 
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) 
+(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda 
+(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda 
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
+(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 
+(\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda 
+(t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 
+t2)) H11 (THead k u1 t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 
+t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k 
+u1 t3) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: 
+T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v 
+t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 
+t9)))))))))) H (THead k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 
+(THead k u2 t4) t6)) (\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 
+t4) (pr0_refl (THead k u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t 
+(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 
+H10 k0) \Rightarrow (\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: 
+(eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T 
+(THead k0 u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda 
+(H13: (eq T (THead k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: 
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 
+t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 
+u3)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead k0 u0 t5) H11) in (let H17 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef 
+_) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 
+u0 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in ((let H19 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+k0 u0 t5) H16) in (\lambda (H20: (eq T u1 u0)).(\lambda (H21: (eq K k 
+k0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead k0 u0 t5) H11) in (eq_ind_r K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: 
+T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) 
+(let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let 
+H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T 
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) 
+t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda 
+(H28: (pr0 u3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
+(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 
+H27 t4 x H25 k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 
+u0 t5) u2 H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k 
+H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 
+t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
+t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
+(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))) (\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in (let H17 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H19 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in (\lambda (H20: (eq T u1 
+v1)).(\lambda (H21: (eq K k (Flat Appl))).(let H22 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t5)) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+v2 t6) t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 
+H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead 
+(Bind Abst) u t5) H19) in (let H25 \def (match H24 with [(pr0_refl t7) 
+\Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda (H26: 
+(eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq T t8 
+t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda 
+(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T (THead 
+(Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: 
+T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 
+t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat 
+Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: 
+(pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)) 
+(THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6 H15) (pr0_comp v2 x H29 
+t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 
+(THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4 H27)) t7 (sym_eq T t7 (THead 
+(Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 H25 t7 t8 H26 k0) \Rightarrow 
+(\lambda (H27: (eq T (THead k0 u0 t7) (THead (Bind Abst) u t5))).(\lambda 
+(H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | 
+(THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) 
+H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) 
+(THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | (TLRef _) 
+\Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) (THead 
+(Bind Abst) u t5) H27) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq T u0 u) 
+\to ((eq T t7 t5) \to ((eq T (THead k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 
+t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: 
+(eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead 
+(Bind Abst) u3 t8) t4) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
+(Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T t7 t5)).(eq_ind T t5 
+(\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to 
+((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) 
+t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda 
+(H34: (eq T (THead (Bind Abst) u3 t8) t4)).(eq_ind T (THead (Bind Abst) u3 
+t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H36: (pr0 t5 
+t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) 
+t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x: 
+T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda 
+(t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: T).(\lambda (H39: (pr0 
+u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x) (pr0_beta u3 u2 x0 H39 t8 x 
+H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind Abbr)))))) (H22 v1 (tlt_head_sx (Flat 
+Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14))))) (H22 t5 (tlt_trans 
+(THead (Bind Abst) u t5) t5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) 
+(tlt_head_dx (Bind Abst) u t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) 
+u t5))) t8 H36 t6 H15)))) t4 H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 
+u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | 
+(pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda 
+(H28: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
+\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u t5) H27) in (False_ind ((eq T (THead (Bind 
+Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25 v0 v3 H26 u0 
+u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat Appl) v0 
+(THead (Bind b) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H30: (eq T 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4)).((let H31 
+\def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Abst) u t5) H29) in (False_ind ((eq T 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not 
+(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
+(THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) | 
+(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead 
+(Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq T (THead 
+(Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7) 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind 
+Abst) u t5) H28) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t4) \to ((pr0 
+u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 H27))) | (pr0_zeta b H25 t7 t8 H26 
+u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind b) u0 (lift (S O) O t7)) 
+(THead (Bind Abst) u t5))).(\lambda (H28: (eq T t8 t4)).((let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) 
+(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let 
+H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 
+| (TLRef _) \Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) 
+u0 (lift (S O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O 
+t7)) (THead (Bind Abst) u t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T 
+u0 u) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) 
+(\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O 
+t7) t5) \to ((eq T t8 t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to 
+(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
+T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S 
+O) O t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to 
+((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
+t6) t10))))))) (\lambda (H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: 
+T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 
+t6) t10)))))) (\lambda (H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 
+t4)).(let H37 \def (match (H35 (refl_equal B Abst)) in False with []) in 
+H37))) t8 (sym_eq T t8 t4 H34))) t5 H33)) u0 (sym_eq T u0 u H32))) b (sym_eq 
+B b Abst H31))) H30)) H29)) H28 H25 H26))) | (pr0_tau t7 t8 H25 u0) 
+\Rightarrow (\lambda (H26: (eq T (THead (Flat Cast) u0 t7) (THead (Bind Abst) 
+u t5))).(\lambda (H27: (eq T t8 t4)).((let H28 \def (eq_ind T (THead (Flat 
+Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abst) u t5) H26) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
+(THead (Bind Abbr) v2 t6) t9))))) H28)) H27 H25)))]) in (H25 (refl_equal T 
+(THead (Bind Abst) u t5)) (refl_equal T t4))))) k H21))))) H18)) H17))))) t2 
+H13)) t H11 H12 H9 H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) 
+\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda 
+(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to 
+((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda 
+(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0 
+u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) H13) in (let H21 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let 
+H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
+| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let H23 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 
+v1)).(\lambda (H25: (eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t5)) H13) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 
+(\lambda (t7: T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def 
+(match H28 with [(pr0_refl t7) \Rightarrow (\lambda (H29: (eq T t7 (THead 
+(Bind b) u0 t5))).(\lambda (H30: (eq T t7 t4)).(eq_ind T (THead (Bind b) u0 
+t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t9))))) (\lambda (H31: (eq T (THead (Bind b) u0 
+t5) t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).(ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))) (ex2_ind T 
+(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t5)) t8)) (\lambda (t8: 
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8))) 
+(\lambda (x: T).(\lambda (H32: (pr0 u2 x)).(\lambda (H33: (pr0 v2 
+x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5 t6 H19 u2 v2 x 
+H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t5)) u2 
+H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5) H29) H30))) | 
+(pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31: (eq T (THead k0 
+u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead k0 u5 t8) 
+t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
+(THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
+\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
+(Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u0 t5) H31) in (eq_ind K 
+(Bind b) (\lambda (k1: K).((eq T u4 u0) \to ((eq T t7 t5) \to ((eq T (THead 
+k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T 
+u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind 
+b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: 
+(eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead (Bind b) u5 t8) t4) 
+\to ((pr0 u0 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T (THead 
+(Bind b) u5 t8) t4)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: T).((pr0 
+u0 u5) \to ((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) 
+u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (pr0 u0 u5)).(\lambda (H40: 
+(pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 
+t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda 
+(H42: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: 
+T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 
+x0)).(\lambda (H44: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) 
+(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda 
+(H45: (pr0 u2 x1)).(\lambda (H46: (pr0 v2 
+x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45 H46 t8 t6 x 
+H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind 
+b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b) u0 t5) u0 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5) 
+(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H26 
+t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b) 
+u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
+b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4 (sym_eq T 
+u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29 H30))) | 
+(pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 t5))).(\lambda 
+(H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def (eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
+\Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T (THead (Bind 
+Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) H32 H29 H30))) | 
+(pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) \Rightarrow (\lambda (H33: 
+(eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead (Bind b) u0 
+t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) 
+O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind 
+b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to 
+((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 H32))) | (pr0_delta 
+u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq T (THead (Bind 
+Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T (THead (Bind Abbr) 
+u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow 
+t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef 
+_) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind Abbr) u4 
+t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) 
+H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) \to ((eq T t7 t5) \to 
+((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to 
+((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 
+t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T u4 u0)).(eq_ind T u0 
+(\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to 
+((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead 
+(Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))))) 
+(\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead 
+(Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 O u5 t8 
+w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda 
+(t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) t4)).(eq_ind T 
+(THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to 
+((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 
+t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 u5)).(\lambda (H41: 
+(pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 \def (eq_ind_r B b 
+(\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind 
+b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v 
+t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 
+t11)))))))))) H26 Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: 
+B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr H36) in (let H45 \def 
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H36) in 
+(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0 t8 x)).(\lambda (H47: (pr0 
+t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) 
+t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H48: (pr0 u5 x0)).(\lambda 
+(H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: 
+T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H50: (pr0 
+u2 x1)).(\lambda (H51: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_delta 
+H45 u5 t8 w H42 u2 v2 x1 H50 H51 t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 
+(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H27 v2 H17))))) 
+(H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5) u0 (THead (Flat Appl) v1 (THead 
+(Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) 
+v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3 H18))))) (H43 t5 (tlt_trans (THead 
+(Bind Abbr) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) 
+(tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind 
+Abbr) u0 t5))) t8 H41 t6 H19)))))))) t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 
+(sym_eq T u4 u0 H37))) b H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 
+H29 t7 t8 H30 u) \Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S 
+O) O t7)) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T t8 t4)).((let H33 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow 
+(lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ t9) 
+\Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 
+t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) 
+(THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in ((let 
+H35 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 
+| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
+b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S 
+O) O t7)) (THead (Bind b) u0 t5) H31) in (eq_ind B b (\lambda (b1: B).((eq T 
+u u0) \to ((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b1 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) 
+u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift 
+(S O) O v2) t6)) t9))))))))) (\lambda (H36: (eq T u u0)).(eq_ind T u0 
+(\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat 
+Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T (lift (S O) O 
+t7) t5)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not 
+(eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H38: (eq T t8 
+t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: 
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) 
+t10)))))) (\lambda (H39: (not (eq B b Abst))).(\lambda (H40: (pr0 t7 
+t4)).(let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(\forall (v: T).((tlt v 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t9))) \to (\forall (t10: T).((pr0 v 
+t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
+t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H26 (lift (S O) O t7) H37) in 
+(let H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T t3 (THead (Bind b) u0 
+t9))) H23 (lift (S O) O t7) H37) in (let H43 \def (eq_ind_r T t5 (\lambda 
+(t9: T).(pr0 t9 t6)) H19 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9: 
+T).(eq T t6 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: 
+T).(\lambda (H44: (eq T t6 (lift (S O) O x))).(\lambda (H45: (pr0 t7 
+x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind 
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)))) (ex2_ind T (\lambda 
+(t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) (\lambda 
+(x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T 
+(\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead 
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t9))) 
+(\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0 v2 
+x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48 H49 
+x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 
+(lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b) u0 
+(lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O 
+t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead 
+(Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44)))) (pr0_gen_lift t7 
+t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0 
+H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_tau t7 t8 
+H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead (Bind 
+b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T (THead 
+(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32)) 
+H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T 
+t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
+(pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
+(Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u3 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 
+t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T 
+(THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to 
+((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
+(\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 u3)).(\lambda (H16: 
+(pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 \def (eq_ind_r T t 
+(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind Abbr) u0 t5) H12) 
+in (let H19 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Bind Abbr) u0 t5) H18) in (\lambda 
+(H22: (eq T u1 u0)).(\lambda (H23: (eq K k (Bind Abbr))).(let H24 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
+u0 t5) H12) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: 
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) 
+t7)))) (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in 
+(let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u3 w) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 
+x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
+(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda 
+(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
+x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x0 H29 H30 t4 x H27 
+H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H15))))) (H24 t5 
+(tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k H23))))) H20)) H19)))))) 
+t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow 
+(\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: 
+(eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: 
+T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) 
+(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b 
+Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in 
+(let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in ((let H18 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef 
+_) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead 
+(Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S 
+O) O t5)) H16) in (\lambda (H20: (eq T u1 u)).(\lambda (H21: (eq K k (Bind 
+b))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Bind b) u (lift (S O) O t5)) H11) in (eq_ind_r K (Bind b) (\lambda 
+(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)))) (let H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 
+u H20) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift 
+(S O) O t5) H19) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O t7))) 
+(\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T 
+t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) 
+(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) 
+(\lambda (t8: T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
+(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+(lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
+T).(\lambda (H27: (pr0 x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) x0 (pr0_zeta b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx 
+(Bind b) u t5 (S O) O) x H26 t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O 
+H24)))) k H21))))) H18)) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 
+H10))) | (pr0_tau t5 t6 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat 
+Cast) u t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u 
+t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: 
+(eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda 
+(H13: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+k u1 t3) t7)) H4 (THead (Flat Cast) u t5) H10) in (let H15 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat 
+Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Cast) u t5) H14) in ((let H17 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) 
+(THead (Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: 
+(eq K k (Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall 
+(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: 
+T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat 
+Cast) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: 
+T).(pr0 t7 u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 
+t7 t4)) H8 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: 
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 
+x)).(\lambda (H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead 
+(Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_tau t4 x H23 u2) 
+H24)))) (H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) 
+H16)) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal 
+T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 
+t4 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead 
+(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T 
+(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda 
+(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda 
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda 
+(t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind 
+T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda 
+(t6: T).(eq T t6 t2)) H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) 
+in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) 
+H9 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def 
+(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: 
+T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind 
+Abbr) v2 t4)) (pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 
+H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) 
+\Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
+k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 
+u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 
+t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ 
+_) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k 
+u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 
+t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow (THead (Bind 
+Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead 
+(Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq T v1 
+u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0: 
+K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: 
+T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda (k0: 
+K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def (eq_ind_r 
+T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H19) in (let 
+H24 \def (match H23 with [(pr0_refl t7) \Rightarrow (\lambda (H24: (eq T t7 
+(THead (Bind Abst) u t3))).(\lambda (H25: (eq T t7 t6)).(eq_ind T (THead 
+(Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t6) t9))))) (\lambda (H26: (eq T (THead (Bind Abst) u t3) 
+t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T 
+(THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u t3) H19) in (let H28 
+\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall 
+(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda 
+(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead 
+(Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let H29 \def (eq_ind T v1 
+(\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t8: T).(pr0 
+v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30: (pr0 v2 x)).(\lambda 
+(H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 
+t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u 
+t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4 t4 (pr0_refl t4) 
+(Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1 (tlt_head_sx (Flat 
+Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6 H26)) t7 (sym_eq T 
+t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3 H24 t7 t8 H25 k0) 
+\Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead (Bind Abst) u 
+t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t7 | (TLRef _) 
+\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u0 t7) (THead 
+(Bind Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t7) 
+(THead (Bind Abst) u t3) H26) in (eq_ind K (Bind Abst) (\lambda (k1: K).((eq 
+T u0 u) \to ((eq T t7 t3) \to ((eq T (THead k1 u3 t8) t6) \to ((pr0 u0 u3) 
+\to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda 
+(H31: (eq T u0 u)).(eq_ind T u (\lambda (t9: T).((eq T t7 t3) \to ((eq T 
+(THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 u3) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T t7 t3)).(eq_ind T 
+t3 (\lambda (t9: T).((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to 
+((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda 
+(H33: (eq T (THead (Bind Abst) u3 t8) t6)).(eq_ind T (THead (Bind Abst) u3 
+t8) (\lambda (t9: T).((pr0 u u3) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
+Appl) u2 t9) t10)))))) (\lambda (_: (pr0 u u3)).(\lambda (H35: (pr0 t3 
+t8)).(let H36 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) 
+u1 t9) t)) H22 (THead (Bind Abst) u t3) H19) in (let H37 \def (eq_ind_r T t 
+(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v 
+t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 
+t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u1 (THead 
+(Bind Abst) u t3)) H36) in (let H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 
+t9 v2)) H7 u1 H20) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: 
+T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) 
+(\lambda (x: T).(\lambda (H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 
+x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x0: 
+T).(\lambda (H41: (pr0 t8 x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T 
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 
+(THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x 
+x0) (pr0_comp v2 x H39 t4 x0 H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 
+H41))))) (H37 t3 (tlt_trans (THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 
+(THead (Bind Abst) u t3)) (tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat 
+Appl) u1 (THead (Bind Abst) u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx 
+(Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 
+(sym_eq T t7 t3 H32))) u0 (sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) 
+H30))) H29)) H28)) H27 H24 H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) 
+\Rightarrow (\lambda (H26: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 
+t8) t6)).((let H28 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t3) 
+H26) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to 
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) 
+| (pr0_upsilon b H24 v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda 
+(H28: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) 
+u t3))).(\lambda (H29: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v3) t8)) t6)).((let H30 \def (eq_ind T (THead (Flat Appl) v0 (THead 
+(Bind b) u0 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+Abst) u t3) H28) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t6) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to 
+((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) 
+H30)) H29 H24 H25 H26 H27))) | (pr0_delta u0 u3 H24 t7 t8 H25 w H26) 
+\Rightarrow (\lambda (H27: (eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) 
+u t3))).(\lambda (H28: (eq T (THead (Bind Abbr) u3 w) t6)).((let H29 \def 
+(eq_ind T (THead (Bind Abbr) u0 t7) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind 
+Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) | 
+(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind 
+b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 
+t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) 
+\Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ 
+t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind 
+Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) 
+\Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u 
+t3) H26) in ((let H30 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead 
+(Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3) H26) in (eq_ind B 
+Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t7) t3) \to ((eq 
+T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat 
+Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda (_: 
+T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B Abst Abst)) 
+\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) (\lambda 
+(H32: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: 
+T).((eq T t8 t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T t8 t6)).(eq_ind T 
+t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to ((pr0 t7 t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u2 t6) t10)))))) (\lambda (H34: (not (eq B Abst 
+Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match (H34 (refl_equal B 
+Abst)) in False with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3 H32)) u0 
+(sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24 H25))) | 
+(pr0_tau t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead (Flat Cast) u0 
+t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8 t6)).((let H27 \def 
+(eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow 
+(match k0 with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) 
+\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26 H24)))]) in (H24 
+(refl_equal T (THead (Bind Abst) u t3)) (refl_equal T t6))))) k H21)))) H18)) 
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0 v0 v3 H9 t5 t6 H10) 
+\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 
+t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind 
+Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: (pr0 t5 t6)).(let H16 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in 
+(let H17 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead 
+(Bind Abst) u0 t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in (\lambda (_: (eq T u 
+u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def (eq_ind_r T t (\lambda (t7: 
+T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall 
+(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: 
+T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) 
+H11) in (let H23 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H21) 
+in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H25: (pr0 t4 
+x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) 
+(\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda 
+(x0: T).(\lambda (H27: (pr0 v2 x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) v3 t6) t7)) (THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 
+t4 x H25 (Bind Abbr)) (pr0_comp v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 
+(tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) 
+(H22 t5 (tlt_trans (THead (Bind Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead 
+(Bind Abst) u0 t5)) (tlt_head_dx (Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) 
+v0 (THead (Bind Abst) u0 t5))) t4 H24 t6 H15)))))))) H18)) H17))))) t2 H13)) 
+t H11 H12 H9 H10))) | (pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) 
+\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b) u1 
+t5)) t)).(\lambda (H14: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v3) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (_: (pr0 v0 
+v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H20 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H13) in (let H21 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | 
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 
+t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7) 
+\Rightarrow (match t7 with [(TSort _) \Rightarrow Abst | (TLRef _) 
+\Rightarrow Abst | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) 
+in ((let H23 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead 
+(Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow 
+t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 
+(THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u u1)).(\lambda (H26: (eq 
+B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst (\lambda (b0: B).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))) (let H28 
+\def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abst H26) in 
+(let H29 \def (match (H28 (refl_equal B Abst)) in False with []) in H29)) b 
+H26))))) H23)) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | 
+(pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead 
+(Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to 
+((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 
+u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) 
+\Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H18) in (False_ind (ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7))) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | 
+(pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) 
+u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind 
+b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b 
+Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 
+t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) 
+in (let H17 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift 
+(S O) O t5)) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))) t6 (sym_eq T t6 t2 
+H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
+(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
+(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: 
+T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: 
+T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) 
+t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) 
+u t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
+False])])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) 
+H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) 
+(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 
+u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 
+v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 
+t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda 
+(t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not 
+(eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 
+u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 with [(pr0_refl t5) 
+\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind 
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r 
+T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b) 
+u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) 
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17 
+\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6: 
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall 
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: 
+T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) 
+in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10 
+(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13) 
+H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T 
+(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T 
+(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda 
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 
+t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Flat Appl) 
+| (TLRef _) \Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H22 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead (Bind b) u1 t3) | 
+(TLRef _) \Rightarrow (THead (Bind b) u1 t3) | (THead _ _ t7) \Rightarrow 
+t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in 
+(\lambda (H24: (eq T v1 u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K 
+(Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
+k0 u3 t6) t7)))) (let H26 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 
+u0 t5) t)) H15 (Flat Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: 
+T).(pr0 t7 t6)) H19 (THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 
+with [(pr0_refl t7) \Rightarrow (\lambda (H28: (eq T t7 (THead (Bind b) u1 
+t3))).(\lambda (H29: (eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda 
+(t8: T).((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u3 t6) t9))))) (\lambda (H30: (eq T (THead (Bind b) u1 t3) 
+t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).(ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t8) t9)))) (let H31 \def 
+(eq_ind_r T t5 (\lambda (t8: T).(eq T (THead (Flat Appl) u0 t8) t)) H26 
+(THead (Bind b) u1 t3) H23) in (let H32 \def (eq_ind_r T t (\lambda (t8: 
+T).(\forall (v: T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall 
+(t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda 
+(t11: T).(pr0 t10 t11)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 
+t3)) H31) in (let H33 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H10 u0 
+H24) in (ex2_ind T (\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u3 
+t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 (THead 
+(Bind b) u1 t3)) t8))) (\lambda (x: T).(\lambda (H34: (pr0 v2 x)).(\lambda 
+(H35: (pr0 u3 x)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 
+t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 
+H34))))) (H32 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H33 
+u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead (Bind b) u1 t3) H28) H29))) | 
+(pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow (\lambda (H30: (eq T (THead k0 
+u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead k0 u5 t8) 
+t6)).((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) 
+(THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u4 | (TLRef _) 
+\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead 
+(Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u4 t7) (THead (Bind b) u1 t3) H30) in (eq_ind K 
+(Bind b) (\lambda (k1: K).((eq T u4 u1) \to ((eq T t7 t3) \to ((eq T (THead 
+k1 u5 t8) t6) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) (\lambda (H35: 
+(eq T u4 u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead 
+(Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) (\lambda 
+(H36: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead (Bind b) u5 
+t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to (ex2 T (\lambda (t10: T).(pr0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda 
+(t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T 
+(THead (Bind b) u5 t8) t6)).(eq_ind T (THead (Bind b) u5 t8) (\lambda (t9: 
+T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) (\lambda (H38: (pr0 u1 
+u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def (eq_ind_r T t5 (\lambda (t9: 
+T).(eq T (THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 t3) H23) in 
+(let H41 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to 
+(\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T 
+(\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H 
+(THead (Flat Appl) u0 (THead (Bind b) u1 t3)) H40) in (let H42 \def (eq_ind T 
+v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: 
+T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x: 
+T).(\lambda (H43: (pr0 v2 x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda 
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) 
+(\lambda (x0: T).(\lambda (H45: (pr0 t8 x0)).(\lambda (H46: (pr0 t4 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind 
+b) u5 t8)) t9))) (\lambda (x1: T).(\lambda (H47: (pr0 u5 x1)).(\lambda (H48: 
+(pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t8))) 
+(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x H44 H43 t8 t4 x0 H45 H46 
+u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead (Bind b) u1 t3) u1 (THead 
+(Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx (Bind b) u1 t3) 
+(tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38 u2 H11))))) (H41 
+t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) u0 (THead (Bind b) 
+u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind 
+b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx (Flat Appl) u0 (THead 
+(Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7 (sym_eq T t7 t3 H36))) u4 
+(sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b) H34))) H33)) H32)) H31 H28 
+H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29) \Rightarrow (\lambda (H30: (eq T 
+(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u1 
+t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H32 \def 
+(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T 
+(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) H32)) H31 
+H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31) \Rightarrow 
+(\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t7)) (THead 
+(Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead (Flat Appl) 
+(lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat Appl) v0 
+(THead (Bind b0) u4 t7)) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 
+with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead 
+(Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat 
+Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) 
+\to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29 H30 H31))) | 
+(pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31: (eq T (THead 
+(Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind 
+Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) 
+\Rightarrow t9])) (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in 
+((let H34 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) 
+(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow Abbr | 
+(TLRef _) \Rightarrow Abbr | (THead k0 _ _) \Rightarrow (match k0 with [(Bind 
+b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) 
+(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) 
+\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) 
+\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda 
+(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4 
+u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind 
+Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to 
+(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
+t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq 
+T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 
+O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat 
+Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w) 
+t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to 
+((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead 
+(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: 
+T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1 
+u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42 
+\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23 
+Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 
+Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T 
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr 
+H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat 
+Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def 
+(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: 
+T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: 
+T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat 
+Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1 
+(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 
+v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead 
+(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: 
+T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: 
+T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda 
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: 
+T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) 
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) 
+(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead 
+(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5 
+x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead 
+(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x 
+H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind 
+Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx 
+(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 
+H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat 
+Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) 
+(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12))))) 
+(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3 
+H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b 
+H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u) 
+\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead 
+(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t7) | (THead _ _ t9) \Rightarrow t9])) (THead (Bind 
+b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H33 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift 
+(S O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
+\Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) 
+O t7)) (THead (Bind b) u1 t3) H30) in (eq_ind B b (\lambda (b1: B).((eq T u 
+u1) \to ((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b1 
+Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u3 t6) t9))))))))) (\lambda (H35: (eq T u u1)).(eq_ind T u1 
+(\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T (lift (S O) O t7) 
+t3)).(eq_ind T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq 
+B b Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 
+(THead (Flat Appl) u3 t6) t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T 
+t6 (\lambda (t9: T).((not (eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda 
+(t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda 
+(H38: (not (eq B b Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def 
+(eq_ind_r T t3 (\lambda (t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift 
+(S O) O t7) H36) in (let H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T 
+(THead (Flat Appl) u0 t9) t)) H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) 
+in (let H42 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) 
+\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to 
+(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 
+t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) 
+H41) in (let H43 \def (eq_ind_r T t3 (\lambda (t9: T).(pr0 t9 t4)) H12 (lift 
+(S O) O t7) H36) in (ex2_ind T (\lambda (t9: T).(eq T t4 (lift (S O) O t9))) 
+(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead 
+(Flat Appl) u3 t6) t9))) (\lambda (x: T).(\lambda (H44: (eq T t4 (lift (S O) 
+O x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: 
+T).(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t9)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) 
+t10)))) (let H46 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) 
+in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
+t6) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) 
+(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) (lift (S O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 
+t6) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 
+x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x)))) 
+(pr0_confluence__pr0_cong_upsilon_zeta b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 
+x1 H49 H50))))) (H42 t7 (tlt_trans (THead (Bind b) u1 (lift (S O) O t7)) t7 
+(THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) (lift_tlt_dx 
+(Bind b) u1 t7 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift 
+(S O) O t7)))) x H45 t6 H39))))) (H42 u0 (tlt_head_sx (Flat Appl) u0 (THead 
+(Bind b) u1 (lift (S O) O t7))) v2 H46 u3 H18))) t4 H44)))) (pr0_gen_lift t7 
+t4 (S O) O H43)))))))) t8 (sym_eq T t8 t6 H37))) t3 H36)) u (sym_eq T u u1 
+H35))) b0 (sym_eq B b0 b H34))) H33)) H32)) H31 H28 H29))) | (pr0_tau t7 t8 
+H28 u) \Rightarrow (\lambda (H29: (eq T (THead (Flat Cast) u t7) (THead (Bind 
+b) u1 t3))).(\lambda (H30: (eq T t8 t6)).((let H31 \def (eq_ind T (THead 
+(Flat Cast) u t7) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+b) u1 t3) H29) in (False_ind ((eq T t8 t6) \to ((pr0 t7 t8) \to (ex2 T 
+(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) H31)) H30 
+H28)))]) in (H28 (refl_equal T (THead (Bind b) u1 t3)) (refl_equal T t6))))) 
+k H25)))) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) | (pr0_beta u v0 v3 
+H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v0 (THead 
+(Bind Abst) u t5)) t)).(\lambda (H16: (eq T (THead (Bind Abbr) v3 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T 
+(THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda 
+(t7: T).((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let 
+H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) 
+in (let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) 
+(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead 
+(Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) 
+\Rightarrow (match t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b 
+| (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
+(Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H23 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 with [(TSort _) 
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow 
+t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
+(THead (Bind Abst) u t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) 
+in (\lambda (_: (eq T u1 u)).(\lambda (H26: (eq B b Abst)).(\lambda (H27: (eq 
+T v1 v0)).(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt 
+v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let 
+H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H27) in (eq_ind_r 
+B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq 
+B b0 Abst))) H9 Abst H26) in (let H31 \def (match (H30 (refl_equal B Abst)) 
+in False with []) in H31)) b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 
+H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) 
+\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 
+t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S 
+O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O 
+v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) 
+(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) 
+t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) 
+(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: 
+(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda 
+(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead 
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) 
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
+in ((let H26 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
+\Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow 
+b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 
+(THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ _ t7) \Rightarrow (match t7 with [(TSort _) \Rightarrow u1 | (TLRef 
+_) \Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) 
+in ((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match 
+t7 with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) 
+\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead 
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 
+u0)).(\lambda (H30: (eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) 
+v0 (THead (Bind b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: 
+T).(pr0 t7 v2)) H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) 
+(lift (S O) O v3) t6)) t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not 
+(eq B b1 Abst))) H9 b0 H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: 
+T).(pr0 t7 u2)) H11 u0 H29) in (let H36 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t4)) H12 t5 H28) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) 
+(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 
+(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda 
+(x: T).(\lambda (H37: (pr0 t4 x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T 
+(\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) 
+O v3) t6)) t7))) (\lambda (x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: 
+(pr0 u3 x0)).(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 
+v3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead 
+(Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 
+v2 x1)).(\lambda (H42: (pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 
+H34 v2 v3 x1 H41 H42 u2 u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 
+(tlt_head_sx (Flat Appl) v0 (THead (Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 
+u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead (Bind 
+b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead 
+(Bind b0) u0 t5))) u2 H35 u3 H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 
+t5) t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) 
+u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 
+H23))))) b H30))))))) H27)) H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 
+H16))) | (pr0_delta u0 u3 H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq 
+T (THead (Bind Abbr) u0 t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T 
+(THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda 
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 
+u3)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t3)) t7)) H6 (THead (Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 
+H17 H13 H14 H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: 
+(eq T (THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 
+t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T 
+t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 
+(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 
+Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind 
+b0) u (lift (S O) O t5)) H15) in (let H21 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b0) u (lift (S O) O t5)) H20) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) H21))))) t6 (sym_eq T t6 t2 H17))) t H15 H16 
+H13 H14))) | (pr0_tau t5 t6 H13 u) \Rightarrow (\lambda (H14: (eq T (THead 
+(Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind T (THead (Flat 
+Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))) H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 
+(refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | 
+(pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead 
+(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) 
+t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to 
+(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) 
+(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 
+t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 
+t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda 
+(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 with [(pr0_refl t5) 
+\Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T t5 t2)).(eq_ind 
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead 
+(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H13: (eq T 
+t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T 
+(THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def 
+(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) 
+H5) in (let H16 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v 
+t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to 
+(ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H 
+(THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead 
+(Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) 
+(THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 
+H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) 
+H12))) | (pr0_comp u0 u3 H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T 
+(THead k u0 t5) t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T 
+(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) 
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) 
+t2)).(eq_ind T (THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 
+t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 
+t3) t7)) H5 (THead k u0 t5) H13) in (let H19 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow 
+(Bind Abbr) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) 
+(THead k u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5) H18) in ((let H21 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead k u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda 
+(H23: (eq K (Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 
+(THead k0 u3 t6) t7)))) (let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T 
+(THead k0 u0 t5) t)) H13 (Bind Abbr) H23) in (let H25 \def (eq_ind_r T t 
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) 
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in 
+(let H26 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let 
+H27 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T 
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda 
+(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u3 t6) t7))) (\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: 
+(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 
+u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 
+t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w 
+H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) 
+u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6 
+H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta 
+u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: 
+T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) 
+u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let 
+H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in 
+(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13 
+H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14) 
+\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 
+t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S 
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 
+t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead 
+(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) 
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15 
+H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow 
+(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T 
+(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda 
+(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) 
+\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead 
+(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: 
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda 
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) 
+(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: 
+(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | 
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in ((let H22 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) 
+(THead (Bind Abbr) u0 t5) H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) 
+u0 t5) H14) in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 
+H23) in (let H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) 
+in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 
+t4 x)).(\lambda (H28: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) 
+(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) 
+u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda 
+(x0: T).(\lambda (H29: (pr0 u2 x0)).(\lambda (H30: (pr0 u3 
+x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19 x0 H29 H30 x 
+H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H25 u3 H17))))) (H24 
+t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H18))))))) H21)))))) t2 H16)) t 
+H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda 
+(H13: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 
+t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 
+t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) 
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b 
+Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: 
+T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O 
+t5)) H13) in (let H19 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let 
+H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 
+| (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind 
+Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18) in ((let H21 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | (TLRef 
+_) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
+t3) (THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1 
+u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0: 
+B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b 
+(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr 
+H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def 
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind 
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T 
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7)) 
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: 
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O 
+x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7: 
+T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7: 
+T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: 
+T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 
+x0)).(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H31 x (pr0_refl 
+(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 
+t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6 
+(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_tau t5 t6 H11 u) 
+\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: 
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 
+w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
+(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17 
+\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17)))) 
+t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t) 
+(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3 
+u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3)) 
+t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O 
+t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) 
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) 
+(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b 
+Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda 
+(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: 
+(pr0 t3 t1)).(let H9 \def (match H1 with [(pr0_refl t5) \Rightarrow (\lambda 
+(H9: (eq T t5 t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: 
+T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: 
+T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: 
+T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let 
+H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u 
+(lift (S O) O t3)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
+t6 t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 
+(THead (Bind b) u (lift (S O) O t3)) H4) in (let H14 \def (eq_ind_r T t 
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) 
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) 
+H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 
+(THead (Bind b) u (lift (S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 
+t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | 
+(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 
+t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) 
+(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) 
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda 
+(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def 
+(f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow (Bind b) | 
+(TLRef _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead 
+(Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
+O) O t3)) (THead k u1 t5) H16) in ((let H19 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x 
+(S O))) O t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S 
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K 
+(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 
+t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r 
+K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23 
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19) 
+in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: 
+T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 
+(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S 
+O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda 
+(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let 
+H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to 
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T 
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H 
+(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x) 
+(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
+(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) 
+(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: 
+T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift 
+(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx 
+(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O) 
+O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta 
+u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 
+(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) 
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: 
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) 
+(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind 
+Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda 
+(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b) 
+u (lift (S O) O t3)) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) 
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 
+H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead 
+(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead 
+(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not 
+(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) 
+u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 
+(THead (Bind b0) u1 t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t6)) t7))) H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta 
+u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) 
+u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T 
+(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda 
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T 
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda 
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T 
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: 
+(pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 
+w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u 
+(lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) in (let H19 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead (Bind Abbr) u1 t5) H18) in ((let H20 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
+(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead 
+(Bind Abbr) u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O 
+t3) | (TLRef _) \Rightarrow (lref_map (\lambda (x: nat).(plus x (S O))) O t3) 
+| (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) 
+(THead (Bind Abbr) u1 t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: 
+(eq B b Abbr)).(let H24 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 
+(lift (S O) O t3) H21) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O 
+t7))) (\lambda (t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda 
+(H25: (eq T t6 (lift (S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def 
+(eq_ind_r T t5 (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift 
+(S O) O t3) H21) in (let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: 
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v 
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 
+\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) 
+H25) in (let H30 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 
+Abbr H23) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 
+t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind 
+Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 
+t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) 
+(pr0_confluence__pr0_delta_tau u2 (lift (S O) O x) w H29 x (pr0_refl (lift (S 
+O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x H26 t1 
+H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2 H14)) t 
+H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda 
+(H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T 
+t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq 
+T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) 
+t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H17 \def (f_equal 
+T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef _) 
+\Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T 
+T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S 
+O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H19 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) \Rightarrow t7])) 
+(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) 
+H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b b0)).(let H22 \def 
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: 
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: 
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) 
+u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3 (\lambda (t7: 
+T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24 \def (eq_ind 
+B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in (ex2_ind T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: 
+(pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26)))) (H22 t5 (lift_tlt_dx (Bind 
+b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18)) H17))))) t6 (sym_eq T t6 t2 
+H13))) t H11 H12 H9 H10))) | (pr0_tau t5 t6 H9 u0) \Rightarrow (\lambda (H10: 
+(eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T 
+(THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) 
+(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T 
+(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) 
+in (let H15 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat 
+_) \Rightarrow False])])) I (THead (Flat Cast) u0 t5) H14) in (False_ind (ex2 
+T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 
+(sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal 
+T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5 H2 H3))) | (pr0_tau t3 t4 H2 u) 
+\Rightarrow (\lambda (H3: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: 
+(eq T t4 t1)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 
+t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
+T).(pr0 t2 t6)))))) (\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: 
+T).((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: 
+T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 with 
+[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5 
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: 
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t 
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6: 
+T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda 
+(t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 
+t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def (eq_ind_r T t (\lambda 
+(t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to 
+(\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) 
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u t3) H3) in 
+(ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Flat 
+Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_tau t3 t1 H6 u)))) t2 H10)) t (sym_eq 
+T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 t5 t6 H8 k) 
+\Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda (H10: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead 
+k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T 
+(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: 
+T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5 
+t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
+t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (Flat Cast) | (TLRef _) \Rightarrow 
+(Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Cast) u t3) 
+(THead k u1 t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) 
+\Rightarrow t7])) (THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t3 | 
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat 
+Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u u1)).(\lambda (H19: 
+(eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0: K).(ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H20 
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9 (Flat Cast) 
+H19) in (let H21 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v 
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to 
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 
+t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def (eq_ind T t3 
+(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 
+t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x: 
+T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6 x)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) 
+t7)) x H23 (pr0_tau t6 x H24 u2))))) (H21 t5 (tlt_head_dx (Flat Cast) u1 t5) 
+t1 H22 t6 H13))))) k H19)))) H16)) H15))))) t2 H11)) t H9 H10 H7 H8))) | 
+(pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq T (THead (Bind 
+Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) 
+(\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
+t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T 
+(THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) 
+(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def (eq_ind_r T 
+t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind 
+Abst) u0 t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 
+H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow 
+(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
+t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) 
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to 
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 
+t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) 
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) 
+(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) 
+u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11) in (let H19 
+\def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
+(Flat Appl) v1 (THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda 
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 
+H10))) | (pr0_delta u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T 
+(THead (Bind Abbr) u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) 
+t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) 
+(\lambda (H12: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 
+t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 
+t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: 
+(subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead 
+(Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def 
+(eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Abbr) u1 t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) 
+t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: 
+(eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 
+t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T 
+t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to 
+(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) 
+(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def 
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead 
+(Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat 
+Cast) u t3) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 
+(lift (S O) O t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) 
+(\lambda (t7: T).(pr0 t2 t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 
+H8))) | (pr0_tau t5 t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
+Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 
+t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: 
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 
+t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u 
+t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) 
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) 
+(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | 
+(THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) 
+u0 t5) H12) in (\lambda (_: (eq T u u0)).(let H16 \def (eq_ind_r T t (\lambda 
+(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to 
+(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) 
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in 
+(let H17 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H14) in 
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: 
+T).(\lambda (H18: (pr0 t1 x)).(\lambda (H19: (pr0 t2 x)).(ex_intro2 T 
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H18 H19)))) 
+(H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17 t2 H11)))))) H13)))) t6 
+(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T 
+t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) 
+(refl_equal T t1))))))))) t0).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/props.ma
new file mode 100644 (file)
index 0000000..ea8ea68
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/props.ma
@@ -0,0 +1,534 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/fwd.ma".
+
+include "basic_1A/subst0/props.ma".
+
+lemma pr0_lift:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall 
+(d: nat).(pr0 (lift h d t1) (lift h d t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) 
+(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: 
+nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 
+(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 
+(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda 
+(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: 
+T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) 
+(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k 
+d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) 
+t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) 
+(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h 
+d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h 
+d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
+(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead 
+(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u 
+t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r 
+T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s 
+(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) 
+(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h 
+d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) 
+(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s 
+(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) 
+u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) 
+d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d 
+(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s 
+(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s 
+(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) 
+(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda 
+(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
+nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: 
+nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: 
+nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: 
+nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) 
+(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) 
+(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) 
+(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead 
+(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead 
+(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) 
+t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O 
+v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead 
+(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift 
+h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) 
+(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h 
+d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d 
+u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) 
+(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat 
+Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) 
+t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) 
+d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d 
+u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) 
+t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d 
+v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) 
+d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S 
+O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) 
+(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) 
+h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) 
+(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 
+t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) 
+(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda 
+(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift 
+h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) 
+u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) 
+d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind 
+Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S 
+d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in 
+(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) 
+(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d 
+(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (le_antisym d 
+d (le_n d) (le_n d)) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind 
+Abbr) u2 w)) (lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) 
+u1 t3)) (lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: 
+nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: 
+nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s 
+(Bind b) d) (lift (S O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) 
+(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d 
+u) (lift h n (lift (S O) O t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O 
+(lift h d t3)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d 
+t4))) (pr0_zeta b H0 (lift h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift 
+h (plus (S O) d) (lift (S O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) 
+(refl_equal nat (S d))) (lift h d (THead (Bind b) u (lift (S O) O t3))) 
+(lift_head (Bind b) u (lift (S O) O t3) h d))))))))))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (H1: ((\forall (h: 
+nat).(\forall (d: nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (u: 
+T).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Flat Cast) (lift h 
+d u) (lift h (s (Flat Cast) d) t3)) (\lambda (t: T).(pr0 t (lift h d t4))) 
+(pr0_tau (lift h (s (Flat Cast) d) t3) (lift h d t4) (H1 h d) (lift h d u)) 
+(lift h d (THead (Flat Cast) u t3)) (lift_head (Flat Cast) u t3 h d))))))))) 
+t1 t2 H))).
+
+lemma pr0_gen_abbr:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) 
+(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda 
+(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S 
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead 
+(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 
+t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind 
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
+Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H5) in ((let H7 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Abbr) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H5) 
+in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind Abbr))).(eq_ind_r 
+K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let H11 \def (eq_ind T 
+t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 t1 H8) in (let 
+H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def 
+(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 
+u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
+(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind 
+Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
+t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind 
+Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O 
+v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
+t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 
+t1) H8) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind 
+Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) 
+(\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: 
+(((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) 
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u0 t0) 
+(THead (Bind Abbr) u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) 
+u1 t1) H6) in (\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda 
+(t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T 
+t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 
+(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: 
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 u1 H9) in (let 
+H13 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or_introl 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S 
+O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda 
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or 
+(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
+u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind Abbr) u2 w)) H13 (or_intror 
+(pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 
+y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
+u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: 
+(not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (H3: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: 
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: 
+T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
+Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) 
+\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in 
+((let H6 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
+(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in ((let 
+H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+(lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow 
+(lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 
+(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) 
+(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0: 
+T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) 
+(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
+H4)))))))) y x H0))) H)))).
+
+lemma pr0_gen_void:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead 
+(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) 
+O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda 
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda 
+(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void) 
+u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq 
+T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead 
+(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 
+t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda 
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal 
+T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind 
+Void) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
+\Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: 
+(eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda (k0: K).(or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Void) 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) 
+(let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))))) H4 t1 
+H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in 
+(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead 
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O u2))))) H2 u1 
+H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
+(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind 
+Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+(lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) u2 
+t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead 
+(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B 
+b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
+(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T 
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow True | 
+Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind 
+Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) 
+u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda 
+(H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T 
+(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1))).(let H5 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b | (TLRef 
+_) \Rightarrow b | (THead k _ _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead 
+_ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind 
+Void) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(TLRef _) \Rightarrow (lref_map (\lambda (x0: nat).(plus x0 (S O))) O t0) | 
+(THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead 
+(Bind Void) u1 t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b 
+Void)).(let H10 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 
+Void H9) in (let H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead 
+(Bind Void) u1 t)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O 
+t2))))) H3 (lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: 
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) 
+(lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) 
+(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: 
+(((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(let H4 \def (eq_ind T 
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind Void) u1 t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x H0))) H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst0.ma
new file mode 100644 (file)
index 0000000..93ca259
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst0.ma
@@ -0,0 +1,1647 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/props.ma".
+
+include "basic_1A/subst0/subst0.ma".
+
+lemma pr0_subst0_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T 
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) 
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 
+v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
+T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) 
+(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 
+(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 
+H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: 
+T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T 
+(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t 
+t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind 
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t 
+(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t 
+(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 
+t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 
+H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+lemma pr0_subst0_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T 
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) 
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v 
+u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: 
+T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) 
+(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: 
+((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) 
+(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 
+u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: 
+T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T 
+(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 
+(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 
+x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda 
+(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to 
+(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 
+t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind 
+T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 
+(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 
+x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 
+(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) 
+(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: 
+T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: 
+T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda 
+(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T 
+(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T 
+(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead 
+k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 
+x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) 
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 
+t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda 
+(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) 
+t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 
+t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
+(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
+v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t2 w2))))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 
+w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
+nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 
+v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd 
+v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: 
+(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 
+w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
+t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: 
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: 
+nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: 
+T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: 
+T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k 
+u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 
+t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
+T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 
+(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) 
+(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) 
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) 
+(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda 
+(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 
+(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind 
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 
+(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: 
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+(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 
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+(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind 
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+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
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+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
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+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
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+(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
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+(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) 
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+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 
+t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind 
+T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
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+(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: 
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+(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead 
+(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 
+H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
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+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) 
+i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda 
+(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind 
+Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 
+i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) 
+w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) 
+x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T 
+x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) 
+x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: 
+(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T 
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s 
+(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
+(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or 
+(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 
+(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) 
+(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 
+v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) 
+H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 
+v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead 
+(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat 
+Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 
+t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 
+x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) 
+x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) 
+(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: 
+T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T 
+(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: 
+T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 
+(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u 
+u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: 
+(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: 
+T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 
+x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 
+H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
+x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) 
+x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) 
+(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind 
+Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: 
+(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind 
+Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead 
+(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) 
+v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda 
+(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) 
+(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 
+t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind 
+Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s 
+(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: 
+(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind 
+Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) 
+(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) 
+(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 
+v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 
+H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 
+t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) 
+x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: 
+(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T 
+x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) 
+x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) 
+w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda 
+(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: 
+(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
+(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
+(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 
+x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) 
+(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) 
+(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: 
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+t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 
+(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda 
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+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x 
+H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead 
+(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 
+t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: 
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+Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
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+H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i 
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+i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda 
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+(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda 
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+(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) 
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+(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat 
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+t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) 
+u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind 
+T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) 
+(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat 
+Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 
+x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: 
+T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x 
+v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind 
+b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda 
+(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 
+x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) 
+u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T 
+(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq 
+T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 
+(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead 
+(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 
+T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 
+(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T 
+(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s 
+(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind 
+b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq 
+T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 
+x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 
+t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 
+w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 
+u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 
+u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) 
+x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda 
+(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
+b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 
+H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S 
+O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 
+H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x 
+(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) 
+i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead 
+(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead 
+(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) 
+(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 
+(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 
+x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) 
+(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind 
+b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 
+x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 
+x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) 
+(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 
+(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s 
+(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind 
+b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 
+u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) 
+v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 
+x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: 
+(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x 
+(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 
+x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
+(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) 
+x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 
+H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 
+w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead 
+(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 
+(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda 
+(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 
+x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind 
+b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst 
+v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) 
+(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
+T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 
+x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
+(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat 
+Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 
+H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S 
+O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat 
+Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead 
+(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 
+x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 
+x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 
+(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
+v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) 
+(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 
+(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S 
+O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) 
+O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 
+(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) 
+(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda 
+(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) 
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead 
+(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) 
+u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 
+x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat 
+Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq 
+T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 
+u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda 
+(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T 
+(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 
+(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead 
+(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or 
+(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda 
+(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) 
+i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat 
+Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or 
+(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: 
+(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) 
+x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 
+t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
+t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda 
+(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 
+H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead 
+(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S 
+O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 
+(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x 
+t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda 
+(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) 
+x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
+v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst 
+v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
+(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift 
+(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both 
+v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat 
+Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O 
+v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat 
+Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) 
+i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T 
+x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) 
+u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) 
+(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: 
+T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s 
+(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: 
+T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in 
+(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or 
+(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 
+x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: 
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) 
+w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x 
+x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind 
+(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i 
+v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 
+(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) 
+u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 
+H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) 
+O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) 
+v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 
+x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s 
+(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 
+x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) 
+(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda 
+(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead 
+(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) 
+v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat 
+Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind 
+b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda 
+(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind 
+T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) 
+x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) 
+(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) 
+O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind 
+(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s 
+(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
+x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) 
+(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) 
+i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat 
+Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead 
+(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 
+v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 
+T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 
+x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd 
+(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift 
+(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind 
+b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) 
+u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: 
+T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s 
+(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 
+x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) 
+(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S 
+O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda 
+(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda 
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead 
+(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 
+(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 
+v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
+T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) 
+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
+b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift 
+(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst 
+v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda 
+(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 
+v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i 
+v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) 
+(\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda (H24: (subst0 i v3 v2 
+x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: 
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+w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind 
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+(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 
+x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind 
+Abbr) i) v1 t3 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or 
+(pr0 t (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda 
+(w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))))) (or_ind (pr0 x1 t4) 
+(ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) 
+i) v2 t4 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) 
+(ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H11: (pr0 x1 
+t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: 
+(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 
+w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: 
+T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 
+w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: 
+T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda 
+(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: 
+T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T 
+(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 
+w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 
+O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def 
+(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in 
+(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w 
+x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x 
+H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) 
+(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 
+H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: 
+T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 
+w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: 
+(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: 
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind 
+Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 
+w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 
+x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x 
+x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 
+(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead 
+(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) 
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) 
+(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd 
+(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind 
+Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: 
+(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
+u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 
+x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O 
+x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead 
+(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind 
+Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 
+x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal 
+nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 
+\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S 
+i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) 
+(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead 
+(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) 
+w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda 
+(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: 
+(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind 
+Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 
+i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 
+H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 
+(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) 
+(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S 
+i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i 
+H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) 
+(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: 
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
+(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift 
+(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
+(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) 
+u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or 
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) 
+u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda 
+(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
+(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u 
+x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda 
+(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda 
+(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) 
+i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: 
+(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift 
+(S O) O t3) x)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: 
+(lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x (S O) O (s (Bind b) 
+i) (le_O_n (s (Bind b) i)) H8 H7 (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 
+w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) (\lambda (Hle: (le (S O) (s 
+(Bind b) i))).(let H_x \def (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S O) 
+O H7 Hle) in (let H8 \def H_x in (ex2_ind T (\lambda (t5: T).(eq T x (lift (S 
+O) O t5))) (\lambda (t5: T).(subst0 (minus i O) v1 t3 t5)) (or (pr0 w1 t4) 
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) O x0))).(\lambda (H10: 
+(subst0 (minus i O) v1 t3 x0)).(let H11 \def (eq_ind T x (\lambda (t: T).(eq 
+T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H9) in (eq_ind_r T (THead 
+(Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda 
+(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (let H12 \def 
+(eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3 x0)) H10 i 
+(minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u (lift (S O) 
+O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H13: (pr0 x0 
+t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H13 u))) (\lambda (H13: (ex2 T 
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 
+t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 
+x1)).(\lambda (H15: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u 
+(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift 
+(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H14 u) H15))))) H13)) (H2 v1 
+x0 i H12 v2 H4))) w1 H11))))) H8)))))))) H5)) (\lambda (H5: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) 
+v1 (lift (S O) O t3) x1)).(lt_le_e (s (Bind b) i) (S O) (or (pr0 w1 t4) (ex2 
+T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (H9: (lt (s (Bind b) i) (S O))).(subst0_gen_lift_false t3 v1 x1 (S 
+O) O (s (Bind b) i) (le_O_n (s (Bind b) i)) H9 H8 (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))) 
+(\lambda (Hle: (le (S O) (s (Bind b) i))).(let H_x \def (subst0_gen_lift_ge 
+v1 t3 x1 (s (Bind b) i) (S O) O H8 Hle) in (let H9 \def H_x in (ex2_ind T 
+(\lambda (t5: T).(eq T x1 (lift (S O) O t5))) (\lambda (t5: T).(subst0 (minus 
+i O) v1 t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H10: (eq T x1 (lift 
+(S O) O x))).(\lambda (H11: (subst0 (minus i O) v1 t3 x)).(let H12 \def 
+(eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift (S O) 
+O x) H10) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda (t: 
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (let H13 \def (eq_ind_r nat (minus i O) (\lambda 
+(n: nat).(subst0 n v1 t3 x)) H11 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or 
+(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2)))) (\lambda (H14: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S 
+O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O 
+x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H14 x0))) 
+(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 
+i v2 t4 w2)) (or (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x 
+x2)).(\lambda (H16: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0 
+(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift 
+(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda 
+(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H15 x0) H16))))) H14)) (H2 v1 
+x i H13 v2 H4))) w1 H12))))) H9)))))))))) H5)) (subst0_gen_head (Bind b) v1 u 
+(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: 
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) 
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda 
+(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) 
+w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda 
+(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u 
+u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda 
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda 
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: 
+(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: 
+T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat 
+Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T 
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda 
+(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: 
+T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))))) (or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2))) (pr0_tau t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: 
+(ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T 
+w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 
+t3 t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat 
+Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T 
+(THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: 
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) 
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) 
+i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) 
+(\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T 
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i 
+v2 t4 w2))) (pr0_tau x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 
+x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T 
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 
+w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead 
+(Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
+T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 
+x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) 
+(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: 
+T).(subst0 i v2 t4 w2)) x0 (pr0_tau x x0 H8 u) H9))))) H7)) (H1 v1 x (s (Flat 
+Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: 
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) 
+i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda 
+(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 
+x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda 
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 
+(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) 
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_tau x1 t4 H8 x0))) 
+(\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 
+(s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) 
+(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat 
+Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) 
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (pr0 x1 
+x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror (pr0 (THead 
+(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) 
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: 
+T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) 
+x (pr0_tau x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat Cast) i) H7 v2 H3)) 
+w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 i H2))))))))))))) t1 
+t2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst1.ma
new file mode 100644 (file)
index 0000000..4026d15
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr0/subst1.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/subst0.ma".
+
+include "basic_1A/subst1/fwd.ma".
+
+lemma pr0_delta1:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
+(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead 
+(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: 
+(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind 
+Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind 
+Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H 
+t1 t2 H0 t0 H2))) w H1)))))))).
+
+lemma pr0_subst1_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
+T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
+(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda 
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda 
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x 
+H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+lemma pr0_subst1_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
+T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
+(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda 
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda 
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x 
+H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+theorem pr0_subst1:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
+(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 
+v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 
+w2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: 
+T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 
+w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to 
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) 
+(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 
+t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) 
+(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda 
+(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 
+T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 
+(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 
+w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 
+w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: 
+(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: 
+T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i 
+v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr1/defs.ma
new file mode 100644 (file)
index 0000000..cdd942d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr1/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/defs.ma".
+
+inductive pr1: T \to (T \to Prop) \def
+| pr1_refl: \forall (t: T).(pr1 t t)
+| pr1_sing: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
+T).((pr1 t2 t3) \to (pr1 t1 t3))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr1/fwd.ma
new file mode 100644 (file)
index 0000000..1dd22f1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr1/fwd.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr1/defs.ma".
+
+implied rec lemma pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t 
+t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: 
+T).((pr1 t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t 
+t0) on p: P t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | 
+(pr1_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr1_ind P f f0) t2 
+t3 p1))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr1/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr1/pr1.ma
new file mode 100644 (file)
index 0000000..8459014
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr1/pr1.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr1/props.ma".
+
+include "basic_1A/pr0/pr0.ma".
+
+lemma pr1_strip:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda 
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: 
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda 
+(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) 
+(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_refl t2))))) (\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda 
+(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T 
+(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: 
+T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda 
+(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
+t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let 
+H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: 
+T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) 
+(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
+H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3 
+t2 H0)))))))))) t0 t1 H))).
+
+theorem pr1_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda 
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: 
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda 
+(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) 
+(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_refl t2))))) (\lambda (t2: 
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: 
+(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda 
+(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda 
+(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def 
+H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t)) 
+(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x: 
+T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x 
+H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda 
+(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 
+t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 
+H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr1/props.ma
new file mode 100644 (file)
index 0000000..496e650
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr1/props.ma
@@ -0,0 +1,108 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr1/fwd.ma".
+
+include "basic_1A/pr0/subst1.ma".
+
+include "basic_1A/subst1/props.ma".
+
+include "basic_1A/T/props.ma".
+
+lemma pr1_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_sing t2 t1 H 
+t2 (pr1_refl t2)))).
+
+theorem pr1_t:
+ \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 
+t3) \to (pr1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) 
+(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda 
+(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_sing t0 t3 H0 t5 (H2 
+t5 H3)))))))))) t1 t2 H))).
+
+lemma pr1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall 
+(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: 
+T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k 
+t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_refl (THead k t0 t))) (\lambda 
+(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda 
+(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_sing 
+(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k 
+t3 t) H2))))))) u1 u2 H))))).
+
+lemma pr1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall 
+(k: K).(pr1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: 
+T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u 
+t) (THead k u t0)))) (\lambda (t: T).(pr1_refl (THead k u t))) (\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_sing 
+(THead k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k 
+u t4) H2))))))) t1 t2 H))))).
+
+theorem pr1_comp:
+ \forall (v: T).(\forall (w: T).((pr1 v w) \to (\forall (t: T).(\forall (u: 
+T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k v t) (THead k w u))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (H: (pr1 v w)).(pr1_ind (\lambda (t: 
+T).(\lambda (t0: T).(\forall (t1: T).(\forall (u: T).((pr1 t1 u) \to (\forall 
+(k: K).(pr1 (THead k t t1) (THead k t0 u)))))))) (\lambda (t: T).(\lambda 
+(t0: T).(\lambda (u: T).(\lambda (H0: (pr1 t0 u)).(\lambda (k: K).(pr1_head_2 
+t0 u H0 t k)))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 
+t2)).(\lambda (t3: T).(\lambda (H1: (pr1 t2 t3)).(\lambda (_: ((\forall (t: 
+T).(\forall (u: T).((pr1 t u) \to (\forall (k: K).(pr1 (THead k t2 t) (THead 
+k t3 u)))))))).(\lambda (t: T).(\lambda (u: T).(\lambda (H3: (pr1 t 
+u)).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t4: T).(pr1 (THead k 
+t1 t0) (THead k t3 t4)))) (\lambda (t0: T).(pr1_head_1 t1 t3 (pr1_sing t2 t1 
+H0 t3 H1) t0 k)) (\lambda (t0: T).(\lambda (t4: T).(\lambda (H4: (pr0 t4 
+t0)).(\lambda (t5: T).(\lambda (_: (pr1 t0 t5)).(\lambda (H6: (pr1 (THead k 
+t1 t0) (THead k t3 t5))).(pr1_sing (THead k t1 t0) (THead k t1 t4) (pr0_comp 
+t1 t1 (pr0_refl t1) t4 t0 H4 k) (THead k t3 t5) H6))))))) t u H3))))))))))) v 
+w H))).
+
+lemma pr1_eta:
+ \forall (w: T).(\forall (u: T).(let t \def (THead (Bind Abst) w u) in 
+(\forall (v: T).((pr1 v w) \to (pr1 (THead (Bind Abst) v (THead (Flat Appl) 
+(TLRef O) (lift (S O) O t))) t)))))
+\def
+ \lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind Abst) w u) in 
+(\lambda (v: T).(\lambda (H: (pr1 v w)).(eq_ind_r T (THead (Bind Abst) (lift 
+(S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr1 (THead (Bind Abst) v 
+(THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w u))) (pr1_comp v w H 
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) 
+(S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) 
+(THead (Flat Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) 
+(S O) u))) (pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef 
+O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) 
+u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind 
+Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) 
+(pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl 
+(lift (S O) (S O) u)) (lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u 
+(pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr 
+not_abbr_abst u u (pr0_refl u) (TLRef O))))) (Bind Abst)) (lift (S O) O 
+(THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/clen.ma
new file mode 100644 (file)
index 0000000..7c9747e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/clen.ma
@@ -0,0 +1,151 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/props.ma".
+
+include "basic_1A/clen/getl.ma".
+
+lemma pr2_gen_ctail:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
+(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 
+(clen c) u t t2)))))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) 
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T 
+(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda 
+(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind 
+Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 
+t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) 
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) 
+(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 
+t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda 
+(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let 
+H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind 
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c 
+(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
+(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d 
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) 
+u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: 
+C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k 
+u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c 
+t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 
+t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 
+H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen 
+c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq 
+nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: 
+nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 
+T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: 
+(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T 
+u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda 
+(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 
+(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind 
+Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda 
+(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 
+(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 
+t2 H0))) H)))))).
+
+lemma pr2_gen_cbind:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
+(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) 
+(THead (Bind b) v t2)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c 
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead 
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y 
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v 
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead 
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) 
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S 
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead 
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) 
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead 
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow 
+c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in ((let H11 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in 
+((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) v) H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C 
+d c)).(let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O 
+H8) in (let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v 
+H12) in (eq_ind B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead 
+(Bind b0) v t))) (pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) 
+(pr0_delta v v (pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) 
+(\lambda (H7: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: 
+nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq 
+nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c 
+(THead (Bind b) v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: 
+(eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 
+\def (f_equal nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def 
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in 
+(pr2_head_2 c v t3 t (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) 
+(getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind 
+Abbr) u) x H9) t3 t4 H2 t H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) 
+H)))))).
+
+lemma pr2_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
+(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c 
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) 
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) 
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) 
+H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/defs.ma
new file mode 100644 (file)
index 0000000..1683cd2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/defs.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/defs.ma".
+
+include "basic_1A/getl/defs.ma".
+
+inductive pr2: C \to (T \to (T \to Prop)) \def
+| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
+(pr2 c t1 t2))))
+| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: 
+T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 
+t)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/fwd.ma
new file mode 100644 (file)
index 0000000..7f8553f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/fwd.ma
@@ -0,0 +1,2801 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/defs.ma".
+
+include "basic_1A/pr0/fwd.ma".
+
+include "basic_1A/getl/clear.ma".
+
+include "basic_1A/getl/drop.ma".
+
+implied lemma pr2_ind:
+ \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c: 
+C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
+(Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
+(\forall (t: T).((subst0 i u t2 t) \to (P c t1 t)))))))))))) \to (\forall (c: 
+C).(\forall (t: T).(\forall (t0: T).((pr2 c t t0) \to (P c t t0)))))))
+\def
+ \lambda (P: ((C \to (T \to (T \to Prop))))).(\lambda (f: ((\forall (c: 
+C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 
+t2))))))).(\lambda (f0: ((\forall (c: C).(\forall (d: C).(\forall (u: 
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to 
+(P c t1 t))))))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2) 
+\Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8) 
+\Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))).
+
+lemma pr2_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to 
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort 
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda 
+(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 
+t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 
+t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda 
+(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t: 
+T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) 
+(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl 
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda 
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 
+t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0)) 
+(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) 
+(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) 
+t1 H4))))))))))))) c y x H0))) H)))).
+
+lemma pr2_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to 
+(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c 
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S 
+n) O u)))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef 
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t: 
+T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead 
+d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O 
+u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u)))))))))) 
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 
+t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda 
+(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t: 
+T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S 
+n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) 
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) 
+(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: 
+T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
+T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref 
+t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda 
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef 
+n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4) 
+in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda 
+(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_: 
+C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2 
+(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in 
+(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) 
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) 
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) 
+(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O 
+u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) 
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) 
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let 
+H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) 
+u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T 
+(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) 
+(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) 
+(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind 
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S 
+n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) 
+(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
+
+lemma pr2_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
+T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in 
+(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1 
+x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) 
+(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1 
+(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b: 
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4)))))) 
+(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda 
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind 
+Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 
+(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead 
+(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1 
+x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead 
+(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead 
+(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda 
+(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind 
+Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2 
+x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2 
+x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d 
+u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 
+(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: 
+T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst) 
+x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0 
+x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead 
+(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1 
+x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2 
+H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind 
+Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 
+t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t 
+(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13: 
+(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3) 
+(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3 
+(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2 
+H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 
+H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) 
+(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+lemma pr2_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c 
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 
+t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0)))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 
+t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind 
+T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2)) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast) 
+x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T 
+(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead 
+(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2 
+H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2 
+H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5 
+\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1) 
+H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast) 
+x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def 
+(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1) 
+H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
+(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead 
+(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2: 
+T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1))) 
+(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 
+c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
+x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12 
+(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11)) 
+(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda 
+(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat 
+Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) 
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda 
+(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12 
+(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11)) 
+(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2 
+x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat 
+Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 
+H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11)) 
+(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
+t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
+(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 
+H5)))))))))))))) c y x H0))) H))))).
+
+lemma pr2_gen_csort:
+ \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) 
+\to (pr0 t1 t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort 
+n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_: 
+C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind 
+(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0 
+t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 
+t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c 
+(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d 
+(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) 
+u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
+
+lemma pr2_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead 
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 
+t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T 
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3 
+\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1) 
+H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1 
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) 
+(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) 
+x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0 
+H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2 
+(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1 
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
+x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
+(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b: 
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2 
+H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not 
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) 
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat 
+Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift 
+(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 
+t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0 
+Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T 
+t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda 
+(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2 
+x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) 
+x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T 
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
+z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0 
+u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 
+H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3)))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
+t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T 
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
+T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2: 
+T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) 
+(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t 
+(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T 
+(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O 
+u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i 
+H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11: 
+(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3: 
+T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t 
+(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
+(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat 
+Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: 
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat 
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) 
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat 
+Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) 
+u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) 
+x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t 
+H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u 
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13: 
+(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1 
+x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) 
+x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 
+H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i 
+H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda 
+(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2 
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in 
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 
+u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda 
+(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead 
+(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 
+i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda 
+(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2 
+x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) 
+(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3 
+H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t 
+(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) 
+(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr) 
+x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T 
+(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) 
+(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
+(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d 
+(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) 
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t 
+(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15: 
+(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) 
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) 
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) 
+(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4 
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u 
+(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) 
+(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12)) 
+(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda 
+(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) 
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda 
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T 
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda 
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind 
+x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) 
+(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 
+x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3: 
+T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O 
+x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) 
+x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t 
+(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead 
+(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) 
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead 
+(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: 
+T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0) 
+u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u 
+x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
+(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) 
+x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
+O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead 
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
+(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda 
+(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3: 
+T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda 
+(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda 
+(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0) 
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind 
+x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) 
+x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat 
+Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
+u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) 
+t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
+O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
+x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T 
+(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
+(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
+x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift 
+(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
+x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H20: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H20 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
+(S O) O H19 Hle) in (let H20 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H21: (eq T x7 (lift (S O) O 
+x8))).(\lambda (H22: (subst0 (minus i O) u x3 x8)).(eq_ind_r T (lift (S O) O 
+x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H23 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x8)) H22 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
+(S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x8 H23) (pr2_free c0 x1 x4 H11) (pr2_free 
+(CHead c0 (Bind x0) x4) x2 x5 H12)))) x7 H21)))) H20))))) x6 H18)))) H17)) 
+(\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S 
+O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O 
+x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 
+x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
+x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: (subst0 (s (Flat 
+Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O 
+x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0 
+u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S 
+i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) 
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda 
+(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u 
+(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat 
+Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda 
+(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat 
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0 
+(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat 
+Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) 
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H21 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
+(S O) O H19 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x7 (lift (S O) O 
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
+(S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_free c0 x1 x4 H11) (pr2_delta 
+(CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) 
+c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 
+H20)))) x7 H22)))) H21))))) x6 H18)))))) H17)) (subst0_gen_head (Flat Appl) u 
+(lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) (\lambda (H14: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind x0) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) 
+O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat 
+Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t 
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: 
+(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
+x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat 
+Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
+u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) 
+t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) 
+O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind 
+x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T 
+(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: 
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s 
+(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T 
+x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift 
+(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
+x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H21 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
+(S O) O H20 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x8 (lift (S O) O 
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
+(S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) 
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat 
+Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12)))) x8 H22)))) H21))))) x7 
+H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat 
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
+(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat 
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
+(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
+(H19: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x8))).(\lambda (H20: 
+(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat 
+Appl) (lift (S O) O x3) x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
+x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
+O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) 
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead 
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda 
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))) 
+x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T 
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0 
+u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 
+(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 
+(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7 
+H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s 
+(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) 
+i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
+(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
+(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20: 
+(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s 
+(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) 
+(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H22: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H22 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
+(S O) O H20 Hle) in (let H22 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x10: T).(\lambda (H23: (eq T x8 (lift (S O) O 
+x10))).(\lambda (H24: (subst0 (minus i O) u x3 x10)).(eq_ind_r T (lift (S O) 
+O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq 
+T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H25 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x10)) H24 i (minus_n_O i)) 
+in (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
+(Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
+O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) 
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T 
+(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) 
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H25) (pr2_delta 
+c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) 
+(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) 
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21)))) x8 H23)))) H22))))) x7 
+H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s 
+(Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 (THead 
+(Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
+(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+lemma pr2_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
+(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e 
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e: 
+C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2))) 
+(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h 
+d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T 
+t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda 
+(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T 
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1 
+x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T 
+t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda 
+(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) 
+x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5)))) 
+(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind 
+Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 
+(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def 
+(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T 
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1 
+x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h 
+d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) 
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def 
+(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i 
+(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d 
+(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i)))) 
+(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d 
+t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: 
+C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl 
+i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 
+x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus 
+i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind 
+T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3))) 
+(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h 
+d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq 
+T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0 
+x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: 
+nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T 
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 
+H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0 
+t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h 
+(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2 
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11 
+H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 
+t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t 
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T 
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) 
+(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13: 
+(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d 
+t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h) 
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 
+H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 
+t2 h d H6)))))))))))))))) c y x H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/pr2.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/pr2.ma
new file mode 100644 (file)
index 0000000..4d7a092
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/pr2.ma
@@ -0,0 +1,236 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/defs.ma".
+
+include "basic_1A/pr0/pr0.ma".
+
+include "basic_1A/getl/fwd.ma".
+
+fact pr2_confluence__pr2_free_free:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 
+t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 
+t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 
+x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) 
+(pr0_confluence t0 t2 H0 t1 H))))))).
+
+fact pr2_confluence__pr2_free_delta:
+ \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall 
+(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to 
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) 
+\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 
+t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 
+t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 
+t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
+(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: 
+(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
+(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 
+c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 
+x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: 
+T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda 
+(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: 
+(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 
+H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) 
+(pr0_confluence t0 t4 H1 t1 H))))))))))))).
+
+fact pr2_confluence__pr2_delta_delta:
+ \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall 
+(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: 
+T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d 
+(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c 
+(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to 
+(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))))))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: 
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i 
+c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 
+i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda 
+(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: 
+T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 
+x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 
+t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 
+x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda 
+(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda 
+(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 
+u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x 
+x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) 
+x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) 
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T 
+(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind 
+T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T 
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: 
+T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 
+t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x 
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) 
+(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) 
+(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H 
+t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) 
+(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 
+w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 
+x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: 
+T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i 
+i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 
+i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: 
+(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d 
+u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 
+(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def 
+(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 
+\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) 
+H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: 
+C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind 
+Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in 
+((let H18 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead 
+d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind 
+Abbr) u0) H15)) in (\lambda (H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 
+(\lambda (t: T).(subst0 i t x x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 
+(\lambda (t: T).(getl i c (CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 
+\def (eq_ind_r C d0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 
+d H19) in (or4_ind (eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) 
+(\lambda (t: T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) 
+(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda 
+(H23: (eq T x1 x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) 
+H11 x0 H23) in (ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda 
+(H23: (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u 
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: 
+T).(subst0 i u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))) (\lambda (x2: T).(\lambda (H24: (subst0 i u x1 
+x2)).(\lambda (H25: (subst0 i u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 
+H25) (pr2_delta c d u i H22 t2 x1 H11 x2 H24))))) H23)) (\lambda (H23: 
+(subst0 i u x1 x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 
+H23))) (\lambda (H23: (subst0 i u x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 
+H23) (pr2_free c t2 x1 H11))) (subst0_confluence_eq x x1 u i H20 x0 H9))))))) 
+H17)))))))))) H10)) (pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) 
+H7)) (pr0_subst0 t3 x H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 
+H3 t3 H0))))))))))))))))))).
+
+theorem pr2_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall 
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: 
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 
+t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H with 
+[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: 
+(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T 
+t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c 
+t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind 
+T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: 
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 
+t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: 
+T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 
+t1)).(let H8 \def (match H0 with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda 
+(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 
+t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 
+t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 
+t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 
+t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
+T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: 
+T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: 
+T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 
+t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 
+H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | 
+(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 
+c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c 
+(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d 
+(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda 
+(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: 
+(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c 
+(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T 
+(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) 
+(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d 
+(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda 
+(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: 
+(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda 
+(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i 
+H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 
+(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) 
+(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T 
+t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 
+H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 
+t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) 
+\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) 
+\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda 
+(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 
+(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to 
+((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 
+t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind 
+T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) 
+\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda 
+(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 
+\def (match H0 with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C 
+c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c 
+(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T 
+(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda 
+(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 
+t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 
+t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) 
+(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 
+(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 
+H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow 
+(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T 
+t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to 
+((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 
+t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 
+t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T 
+t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to 
+((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
+(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 
+(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to 
+((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda 
+(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) 
+u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 
+t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 
+H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 
+(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) 
+(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T 
+t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C 
+c) (refl_equal T t0) (refl_equal T t1)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/props.ma
new file mode 100644 (file)
index 0000000..d449b91
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/props.ma
@@ -0,0 +1,998 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/fwd.ma".
+
+include "basic_1A/pr0/subst0.ma".
+
+lemma pr2_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) 
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 
+t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u 
+(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 
+t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i 
+H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 
+t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 
+u)))))))))))) c t1 t2 H)))))).
+
+lemma pr2_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
+u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: 
+T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 
+(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 
+t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 
+t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c 
+u1 u2 H)))))).
+
+lemma pr2_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u) 
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead 
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c 
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k 
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3 
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d: 
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0)) 
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: 
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u 
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u)) 
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1: 
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0 
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b) 
+u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) 
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u 
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H7 
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k0 _) \Rightarrow (match k0 with [(Bind b0) \Rightarrow b0 | (Flat 
+_) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) 
+(clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) 
+u) (CHead d (Bind Abbr) u0) H5))) in ((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) 
+H5))) in (\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def 
+(eq_ind T u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr 
+(\lambda (b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) 
+(pr2_free c (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u 
+(pr0_refl u) t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: 
+nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: 
+T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to 
+((eq C c0 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind 
+b) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) 
+u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c 
+(Bind b) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 
+(CHead d (Bind Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def 
+(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to 
+(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 
+n u0 t6 t0) \to ((eq C c1 (CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u 
+t5) (THead (Bind b) u t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta 
+c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) 
+(THead (Bind b) u t3) (THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 
+H3 (Bind b)) (THead (Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) 
+H4 u))))))))))))) i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) 
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) 
+\to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: 
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 
+t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) 
+u) H4) in (pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c 
+(drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O 
+(CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) 
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead 
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: 
+nat).(\lambda (H1: (((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: 
+T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to 
+((eq C c0 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat 
+f) u t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) 
+u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c 
+(Flat f) u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 
+(CHead d (Bind Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def 
+(eq_ind C c0 (\lambda (c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to 
+(\forall (t5: T).(\forall (t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 
+n u0 t6 t0) \to ((eq C c1 (CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u 
+t5) (THead (Flat f) u t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta 
+c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) 
+(THead (Flat f) u t3) (THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 
+H3 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) 
+H4 u))))))))))))) i)))))) k) y t1 t2 H0))) H)))))).
+
+lemma clear_pr2_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to 
+(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1: 
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_: 
+(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1: 
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c 
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
+
+lemma pr2_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free 
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u 
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c 
+t1 t2 H)))))).
+
+lemma pr2_ctail:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) 
+t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: 
+(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail 
+Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
+
+lemma pr2_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
+T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to 
+(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
+(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind 
+b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda 
+(c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 (CHead c (Bind b) v2) t1 t2)) 
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 
+(CHead c (Bind b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) 
+v1))).(pr2_free (CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: 
+(eq C c0 (CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in 
+(nat_ind (\lambda (n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind 
+Abbr) u)) \to ((subst0 n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) 
+(\lambda (H7: (getl O (CHead c (Bind b) v1) (CHead d (Bind Abbr) 
+u))).(\lambda (H8: (subst0 O u t4 t)).(let H9 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) 
+H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 
+(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11 
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) 
+v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind 
+b) v1) (CHead d (Bind Abbr) u) H7))) in (\lambda (H12: (eq B Abbr 
+b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind T u (\lambda (t0: 
+T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def (eq_ind_r B b (\lambda 
+(b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B Abbr (\lambda (b0: 
+B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match (H15 (refl_equal B 
+Abbr)) in False with []) in H16) b H12)))))) H10)) H9)))) (\lambda (i0: 
+nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda 
+(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda 
+(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) d u (S i0) 
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
+(CHead d (Bind Abbr) u) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4))))))))))))) 
+y t1 t2 H1))) H0)))))))).
+
+lemma pr2_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift 
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c 
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) 
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d))))))) 
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 
+t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1: 
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c 
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def 
+(drop_getl_trans_le i d (le_S_n i d (le_S_n (S i) (S d) (le_S (S (S i)) (S d) 
+(le_n_S (S i) d H7)))) c e h H (CHead d0 (Bind Abbr) u) H6) in (ex3_2_ind C C 
+(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda 
+(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear 
+e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t3) (lift h d t)) (\lambda 
+(x0: C).(\lambda (x1: C).(\lambda (H9: (drop i O c x0)).(\lambda (H10: (drop 
+h (minus d i) x0 x1)).(\lambda (H11: (clear x1 (CHead d0 (Bind Abbr) 
+u))).(let H12 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 
+x1)) H10 (S (minus d (S i))) (minus_x_Sy d i H7)) in (let H13 \def 
+(drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u H11) in (ex2_ind C 
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) 
+u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t3) 
+(lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x (Bind Abbr) 
+(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x 
+d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x 
+(Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d 
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7 
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h) 
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3) 
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h 
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
+
+lemma pr2_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: 
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T 
+T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) 
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T 
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
+t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
+Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
+(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
+t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in 
+(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
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+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
+(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 
+(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: 
+T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) 
+u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl 
+Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S 
+i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl 
+x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda 
+(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t 
+(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: 
+(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 
+x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
+(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 
+(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
+c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O 
+(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) 
+u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 
+x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) 
+H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) 
+H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
+(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) 
+H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) 
+(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
+
+lemma pr2_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
+(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
+(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift 
+(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 
+t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind 
+Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
+t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
+b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) 
+t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) 
+(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 
+x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 
+H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
+u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
+(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 
+(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
+t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T 
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
+T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
+(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) 
+(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t 
+(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind 
+Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 
+x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 
+x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead 
+(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda 
+(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: 
+T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 
+(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t 
+(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: 
+(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) 
+(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
+T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
+T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: 
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S 
+O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 
+t2 H5)))))))))))))) c y x H0))) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr2/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr2/subst1.ma
new file mode 100644 (file)
index 0000000..95a98ab
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr2/subst1.ma
@@ -0,0 +1,265 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/fwd.ma".
+
+include "basic_1A/pr0/subst1.ma".
+
+include "basic_1A/csubst1/getl.ma".
+
+include "basic_1A/subst1/subst1.ma".
+
+lemma pr2_delta1:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) 
+\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 
+t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) 
+(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 
+H0 t0 H2))) t H1)))))))))).
+
+lemma pr2_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c 
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1 
+t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T 
+(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))) 
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1: 
+T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda 
+(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1: 
+T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T 
+(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))) 
+(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)) 
+(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))) 
+(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4 
+x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v 
+t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v 
+(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda 
+(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda 
+(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda 
+(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c 
+(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda 
+(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda 
+(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 
+x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: 
+T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not 
+(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: 
+T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t 
+x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c 
+w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x 
+H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t 
+H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let 
+H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in 
+(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind 
+Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v) 
+(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead 
+e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal 
+C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow e | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono 
+c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in ((let H15 \def 
+(f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow v | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) 
+u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in 
+(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl 
+i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u 
+(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C 
+d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in 
+(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x 
+t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t 
+w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21: 
+(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: 
+T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20)))) 
+(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9))))))) 
+H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0 
+H5))))))))))))))) y t1 t2 H1))) H0)))))))).
+
+lemma pr2_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: 
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to 
+(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 
+a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda 
+(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 
+x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
+d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d 
+x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: 
+T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 
+(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda 
+(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T 
+(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a 
+x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda 
+(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) 
+H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 
+(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 
+H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S 
+O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: 
+T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda 
+(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) 
+(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: 
+T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) 
+x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x 
+(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: 
+T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) 
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: 
+(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) 
+H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 
+d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: 
+(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: 
+T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 
+t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: 
+T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) 
+d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d 
+(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: 
+T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) 
+(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) 
+u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) 
+(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 
+(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) 
+d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
+(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 
+t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: 
+T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) 
+x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 
+(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl 
+i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 
+(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) 
+(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 
+u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: 
+T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) 
+x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop 
+(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: 
+T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) 
+H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) 
+x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S 
+i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 
+x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: 
+(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda 
+(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind 
+nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S 
+O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in 
+(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S 
+i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: 
+T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) 
+(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S 
+i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 
+(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift 
+(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat 
+(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S 
+i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let 
+H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: 
+nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) 
+in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) 
+(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) 
+d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) 
+(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) 
+(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S 
+i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i 
+H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 
+c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i 
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) 
+(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: 
+nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def 
+(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 
+\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in 
+(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind 
+Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: 
+T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let 
+H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) 
+H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead 
+e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match 
+e0 with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d 
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) 
+i H0 (CHead e (Bind Abbr) u0) H16)) in ((let H19 \def (f_equal C T (\lambda 
+(e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow 
+t0])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d 
+(Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d 
+e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind 
+Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: 
+T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r 
+T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u 
+(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) 
+(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: 
+C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda 
+(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) 
+t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: 
+T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t 
+x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind 
+T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) 
+(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) 
+(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i 
+(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i 
+x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) 
+(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) 
+H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T 
+(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S 
+O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) 
+(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 
+u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T 
+(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 
+(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) 
+d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq 
+T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 
+x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 
+(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t 
+(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u 
+(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 
+(csubst1_getl_ge d0 i (le_S_n d0 i (le_S_n (S d0) (S i) (le_S (S (S d0)) (S 
+i) (le_n_S (S d0) i H12)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) 
+d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 
+(S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3 H16)))))) (subst1_gen_lift_ge u x0 
+x2 i (S O) d0 H14 (eq_ind_r nat (plus (S O) d0) (\lambda (n: nat).(le n i)) 
+H12 (plus d0 (S O)) (plus_sym d0 (S O)))))))) (subst1_confluence_neq t4 t u i 
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i 
+(lt_neq d0 i H12)))))))))) (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 
+t4 H1 u0 (lift (S O) d0 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 
+t2 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/defs.ma
new file mode 100644 (file)
index 0000000..57297e8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr2/defs.ma".
+
+inductive pr3 (c: C): T \to (T \to Prop) \def
+| pr3_refl: \forall (t: T).(pr3 c t t)
+| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/fwd.ma
new file mode 100644 (file)
index 0000000..99999ba
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/fwd.ma
@@ -0,0 +1,334 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/defs.ma".
+
+include "basic_1A/pr2/fwd.ma".
+
+implied rec lemma pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: 
+T).(P t t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to 
+(\forall (t3: T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) 
+(t0: T) (p: pr3 c t t0) on p: P t t0 \def match p with [(pr3_refl t1) 
+\Rightarrow (f t1) | (pr3_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 
+((pr3_ind c P f f0) t2 t3 p1))].
+
+lemma pr3_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to 
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort 
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda 
+(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: 
+T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
+(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 
+t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda 
+(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: 
+T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) 
+\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T 
+(TSort n)))) t1 H4))))))))) y x H0))) H)))).
+
+lemma pr3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: 
+T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead 
+(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: 
+T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c 
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
+(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 
+x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl 
+(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
+(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
+(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) 
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 
+t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 
+x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: 
+T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 
+t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 
+(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) 
+x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) 
+x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: 
+T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) 
+H11)))))))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c 
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
+t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T 
+t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall 
+(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 
+t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 
+x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) 
+x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T 
+(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
+T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: 
+(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: 
+T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def 
+(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c 
+x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 
+c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda 
+(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) 
+x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 
+x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T 
+t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: 
+(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat 
+Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead 
+(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 
+(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) 
+(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 
+t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c 
+x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 
+H2))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) 
+(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) 
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e 
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
+(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T 
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) 
+(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h 
+d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T 
+t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: 
+T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: 
+C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h 
+d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 
+(lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: 
+T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda 
+(x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: 
+(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 
+(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in 
+(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e 
+x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: 
+T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d 
+x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift 
+h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 
+(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda 
+(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T 
+(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 
+H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x 
+H0)))) H)))))).
+
+lemma pr3_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to 
+(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T x (lift (S n) O v))))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef 
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: 
+T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
+x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or 
+(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T 
+t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) 
+(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: 
+T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or 
+(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef 
+n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) 
+in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T 
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
+v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef 
+n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T 
+t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef 
+n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or 
+(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def 
+(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 
+(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: 
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: 
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) 
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead 
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u 
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
+v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 
+(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def 
+(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C 
+T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind 
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O 
+v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda 
+(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift 
+c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda 
+(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq 
+T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T 
+t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 
+(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: 
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 
+(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) 
+H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/iso.ma
new file mode 100644 (file)
index 0000000..dca56ee
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/iso.ma
@@ -0,0 +1,1125 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/props.ma".
+
+include "basic_1A/iso/props.ma".
+
+include "basic_1A/tlist/fwd.ma".
+
+lemma pr3_iso_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat 
+Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w)) 
+u2))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
+(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall 
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
+(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2: 
+T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to 
+(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq 
+T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda 
+(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso 
+(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T 
+(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef 
+i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T 
+(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind 
+Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O 
+v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_: 
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u: 
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0 
+(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift 
+(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t) 
+\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S 
+i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C 
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind 
+Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) 
+H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 
+(Bind Abbr) x1) H4)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H 
+(CHead x0 (Bind Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
+(eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) 
+in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in 
+(let H14 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
+w))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w 
+x2)) H13 d H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 
+H15))))))) H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef 
+i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) 
+u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3 
+\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) in (or3_ind 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))))).(ex3_2_ind T T (\lambda 
+(u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t 
+x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def 
+(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
+t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1) 
+H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t 
+x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1))) 
+u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t 
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O 
+w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t 
+x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind 
+Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c 
+(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t 
+(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
+(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead 
+(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso 
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P: 
+Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead 
+(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) 
+(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) 
+x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 
+(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t 
+x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5)))))))))) 
+H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat 
+Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) 
+(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda 
+(H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0) 
+x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads 
+(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind 
+x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c 
+(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1 
+x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0 
+(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P: 
+Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead 
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat 
+Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift 
+(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0 
+H5 t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) (THead (Bind 
+x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr3_head_12 c x1 x1 
+(pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift (S O) O t) x2) (THead 
+(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift 
+(S O) O t) (lift (S O) O x4) (pr3_lift (CHead c (Bind x0) x1) c (S O) O 
+(drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H8) (Flat Appl) x2 x2 
+(pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) 
+u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 
+c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat 
+Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 
+(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
+
+lemma pr3_iso_appls_cast:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 
+\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: 
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THeads (Flat Appl) vs t) u2))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0 
+(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2)))))) 
+(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda 
+(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1 
+\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
+t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0 
+x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def 
+(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall 
+(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat 
+Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head v x0 t x1 (Flat 
+Cast)) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
+(pr3 c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: 
+((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) 
+\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall 
+(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2: 
+T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead 
+(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P: 
+Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead 
+(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
+(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl) 
+x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) 
+t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2: 
+T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v 
+t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in 
+(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head t0 x0 (THeads (Flat 
+Appl) t1 (THead (Flat Cast) v t)) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) 
+t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3)) 
+(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) 
+(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6: 
+(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0 
+x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
+u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads 
+(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6 
+(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead 
+(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast 
+Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c 
+(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) 
+(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind 
+Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c 
+t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) 
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat 
+Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) 
+(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda 
+(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat 
+Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) 
+c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead 
+(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads 
+(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda 
+(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0 
+Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2 
+c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead 
+(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead 
+(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
+x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2 
+(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) 
+x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift 
+(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead 
+c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind 
+x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7) 
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
+
+lemma pr3_iso_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: 
+T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t)) 
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda 
+(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c 
+(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead 
+(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let 
+H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 t) u2 H0) in (or3_ind (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
+t2)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) 
+(\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 
+z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c (THead (Bind b) v2 t) t2))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq 
+T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_: 
+(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0: 
+T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P: 
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S 
+O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl)) 
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Flat Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1 
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c (THead (Bind Abbr) 
+x2 x3) u2)).(\lambda (H5: (pr3 c v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b) 
+v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3) 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def 
+(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in 
+(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) 
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind 
+Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
+x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) 
+t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
+(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 
+x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind b) x4 
+x5) H10) in (\lambda (H16: (eq T x0 x4)).(\lambda (H17: (eq B Abst b)).(let 
+H18 \def (eq_ind_r T x5 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) 
+H12 x1 H15) in (let H19 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) 
+H11 x0 H16) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c 
+(Bind b0) v2) t x1)) H18 Abst H17) in (let H21 \def (eq_ind_r B b (\lambda 
+(b0: B).(not (eq B b0 Abst))) H Abst H17) in (eq_ind B Abst (\lambda (b0: 
+B).(pr3 c (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Bind Abbr) x2 x3))) (let H22 \def (match (H21 (refl_equal B Abst)) in False 
+with []) in H22) b H17)))))))) H14)) H13))))))) H9)) (\lambda (H9: (pr3 
+(CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1)))).(pr3_t 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead 
+(Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
+Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) (Bind b) 
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x2) (pr3_lift 
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
+v1 x2 H5) t (lift (S O) O (THead (Bind Abst) x0 x1)) H9 Appl)) (THead (Bind 
+Abbr) x2 x3) (eq_ind T (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) 
+x0 x1))) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 t0) (THead (Bind Abbr) x2 
+x3))) (pr3_sing c (THead (Bind Abbr) x2 x1) (THead (Bind b) v2 (lift (S O) O 
+(THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (pr2_free c (THead (Bind 
+b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (THead 
+(Bind Abbr) x2 x1) (pr0_zeta b H (THead (Flat Appl) x2 (THead (Bind Abst) x0 
+x1)) (THead (Bind Abbr) x2 x1) (pr0_beta x0 x2 x2 (pr0_refl x2) x1 x1 
+(pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) (pr3_head_12 c x2 x2 (pr3_refl 
+c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead (Flat Appl) (lift (S O) O x2) 
+(lift (S O) O (THead (Bind Abst) x0 x1))) (lift_flat Appl x2 (THead (Bind 
+Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) H3)) (\lambda (H3: (ex6_6 B T T 
+T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: 
+T).(\lambda (y2: T).(pr3 c (THead (Bind b0) y2 (THead (Flat Appl) (lift (S O) 
+O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v1 u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: 
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 Abst))).(\lambda (H5: (pr3 c 
+(THead (Bind b) v2 t) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: 
+(pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind 
+x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+x4) x3)) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let 
+H_x \def (pr3_gen_bind b H c v2 t (THead (Bind x0) x1 x2) H5) in (let H10 
+\def H_x in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
+(Bind x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) 
+v2) t t2)))) (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 
+x2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (H11: 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) 
+(THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind 
+x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c 
+v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))) 
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H12: (eq T (THead (Bind x0) x1 x2) (THead (Bind 
+b) x6 x7))).(\lambda (H13: (pr3 c v2 x6)).(\lambda (H14: (pr3 (CHead c (Bind 
+b) v2) t x7)).(let H15 \def (f_equal T B (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) 
+\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in ((let H16 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x1 | (TLRef 
+_) \Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) 
+(THead (Bind b) x6 x7) H12) in ((let H17 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 
+x7) H12) in (\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 
+\def (eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 
+H17) in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 
+H18) in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) 
+x5) x2 x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq 
+B b0 Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead 
+(Flat Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 
+x5 H8) (Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) 
+(lift (S O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift 
+(S O) O x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c 
+c (drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 
+(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 
+(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: 
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t 
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
+(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat 
+Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b) 
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift 
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) 
+v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind 
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O 
+(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c 
+(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O 
+x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
+x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind 
+x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) 
+x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2)) 
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0 
+H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead 
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (pr3_head_12 c x1 x5 
+H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) 
+(lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S 
+O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead 
+(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) 
+H10))) u2 H6))))))))))))) H3)) H2)))))))))).
+
+lemma pr3_iso_appls_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: 
+T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs 
+(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2: 
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
+t))) u2)))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda 
+(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: 
+TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
+b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead 
+(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c: 
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b) 
+u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t)) 
+u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1))))) 
+(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall 
+(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
+b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda 
+(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda 
+(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v 
+(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def 
+(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind 
+b) u t))) u2 H1) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind b) u t))) t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
+Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) 
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat 
+Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat 
+Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def 
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) 
+t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P: 
+Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8 
+(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat 
+Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))))).(ex4_4_ind T T 
+T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: 
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) 
+u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1 
+(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 
+x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind 
+b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 
+(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) 
+t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat 
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S 
+O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u 
+(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c 
+(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P: 
+Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t) 
+t1 H9 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c (THead (Flat Appl) 
+t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr2_free c (THead 
+(Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1) (pr0_beta 
+x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) 
+x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c t0 x2 H6 (Bind Abbr) x1 x3 
+(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u 
+t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t0 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c 
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat 
+Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not 
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat 
+Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
+(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (pr3 (CHead c 
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u 
+(THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t (THead (Bind x0) x1 (THead 
+(Flat Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) 
+t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t 
+(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads 
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c 
+(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2) 
+H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead 
+(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P: 
+Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1 
+H11 P)))) t0 Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) 
+x2)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind 
+x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat 
+Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
+(lift (S O) O t0) x2)) (pr0_upsilon x0 H5 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl 
+x1) x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat 
+Appl) (lift (S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(pr3_head_12 (CHead c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) 
+(pr3_lift (CHead c (Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c 
+(drop_refl c) x1) t0 x4 H8) (Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind 
+x0) x1) (Flat Appl) (lift (S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 
+(THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat 
+Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat 
+Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) 
+(pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 
+H7)))))))))))))) H4)) H3))))))))) vs)))))).
+
+lemma pr3_iso_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: 
+T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t)) 
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) 
+(lifts (S O) O vs) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs: 
+TList).(tlist_ind_rev (\lambda (t: TList).(\forall (u: T).(\forall (t0: 
+T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c: 
+C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t) 
+t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda 
+(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso 
+(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts: 
+TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0: 
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead 
+(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0)) 
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat 
+Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0: 
+T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl) 
+(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat 
+Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P: 
+Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t)) 
+(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0)) 
+u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl) 
+(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let 
+H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0)) 
+(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl) 
+t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) t (THead (Bind b) u t0) 
+ts)) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind 
+b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2 
+(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0))) 
+(theads_tapp (Flat Appl) t (THead (Bind b) u t0) ts)) in (TList_ind (\lambda 
+(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall 
+(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to 
+((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P: 
+Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O 
+t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat 
+Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c 
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl) 
+(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1: 
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead 
+(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0 
+t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads 
+(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c 
+(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0))) 
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b 
+H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_: 
+((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3 
+c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads 
+(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to 
+(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2)) 
+u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
+(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t 
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead 
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift 
+(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2: 
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1 
+ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1 
+ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 
+(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2)) 
+u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead 
+(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat 
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to 
+(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2 
+(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8: 
+(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat 
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads 
+(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O 
+t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead 
+(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl) 
+(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads 
+(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat 
+Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) 
+(lifts_tapp (S O) O t ts))))))))))) vs))).
+
+lemma pr3_iso_beta:
+ \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat 
+Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c 
+u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind 
+Abbr) v t) u2))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: 
+T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t)) 
+u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2) 
+\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind 
+Abst) w t) u2 H) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w t) t2)))) 
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c v u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
+(THead (Bind Abbr) v t) u2) (\lambda (H2: (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abst) w t) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) 
+w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v 
+x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T 
+u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0) 
+\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t) 
+t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead 
+(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: 
+(ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: 
+T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) 
+w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Bind Abbr) v t) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v 
+x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0 
+x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) 
+u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5) 
+in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) x4 x5) H8) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) 
+x4 x5) H8) in (\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
+t0)))) H10 x1 H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w 
+t0)) H9 x0 H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c 
+(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) 
+(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) 
+(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) w t) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v 
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind Abst) w t) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c v u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda 
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind 
+Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5 
+(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v 
+x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2 
+x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in 
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w 
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead 
+(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def 
+(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead k _ _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead 
+(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in 
+((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq 
+T x1 x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t 
+t0)))) H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w 
+t0)) H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c 
+(Bind b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: 
+B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) 
+H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b 
+Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in 
+False with []) in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) 
+H1)))))))).
+
+lemma pr3_iso_appls_beta:
+ \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 
+\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in 
+(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to 
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr) 
+v t)) u2)))))))))
+\def
+ \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall 
+(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl) 
+v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 
+u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat 
+Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w: 
+T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c 
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso 
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P: 
+Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda 
+(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1: 
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead 
+(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P: 
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)) 
+u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c: 
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1: 
+(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v 
+(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def 
+(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind 
+Abst) w t1))) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (pr3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) 
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) t2))))).(ex3_2_ind T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c 
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2))) 
+(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat 
+Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def 
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl) 
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) t2) \to (\forall (P: 
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl) 
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
+(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0 
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (THead (Flat 
+Appl) x0 x1))) u2 H4))))))) H3)) (\lambda (H3: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T 
+T (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c 
+(THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) y1 z1)))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))) (pr3 c (THead (Flat 
+Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c 
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t x2)).(\lambda (H6: (pr3 
+c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 
+(THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: 
+T).(pr3 (CHead c (Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
+(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1 
+c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0 
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0 
+x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 
+(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1) 
+(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) 
+t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead 
+(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2 
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t 
+x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) 
+w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat 
+Appl) v (THead (Bind Abst) w t1))) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2)) 
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t u3))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead 
+(Bind Abbr) v t1))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq 
+B x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (THead (Flat Appl) v 
+(THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda 
+(H7: (pr3 c t x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c 
+(Bind x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S 
+O) O x4) x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) 
+v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c 
+(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t 
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads 
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c 
+(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead 
+(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda 
+(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst) 
+w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) 
+O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead 
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead 
+(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) 
+(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) x1 x1 (pr0_refl x1) 
+x2 x2 (pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O 
+x4) x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) 
+(lift (S O) O t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 
+(CHead c (Bind x0) x1) (lift (S O) O t) (lift (S O) O x4) (pr3_lift (CHead c 
+(Bind x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t x4 H7) 
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift 
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S 
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c 
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) 
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) 
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr1.ma
new file mode 100644 (file)
index 0000000..59d537d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr1.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/defs.ma".
+
+include "basic_1A/pr1/fwd.ma".
+
+lemma pr3_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: 
+T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 
+t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: 
+C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr3.ma
new file mode 100644 (file)
index 0000000..5369615
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/pr3.ma
@@ -0,0 +1,68 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/props.ma".
+
+include "basic_1A/pr2/pr2.ma".
+
+lemma pr3_strip:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t 
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t 
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
+t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 
+t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: 
+T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda 
+(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: 
+T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c 
+t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c 
+x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) 
+t0 t1 H)))).
+
+theorem pr3_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall 
+(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t 
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t 
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 
+t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda 
+(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 
+t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c 
+t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 
+t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) 
+(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) 
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda 
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T 
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 
+c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 
+H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/props.ma
new file mode 100644 (file)
index 0000000..ee6ad89
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/props.ma
@@ -0,0 +1,1604 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/fwd.ma".
+
+include "basic_1A/pr3/pr1.ma".
+
+include "basic_1A/pr2/props.ma".
+
+include "basic_1A/pr1/props.ma".
+
+lemma clear_pr3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
+(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1 
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t: 
+T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
+t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
+
+lemma pr3_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
+
+theorem pr3_t:
+ \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall 
+(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0 
+t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3 
+c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall 
+(t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
+(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+lemma pr3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t: 
+T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 
+t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u 
+t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
+t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
+
+lemma pr3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall 
+(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda 
+(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda 
+(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c 
+(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing 
+c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
+(H2 k t)))))))))) u1 u2 H)))).
+
+lemma pr3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u 
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u) 
+(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0)))) 
+(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u 
+t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) 
+(THead k u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c 
+u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pr3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
+(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
+u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
+
+lemma pr3_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v: 
+T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f: 
+F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead 
+c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c 
+(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
+
+theorem pr3_flat:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead 
+(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
+(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
+u2))))))))).
+
+lemma pr3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0 
+t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda 
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: 
+(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k 
+u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: 
+(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead 
+c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d 
+(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t)))) 
+(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8: 
+(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d 
+(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda 
+(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
+c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) 
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) 
+H9))) in ((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda 
+(H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u 
+(\lambda (t0: T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda 
+(b0: B).(pr3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind 
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda 
+(H17: (pr0 x t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c 
+(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 
+(CHead c (Bind Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
+(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: 
+F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
+u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O 
+(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c 
+(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind 
+Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda 
+(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 
+t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k 
+u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind 
+(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to 
+((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) 
+\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda 
+(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) 
+u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2 
+(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0) 
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c 
+(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f: 
+F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
+u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) 
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2 
+(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) 
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
+u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
+
+lemma pr3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 
+u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1: 
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3 
+(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: 
+K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0 
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2: 
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda 
+(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1: 
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y: 
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4: 
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4 
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0 
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5 
+(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0: 
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0 
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4 
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C 
+c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2 
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n: 
+nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5 
+t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 
+(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
+(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 
+(CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
+C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
+H11 t H15) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: 
+T).(subst0 (S (plus i O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) 
+(\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S 
+(plus i O)) u x t6)).(let H21 \def (f_equal nat nat S (plus i O) i (sym_eq 
+nat i (plus i O) (plus_n_O i))) in (let H22 \def (eq_ind nat (S (plus i O)) 
+(\lambda (n: nat).(subst0 n u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda 
+(t7: T).(subst0 O t1 t5 t7)) (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 
+(Bind Abbr) t1) t4 t6) (\lambda (x0: T).(\lambda (H23: (subst0 O t1 t5 
+x0)).(\lambda (H24: (pr0 x0 x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 
+(pr2_delta (CHead c0 (Bind Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 
+x0 H23) t6 (pr3_pr2 (CHead c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 
+(Bind Abbr) t1) d u (S i) (getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 
+t1 (clear_bind Abbr c0 t1) (CHead d (Bind Abbr) u) i H0) x0 x H24 t6 
+H22)))))) (pr0_subst0_back t2 t5 x O H19 t1 H1))))))) (subst0_subst0 t5 t6 t 
+O H18 t2 u i H2)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: 
+(clear (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 O (getl_intro O c0 
+(CHead d0 (Bind Abbr) u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead d0 
+(Bind Abbr) u0) t H12)) t4 t5 H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k 
+t) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda (_: (((getl 
+i1 (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to 
+(pr3 (CHead c0 k t1) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((getl (S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to 
+(pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) 
+(CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) 
+t1) t4 t6 (pr2_delta (CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) 
+i1 c0 (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) 
+u0) t i1 H12) t1) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S 
+i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) 
+(getl_gen_S (Flat f) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) 
+f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c 
+u1 u2 H)))).
+
+lemma pr3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to 
+(pr3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2) 
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3 
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c 
+u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: 
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2) 
+\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 
+u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pr3_pr3_pr3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall 
+(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1 
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: 
+K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 
+t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 
+(CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
+t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
+
+lemma pr3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift 
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1 
+t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h 
+d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0 
+t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d 
+t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
+H0)))))))).
+
+lemma pr3_eta:
+ \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind 
+Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v 
+(THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(let t \def (THead (Bind 
+Abst) w u) in (\lambda (v: T).(\lambda (H: (pr3 c v w)).(eq_ind_r T (THead 
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u)) (\lambda (t0: T).(pr3 c 
+(THead (Bind Abst) v (THead (Flat Appl) (TLRef O) t0)) (THead (Bind Abst) w 
+u))) (pr3_head_12 c v w H (Bind Abst) (THead (Flat Appl) (TLRef O) (THead 
+(Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u (pr3_pr1 (THead (Flat 
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) u 
+(pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (THead (Flat 
+Appl) (TLRef O) (THead (Bind Abst) (lift (S O) O w) (lift (S O) (S O) u))) 
+(pr0_beta (lift (S O) O w) (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S 
+O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing 
+(THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) 
+(lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) 
+(lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) 
+(lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u (pr1_pr0 (THead (Bind 
+Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u 
+(pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead 
+(Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
+
+lemma pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
+x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c 
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
+(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind 
+Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) 
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
+u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c 
+t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
+x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
+(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) 
+O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind 
+Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 
+(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
+Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
+T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
+t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
+(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
+t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
+c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
+(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
+T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
+(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
+(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
+(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 
+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
+(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda 
+(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
+t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
+Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead 
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
+(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
+(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
+x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
+\def (match (H (refl_equal B Abst)) in False with []) in H1))))))) (\lambda 
+(_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/subst1.ma
new file mode 100644 (file)
index 0000000..648ce77
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/subst1.ma
@@ -0,0 +1,89 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/fwd.ma".
+
+include "basic_1A/pr2/subst1.ma".
+
+lemma pr3_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) 
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c 
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: 
+T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v 
+t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i 
+v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T 
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T 
+(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) 
+(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 
+x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 
+w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 
+w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v 
+t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 
+i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
+e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
+
+lemma pr3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T 
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a 
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 
+d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) 
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d 
+x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) 
+(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
+T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: 
+T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 
+x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d 
+x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda 
+(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 
+d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda 
+(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) 
+(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: 
+T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) 
+d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d 
+u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x 
+x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 
+e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1A/pr3/wcpr0.ma
new file mode 100644 (file)
index 0000000..1b59a29
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/pr3/wcpr0.ma
@@ -0,0 +1,63 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/props.ma".
+
+include "basic_1A/wcpr0/getl.ma".
+
+lemma pr3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: 
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 
+t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: 
+T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) 
+(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: 
+T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead 
+c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 
+(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) 
+(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: 
+(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq 
+C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: 
+C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: 
+(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 
+k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) 
+u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda 
+(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k 
+u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind 
+Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: 
+C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: 
+T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T 
+(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 
+(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 
+x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta 
+(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t 
+(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 
+H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 
+H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) 
+H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 
+H2)))))))))))))) c2 c1 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/preamble.ma b/matita/matita/contribs/lambdadelta/basic_1A/preamble.ma
new file mode 100644 (file)
index 0000000..6b4b1d5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/preamble.ma
@@ -0,0 +1,15 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_1A/theory.ma".

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/r/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/r/defs.ma
new file mode 100644 (file)
index 0000000..6018cdd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/r/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+definition r:
+ K \to (nat \to nat)
+\def
+ \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | 
+(Flat _) \Rightarrow (S i)])).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/r/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/r/props.ma
new file mode 100644 (file)
index 0000000..9b990e0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/r/props.ma
@@ -0,0 +1,153 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/r/defs.ma".
+
+include "basic_1A/s/defs.ma".
+
+lemma r_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S 
+i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r 
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat 
+f) i))))) k).
+
+lemma r_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
+(plus (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda 
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j))))) 
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r 
+(Flat f) i) j))))) k).
+
+lemma r_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) 
+(plus i (r k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda 
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_: 
+F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
+
+lemma r_minus:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat 
+(minus (r k i) (S n)) (r k (minus i (S n)))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k: 
+K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S 
+n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_: 
+F).(minus_x_Sy i n H)) k)))).
+
+lemma r_dis:
+ \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) 
+\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i: 
+nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i) 
+(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H: 
+((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_: 
+((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i: 
+nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_: 
+((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0: 
+((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda 
+(i: nat).(refl_equal nat (S i)))))))) k).
+
+lemma s_r:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 
+i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i)))) 
+(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
+
+lemma r_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n: 
+nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n: 
+nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O)) 
+(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
+
+lemma r_arith1:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S 
+i)) (S j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i)) 
+(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat 
+(minus (r k i) j)) (r k (S i)) (r_S k i)))).
+
+lemma r_arith2:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to 
+(le (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda 
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S i) (S j))).(let H_y \def 
+(le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (le (S i) j)).H)))) k).
+
+lemma r_arith3:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to 
+(le j (r k i)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda 
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S j) (S i))).(let H_y \def 
+(le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (le j (S i))).H)))) k).
+
+lemma r_arith4:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k 
+j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (minus (S i) (s k0 j)) (minus (r k0 i) j))))) (\lambda (b: 
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i) 
+j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
+(minus (r (Flat f) i) j))))) k).
+
+lemma r_arith5:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to 
+(lt j (r k i)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda 
+(i: nat).(\lambda (j: nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) 
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S 
+i))).H)))) k).
+
+lemma r_arith6:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S 
+j)) (minus i (s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (minus (r k0 i) (S j)) (minus i (s k0 j)))))) (\lambda (b: 
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b) 
+j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat 
+(minus i (s (Flat f) j)))))) k).
+
+lemma r_arith7:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j)) 
+\to (eq nat (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((eq nat (S i) (s k0 j)) \to (eq nat (r k0 i) j))))) (\lambda (_: 
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (S i) (S 
+j))).(eq_add_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (eq nat (S i) j)).H)))) k).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/rect.txt b/matita/matita/contribs/lambdadelta/basic_1A/rect.txt
new file mode 100644 (file)
index 0000000..cba309a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/rect.txt
@@ -0,0 +1,3 @@
+T_rect
+A_rect
+C_rect

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/s/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/s/defs.ma
new file mode 100644 (file)
index 0000000..3b8980e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/s/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+definition s:
+ K \to (nat \to nat)
+\def
+ \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | 
+(Flat _) \Rightarrow i])).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/s/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/s/fwd.ma
new file mode 100644 (file)
index 0000000..12290a1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/s/fwd.ma
@@ -0,0 +1,48 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/s/defs.ma".
+
+lemma s_inj:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) 
+\to (eq nat i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((eq nat (s k0 i) (s k0 j)) \to (eq nat i j))))) (\lambda (b: 
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (s (Bind b) i) (s 
+(Bind b) j))).(eq_add_S i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda 
+(j: nat).(\lambda (H: (eq nat (s (Flat f) i) (s (Flat f) j))).H)))) k).
+
+lemma s_le_gen:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k i) (s k j)) \to 
+(le i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((le (s k0 i) (s k0 j)) \to (le i j))))) (\lambda (b: B).(\lambda (i: 
+nat).(\lambda (j: nat).(\lambda (H: (le (s (Bind b) i) (s (Bind b) 
+j))).(le_S_n i j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (le (s (Flat f) i) (s (Flat f) j))).H)))) k).
+
+lemma s_lt_gen:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k i) (s k j)) \to 
+(lt i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((lt (s k0 i) (s k0 j)) \to (lt i j))))) (\lambda (b: B).(\lambda (i: 
+nat).(\lambda (j: nat).(\lambda (H: (lt (s (Bind b) i) (s (Bind b) 
+j))).(le_S_n (S i) j H))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
+nat).(\lambda (H: (lt (s (Flat f) i) (s (Flat f) j))).H)))) k).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/s/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/s/props.ma
new file mode 100644 (file)
index 0000000..6f57eeb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/s/props.ma
@@ -0,0 +1,109 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/s/defs.ma".
+
+lemma s_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S 
+i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s 
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat 
+f) i))))) k).
+
+lemma s_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
+(plus (s k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda 
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j))))) 
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s 
+(Flat f) i) j))))) k).
+
+lemma s_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) 
+(plus i (s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda 
+(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq 
+nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j)) 
+(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: 
+nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
+
+lemma s_minus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s 
+k (minus i j)) (minus (s k i) j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j)))))) 
+(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j 
+i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i) 
+j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H)))))) 
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j 
+i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
+
+lemma minus_s_s:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s 
+k j)) (minus i j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_: 
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j))))) 
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i 
+j))))) k).
+
+lemma s_le:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) 
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: 
+nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_: 
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
+
+lemma s_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) 
+(s k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j: 
+nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i: 
+nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(lt_n_S i j H))))) (\lambda (_: 
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
+
+lemma s_inc:
+ \forall (k: K).(\forall (i: nat).(le i (s k i)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i)))) 
+(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S_n (S i) (S 
+(s (Bind b) i)) (le_S_n (S (S i)) (S (S (s (Bind b) i))) (le_S (S (S (S i))) 
+(S (S (s (Bind b) i))) (le_n (S (S (s (Bind b) i)))))))))) (\lambda (f: 
+F).(\lambda (i: nat).(le_n (s (Flat f) i)))) k).
+
+lemma s_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n: 
+nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal 
+nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
+
+lemma s_arith1:
+ \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
+\def
+ \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n 
+i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sc3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1A/sc3/arity.ma
new file mode 100644 (file)
index 0000000..abeeb54
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sc3/arity.ma
@@ -0,0 +1,313 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubc/arity.ma".
+
+include "basic_1A/csubc/getl.ma".
+
+include "basic_1A/csubc/drop1.ma".
+
+include "basic_1A/csubc/props.ma".
+
+lemma sc3_arity_csubc:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 
+t a) \to (\forall (d1: C).(\forall (is: PList).((drop1 is d1 c1) \to (\forall 
+(c2: C).((csubc g d1 c2) \to (sc3 g a c2 (lift1 is t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0: 
+A).(\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0)))))))))) (\lambda (c: 
+C).(\lambda (n: nat).(\lambda (d1: C).(\lambda (is: PList).(\lambda (_: 
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (_: (csubc g d1 c2)).(eq_ind_r T 
+(TSort n) (\lambda (t0: T).(land (arity g c2 t0 (ASort O n)) (sn3 c2 t0))) 
+(conj (arity g c2 (TSort n) (ASort O n)) (sn3 c2 (TSort n)) (arity_sort g c2 
+n) (sn3_nf2 c2 (TSort n) (nf2_sort c2 n))) (lift1 is (TSort n)) (lift1_sort n 
+is))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: 
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall 
+(is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
+a0 c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda 
+(H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let 
+H_x \def (drop1_getl_trans is c d1 H3 Abbr d u i H0) in (let H5 \def H_x in 
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: 
+C).(getl (trans is i) d1 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) u)))) 
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (_: (drop1 
+(ptrans is i) x d)).(\lambda (H7: (getl (trans is i) d1 (CHead x (Bind Abbr) 
+(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x 
+(Bind Abbr) (lift1 (ptrans is i) u)) (trans is i) H7 c2 H4) in (let H8 \def 
+H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2: 
+C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2 
+(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2 
+x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) 
+x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind 
+Abbr) H10) in (let H11 \def H_x1 in (or3_ind (ex2 C (\lambda (c3: C).(eq C x0 
+(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K 
+(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (ex4_3 B C T (\lambda 
+(b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3))))) 
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq 
+C x0 (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc 
+g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1 
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr) 
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind 
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) 
+(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in 
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y 
+(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O 
+u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i) 
+O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O) 
+(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4) 
+(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans 
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef 
+i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_: 
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda 
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
+K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13: 
+(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr) 
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1 
+(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0 
+(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14) 
+in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee with [(Bind 
+b) \Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False 
+| Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13) 
+in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12)) 
+(\lambda (H12: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abbr) (Bind Void))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: 
+B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) 
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (H13: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H14: (eq 
+K (Bind Abbr) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: 
+(csubc g x x2)).(let H17 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is 
+i) c2 c0)) H9 (CHead x2 (Bind x1) x3) H13) in (let H18 \def (eq_ind K (Bind 
+Abbr) (\lambda (ee: K).(match ee with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])) I (Bind Void) H14) in (False_ind (sc3 g a0 c2 
+(lift1 is (TLRef i))) H18)))))))))) H12)) H11)))))) H8)))))) 
+H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0: 
+A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1: 
+C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1 
+c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda 
+(is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4: 
+(csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1 
+H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1 
+(ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind 
+Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda 
+(x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is 
+i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def 
+(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is 
+i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans 
+is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans 
+is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda 
+(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) 
+(lift1 (ptrans is i) u)) x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 
+(ptrans is i) u) (Bind Abst) H11) in (let H12 \def H_x1 in (or3_ind (ex2 C 
+(\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) 
+(\lambda (c3: C).(csubc g x c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
+w))))) (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: T).(eq C 
+x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: 
+T).(csubc g x c3))))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (H13: (ex2 
+C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) 
+(\lambda (c3: C).(csubc g x c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 
+(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x 
+c3)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C 
+x0 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x 
+x1)).(let H16 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) 
+H10 (CHead x1 (Bind Abst) (lift1 (ptrans is i) u)) H14) in (let H_y \def 
+(sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0: 
+T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef 
+(trans is i)) a0 (eq_ind T (lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 
+t0 a0)) (arity_lift1 g a0 c is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 
+H1)) (TLRef (trans is i)) (lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 
+(ptrans is i) u) (trans is i) H16) I) (lift1 is (TLRef i)) (lift1_lref is 
+i))))))) H13)) (\lambda (H13: (ex5_3 C T A (\lambda (_: C).(\lambda (_: 
+T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w))))) 
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans 
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq 
+K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: 
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: 
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: 
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is 
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (_: 
+(eq K (Bind Abst) (Bind Abst))).(\lambda (H15: (eq C x0 (CHead x1 (Bind Abbr) 
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (H17: (sc3 g (asucc g x3) x 
+(lift1 (ptrans is i) u))).(\lambda (H18: (sc3 g x3 x1 x2)).(let H19 \def 
+(eq_ind C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind 
+Abbr) x2) H15) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef 
+(trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 
+(let H_y0 \def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let 
+H_y1 \def (sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g x3) H17) in 
+(sc3_repl g x3 c2 (lift (S (trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S 
+(trans is i)) O (getl_drop Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g 
+x3 a0 (arity_mono g x (lift1 (ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) 
+H_y0))))) H19) (lift1 is (TLRef i)) (lift1_lref is i)))))))))))) H13)) 
+(\lambda (H13: (ex4_3 B C T (\lambda (b: B).(\lambda (c3: C).(\lambda (v2: 
+T).(eq C x0 (CHead c3 (Bind b) v2))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (_: T).(eq K (Bind Abst) (Bind Void))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (_: T).(not (eq B b Void))))) (\lambda (_: B).(\lambda (c3: 
+C).(\lambda (_: T).(csubc g x c3)))))).(ex4_3_ind B C T (\lambda (b: 
+B).(\lambda (c3: C).(\lambda (v2: T).(eq C x0 (CHead c3 (Bind b) v2))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: T).(eq K (Bind Abst) (Bind 
+Void))))) (\lambda (b: B).(\lambda (_: C).(\lambda (_: T).(not (eq B b 
+Void))))) (\lambda (_: B).(\lambda (c3: C).(\lambda (_: T).(csubc g x c3)))) 
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: B).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (H14: (eq C x0 (CHead x2 (Bind x1) x3))).(\lambda (H15: (eq 
+K (Bind Abst) (Bind Void))).(\lambda (_: (not (eq B x1 Void))).(\lambda (_: 
+(csubc g x x2)).(let H18 \def (eq_ind C x0 (\lambda (c0: C).(getl (trans is 
+i) c2 c0)) H10 (CHead x2 (Bind x1) x3) H14) in (let H19 \def (eq_ind K (Bind 
+Abst) (\lambda (ee: K).(match ee with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])) I (Bind Void) H15) in (False_ind (sc3 g a0 c2 
+(lift1 is (TLRef i))) H19)))))))))) H13)) H12)))))) H9)))))) 
+H6))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b 
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity 
+g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 
+c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is 
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c 
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is: 
+PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1 
+c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: 
+PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g 
+d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead 
+(Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) 
+(H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u)) 
+(Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u)) 
+(csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is 
+(THead (Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g 
+a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) 
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is 
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c 
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is: 
+PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g 
+d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: 
+PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g 
+d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) 
+(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall 
+(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g 
+a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead 
+(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d 
+c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1 
+is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind 
+Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1 
+is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 
+(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst) 
+(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d: 
+C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0: 
+PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1 
+is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3 
+g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8 
+d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr 
+not_abbr_abst a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) 
+(let H_x \def (csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x 
+in (ex2_ind C (\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g 
+c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) 
+(\lambda (x: C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x 
+d)).(eq_ind_r T (lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 
+(CHead d (Bind Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: 
+PList).(sc3 g a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind 
+Abst) (lift1 (papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c 
+(papp is0 is) x u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) 
+(csubc_abst g x d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) 
+(drop1_trans is0 x d1 H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) 
+(Ss is)) (papp_ss is0 is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 
+(Ss is0) (Ss is) t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g 
+c2 (asucc g a1) is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 
+(THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 
+(lift1 is u) (lift1 (Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) 
+(lift1_bind Abst is u t0)))))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall 
+(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g 
+d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: 
+A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: 
+C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 
+c2) \to (sc3 g (AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: 
+C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: 
+C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let 
+H_y0 \def (H3 d1 is H4 c2 H5) in (let H6 \def H_y0 in (land_ind (arity g c2 
+(lift1 is t0) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) 
+\to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat 
+Appl) w (lift1 is0 (lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat 
+Appl) u t0))) (\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda 
+(H8: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: 
+PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 
+(lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in 
+(eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
+T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u 
+t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g 
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) 
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is 
+u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3: 
+((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: 
+C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1: 
+C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: 
+C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in 
+(eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1: 
+T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0) 
+(H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is 
+u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall 
+(is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g 
+a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 
+a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1 
+c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1 
+is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
+
+lemma sc3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to (sc3 g a c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(let H_y \def (sc3_arity_csubc g c t a H c PNil) in (H_y 
+(drop1_nil c) c (csubc_refl g c))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sc3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/sc3/defs.ma
new file mode 100644 (file)
index 0000000..fb12ab2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sc3/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sn3/defs.ma".
+
+include "basic_1A/arity/defs.ma".
+
+include "basic_1A/drop1/defs.ma".
+
+rec definition sc3 (g: G) (a: A) on a: C \to (T \to Prop) \def \lambda (c: 
+C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t 
+(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead 
+a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is 
+t)))))))))])).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/sc3/props.ma
new file mode 100644 (file)
index 0000000..e77002e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sc3/props.ma
@@ -0,0 +1,697 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sc3/defs.ma".
+
+include "basic_1A/sn3/lift1.ma".
+
+include "basic_1A/nf2/lift1.ma".
+
+include "basic_1A/csuba/arity.ma".
+
+include "basic_1A/arity/lift1.ma".
+
+include "basic_1A/arity/aprem.ma".
+
+include "basic_1A/llt/props.ma".
+
+include "basic_1A/llt/fwd.ma".
+
+include "basic_1A/drop1/getl.ma".
+
+include "basic_1A/drop1/props.ma".
+
+include "basic_1A/lift1/drop1.ma".
+
+lemma sc3_arity_gen:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c 
+t) \to (arity g c t a)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind 
+(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity 
+g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: 
+(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to 
+(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity 
+g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in 
+(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g 
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
+Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity 
+g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g 
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat 
+Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
+
+lemma sc3_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c 
+t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: 
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 
+g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: 
+A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to 
+(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 
+g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall 
+(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 
+c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda 
+(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 
+in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda 
+(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def 
+(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n 
+n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) 
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: 
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort 
+n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 
+x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in 
+(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) 
+H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity 
+g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) 
+(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to 
+(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: 
+A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: 
+C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to 
+(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) 
+\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: 
+((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: 
+T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c 
+t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t 
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is 
+t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 
+\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: 
+T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity 
+g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a 
+d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat 
+Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in 
+(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a 
+a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: 
+A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: 
+A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 
+x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A 
+(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) 
+(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall 
+(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d 
+c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t 
+(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: 
+C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: 
+PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: 
+(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 
+t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 
+(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat 
+Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) 
+(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 
+H11))))))) H7))))) H4)))))))))))) a2)) a1)).
+
+lemma sc3_lift:
+ \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e 
+t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) 
+\to (sc3 g a c (lift h d t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: 
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda 
+(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in 
+(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) 
+(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n 
+n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) 
+(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e 
+t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: 
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d 
+t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: 
+T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: 
+C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall 
+(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d 
+e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: 
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 
+\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall 
+(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g 
+a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) 
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda 
+(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
+PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) 
+(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: 
+T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 
+is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 
+(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w 
+t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) 
+(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
+
+lemma sc3_lift1:
+ \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: 
+PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) 
+\to (sc3 g a c (lift1 hds t)))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: 
+PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g 
+a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: 
+C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c 
+e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: 
+C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 
+g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: 
+C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n 
+n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x 
+in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 
+e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n 
+n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 
+H4) c n n0 H3)))) H2))))))))))) hds)))).
+
+lemma sc3_abbr:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: 
+nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads 
+(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to 
+(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
+C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
+(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef 
+i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: 
+C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) 
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda 
+(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g 
+c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat 
+Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef 
+i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: 
+(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda 
+(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c 
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs 
+(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) 
+(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda 
+(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: 
+T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to 
+((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs 
+(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: 
+TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: 
+C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c 
+(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef 
+i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda 
+(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs 
+(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 
+d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat 
+Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda 
+(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity 
+g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
+(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead 
+a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: 
+PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads 
+(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: 
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) 
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift 
+(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) 
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall 
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs 
+(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 
+w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def 
+(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C 
+(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is 
+i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: 
+C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) 
+d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w 
+(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is 
+(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r 
+T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w 
+(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans 
+is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 
+d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T 
+(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 
+d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) 
+vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) 
+H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs 
+(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) 
+H3))))))))))))) a)).
+
+theorem sc3_cast:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
+(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: 
+T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) 
+vs (THead (Flat Cast) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: 
+TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat 
+Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to 
+(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | 
+(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: 
+T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) 
+(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g 
+(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow 
+(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads 
+(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land 
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) 
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: 
+(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: 
+(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat 
+Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs 
+u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads 
+(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat 
+Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs 
+t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat 
+Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat 
+Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs 
+t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort 
+O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) 
+(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t 
+H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with 
+[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c 
+(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) 
+(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 
+n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads 
+(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let 
+H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) 
+(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) 
+(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def 
+H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 
+c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort 
+(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g 
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs 
+(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n 
+H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall 
+(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to 
+(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: 
+A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 
+g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c 
+(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead 
+(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: 
+T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc 
+g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
+is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land 
+(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall 
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
+d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 
+\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g 
+a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 
+is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 
+g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u 
+t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 
+(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d 
+w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead 
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 
+in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity 
+g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) 
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c 
+u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: 
+(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y 
+\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d 
+(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 
+is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) 
+(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat 
+Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w 
+t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) 
+(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat 
+Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w 
+H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl 
+is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) 
+(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl 
+is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
+
+fact sc3_props__sc3_sn3_abst:
+ \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g 
+a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def 
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to 
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: 
+C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: 
+TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in 
+(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
+(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall 
+(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 
+c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c 
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to 
+((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n 
+n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: 
+C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c 
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c 
+t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) 
+H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: 
+(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: 
+(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat 
+Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H 
+(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land 
+(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall 
+(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) 
+vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c 
+(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: 
+(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
+(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to 
+(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: 
+C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads 
+(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c 
+vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: 
+T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall 
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 
+d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind 
+(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) 
+(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads 
+(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to 
+(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: 
+((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 
+t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) 
+\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef 
+i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: 
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to 
+((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs 
+(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: 
+T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: 
+TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs 
+(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 
+(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind 
+(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) 
+\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
+(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 
+a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to 
+(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
+(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) 
+in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: 
+C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 
+O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 
+(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) 
+(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 
+H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) 
+I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) 
+(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 
+(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil 
+(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) 
+x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 
+(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S 
+x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef 
+O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O 
+t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop 
+(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) 
+(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g 
+c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c 
+(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) 
+vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) 
+\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w 
+(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: 
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: 
+PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall 
+(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
+vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) 
+w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: 
+C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: 
+((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 
+(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 
+c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 
+\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to 
+(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef 
+i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef 
+i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs 
+(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) 
+\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) 
+\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat 
+Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) 
+in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) 
+(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef 
+(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat 
+Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) 
+(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 
+is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) 
+(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 
+(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 
+(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) 
+(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is 
+(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is 
+(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) 
+(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is 
+vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) 
+(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat 
+Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)).
+
+lemma sc3_sn3:
+ \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c 
+t) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: 
+(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def 
+H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 
+c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g 
+c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 
+vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) 
+(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 
+t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: 
+C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) 
+\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef 
+i))))))))))).(H1 c t H))) H0))))))).
+
+lemma sc3_abst:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall 
+(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef 
+i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda 
+(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) 
+a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def 
+(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: 
+C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: 
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) 
+vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a 
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) 
+vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) 
+\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: 
+nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to 
+((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) 
+vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))).
+
+theorem sc3_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: 
+A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: 
+T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts 
+(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs 
+(THead (Bind b) v t)))))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda 
+(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall 
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads 
+(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads 
+(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: 
+T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat 
+Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 
+in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O 
+vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S 
+O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) 
+(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda 
+(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) 
+(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind 
+b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) 
+(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) 
+H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) 
+(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall 
+(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) 
+vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall 
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead 
+c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) 
+\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v 
+t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda 
+(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) 
+(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a 
+d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g 
+a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) 
+t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind 
+(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) 
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall 
+(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat 
+Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land 
+(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c 
+(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda 
+(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
+PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) 
+w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity 
+g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: 
+C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) 
+\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead 
+(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 
+H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 
+g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def 
+(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is 
+vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead 
+(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) 
+t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) 
+(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList 
+(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d 
+(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat 
+Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) 
+(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is 
+v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is 
+v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S 
+O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) 
+(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts 
+(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O 
+vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is 
+d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is 
+(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead 
+(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))).
+
+theorem sc3_appl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: 
+TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: 
+T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead 
+(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: 
+A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 
+g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) 
+\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat 
+Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: 
+T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead 
+(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda 
+(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat 
+Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs 
+(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads 
+(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: 
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n 
+n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v 
+t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead 
+(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat 
+Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen 
+g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) 
+(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) 
+H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall 
+(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g 
+(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall 
+(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c 
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to 
+(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) 
+vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: 
+TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land 
+(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) 
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads 
+(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c 
+v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 
+in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) 
+(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall 
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
+(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c 
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead 
+a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: 
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is 
+(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w 
+t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind 
+Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: 
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v 
+t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: 
+T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d 
+(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v 
+(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g 
+c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) 
+(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: 
+PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 
+is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda 
+(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat 
+Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 
+g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) 
+(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: 
+T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) 
+(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 
+is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead 
+(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads 
+(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs 
+(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 
+t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead 
+(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead 
+(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) 
+(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d 
+w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) 
+(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is 
+v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat 
+Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v 
+(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sn3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/sn3/defs.ma
new file mode 100644 (file)
index 0000000..85a0b7d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sn3/defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr3/defs.ma".
+
+inductive sn3 (c: C): T \to Prop \def
+| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
+
+rec definition sns3 (c: C) (ts: TList) on ts: Prop \def match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sn3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/sn3/fwd.ma
new file mode 100644 (file)
index 0000000..aab2890
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sn3/fwd.ma
@@ -0,0 +1,189 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sn3/defs.ma".
+
+include "basic_1A/pr3/props.ma".
+
+implied rec lemma sn3_ind (c: C) (P: (T \to Prop)) (f: (\forall (t1: 
+T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P0: Prop).P0))) \to ((pr3 
+c t1 t2) \to (sn3 c t2))))) \to (((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P0: Prop).P0))) \to ((pr3 c t1 t2) \to (P t2))))) \to (P t1))))) 
+(t: T) (s0: sn3 c t) on s0: P t \def match s0 with [(sn3_sing t1 s1) 
+\Rightarrow (f t1 s1 (\lambda (t2: T).(\lambda (p: (((eq T t1 t2) \to 
+(\forall (P0: Prop).P0)))).(\lambda (p0: (pr3 c t1 t2)).((sn3_ind c P f) t2 
+(s1 t2 p p0))))))].
+
+lemma sn3_gen_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0: 
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 (CHead c (Bind b) u) t))) 
+(\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T 
+y (THead (Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) 
+(unintro T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) 
+\to (land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda 
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to 
+(land (sn3 c x) (sn3 (CHead c (Bind b) x) x0)))))) (\lambda (t1: T).(\lambda 
+(H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
+c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall 
+(x0: T).((eq T t2 (THead (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c 
+(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T 
+t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1 
+x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead 
+(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall 
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to 
+(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c 
+(Bind b) x) x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x t2) 
+\to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4 
+(THead (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind 
+b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead 
+_ t0 _) \Rightarrow t0])) (THead (Bind b) x x0) (THead (Bind b) t2 x0) H8) in 
+(let H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x t0)) H7 x H9) in (let 
+H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7 
+(Bind b) x0 x0 (pr3_refl (CHead c (Bind b) t2) x0)) t2 x0 (refl_equal T 
+(THead (Bind b) t2 x0))) in (land_ind (sn3 c t2) (sn3 (CHead c (Bind b) t2) 
+x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 (CHead c (Bind b) 
+t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b) x) x0 (\lambda (t2: 
+T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
+(pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4 (THead (Bind b) x t2) 
+(\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind b) x t2))).(\lambda 
+(P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind b) x x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T 
+t2 (\lambda (t0: T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x 
+(pr3_refl c x) (Bind b) x0 t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) 
+in (land_ind (sn3 c x) (sn3 (CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) 
+x) t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) 
+t2)).H10)) H8))))))))))))))) y H0))))) H))))).
+
+lemma sn3_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0: 
+T).(sn3 c t0)) (\lambda (_: T).(land (sn3 c u) (sn3 c t))) (\lambda (y: 
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead 
+(Flat f) u t0)) \to (land (sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: 
+T).(\forall (x: T).((eq T y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 
+c x))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T 
+t0 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: 
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land 
+(sn3 c x) (sn3 c x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: 
+(eq T t1 (THead (Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0: 
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Flat f) x1 
+x2)) \to (land (sn3 c x1) (sn3 c x2))))))))) H2 (THead (Flat f) x x0) H3) in 
+(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead 
+(Flat f) x x0) H3) in (conj (sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: 
+T).(\lambda (H6: (((eq T x t2) \to (\forall (P: Prop).P)))).(\lambda (H7: 
+(pr3 c x t2)).(let H8 \def (H4 (THead (Flat f) t2 x0) (\lambda (H8: (eq T 
+(THead (Flat f) x x0) (THead (Flat f) t2 x0))).(\lambda (P: Prop).(let H9 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) x 
+x0) (THead (Flat f) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 
+T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: 
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T 
+x) P)))))) (pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl (CHead c (Flat f) 
+t2) x0)) t2 x0 (refl_equal T (THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) 
+(sn3 c x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) 
+H8)))))) (sn3_sing c x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead 
+(Flat f) x t2) (\lambda (H8: (eq T (THead (Flat f) x x0) (THead (Flat f) x 
+t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) 
+\Rightarrow t0])) (THead (Flat f) x x0) (THead (Flat f) x t2) H8) in (let H10 
+\def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 
+\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: 
+Prop).P0))) H6 x0 H9) in (H11 (refl_equal T x0) P)))))) (pr3_thin_dx c x0 t2 
+H7 x f) x t2 (refl_equal T (THead (Flat f) x t2))) in (land_ind (sn3 c x) 
+(sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c x)).(\lambda (H10: (sn3 c 
+t2)).H10)) H8))))))))))))))) y H0))))) H))))).
+
+lemma sn3_gen_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c 
+(THead k u t)) \to (sn3 c u)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u: 
+T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b: 
+B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in 
+(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3 
+c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f: 
+F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in 
+(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_: 
+(sn3 c t)).H1)) H0)))))))) k).
+
+lemma sn3_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead 
+c (Flat f) u) t) \to (sn3 c t)))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: 
+(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0: 
+T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
+(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T 
+t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to 
+(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) 
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2 
+(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
+
+lemma sn3_gen_lift:
+ \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 
+(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1 
+t0)) (\lambda (_: T).(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))) 
+(\lambda (y: T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq 
+T y (lift h d t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) 
+(sn3_ind c1 (\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to 
+(\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: 
+T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c1 t1 t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to 
+(\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2: C).((drop h d c1 
+c2) \to (sn3 c2 x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d 
+x))).(\lambda (c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T 
+t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c1 t0 t2) \to (\forall (x0: T).((eq T t2 (lift h d x0)) 
+\to (\forall (c3: C).((drop h d c1 c3) \to (sn3 c3 x0))))))))) H2 (lift h d 
+x) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq 
+T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t0 t2) \to (sn3 c1 t2))))) 
+H1 (lift h d x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T 
+x t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d 
+t2) (\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let 
+H10 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c2 x t0)) H8 x (lift_inj x t2 h 
+d H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to 
+(\forall (P0: Prop).P0))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T 
+x) P))))) (pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2 
+H4)))))))))))))) y H0)))) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sn3/lift1.ma b/matita/matita/contribs/lambdadelta/basic_1A/sn3/lift1.ma
new file mode 100644 (file)
index 0000000..67e0238
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sn3/lift1.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sn3/props.ma".
+
+include "basic_1A/drop1/fwd.ma".
+
+include "basic_1A/lift1/props.ma".
+
+lemma sns3_lifts1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to 
+(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c 
+(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda 
+(ts: TList).(\lambda (H0: (sns3 e ts)).(let H_y \def (drop1_gen_pnil c e H) 
+in (eq_ind_r C e (\lambda (c0: C).(sns3 c0 (lifts1 PNil ts))) (eq_ind_r TList 
+ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c 
+H_y)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to 
+(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 
+p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H_x \def 
+(drop1_gen_pcons c e p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 e)) (sns3 c (lifts1 
+(PCons n n0 p) ts)) (\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda 
+(H4: (drop1 p x e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: 
+TList).(sns3 c t)) (sns3_lifts c x n n0 H3 (lifts1 p ts) (H x H4 ts H1)) 
+(lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts))))) H2))))))))))) hds)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1A/sn3/nf2.ma
new file mode 100644 (file)
index 0000000..e00ecab
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sn3/nf2.ma
@@ -0,0 +1,60 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sn3/fwd.ma".
+
+include "basic_1A/nf2/dec.ma".
+
+include "basic_1A/nf2/pr3.ma".
+
+lemma sn3_nf2:
+ \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
+(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 
+H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) 
+in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: 
+Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
+(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
+
+lemma nf2_sn3:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
+t u)) (\lambda (u: T).(nf2 c u)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda 
+(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let 
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 
+c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c 
+t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 
+x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 
+x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: 
+T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 
+u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x 
+x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) 
+H2)))))) t H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sn3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/sn3/props.ma
new file mode 100644 (file)
index 0000000..fdab5af
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sn3/props.ma
@@ -0,0 +1,2403 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sn3/nf2.ma".
+
+include "basic_1A/nf2/iso.ma".
+
+include "basic_1A/pr3/iso.ma".
+
+lemma sn3_pr3_trans:
+ \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 
+t2) \to (sn3 c t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda 
+(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2: 
+T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall 
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
+(\forall (t4: T).((pr3 c t3 t4) \to (sn3 c t4)))))))).(\lambda (t3: 
+T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3: 
+(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let 
+H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T 
+t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let 
+H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8 
+\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P))) 
+H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2 
+H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P: 
+Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
+
+lemma sn3_pr2_intro:
+ \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c 
+t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to 
+(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in 
+((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall 
+(t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) \to ((pr2 c t t3) \to (sn3 
+c t3))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0))))) 
+(\lambda (t: T).(\lambda (H4: ((\forall (t3: T).((((eq T t t3) \to (\forall 
+(P: Prop).P))) \to ((pr2 c t t3) \to (sn3 c t3)))))).(\lambda (H5: (((eq T t 
+t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t)))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5: 
+T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t6: T).((((eq T t3 
+t6) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t6) \to (sn3 c t6))))) \to 
+((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7: 
+((\forall (t6: T).((((eq T t4 t6) \to (\forall (P: Prop).P))) \to ((pr2 c t4 
+t6) \to (sn3 c t6)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P: 
+Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq 
+T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10: 
+(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to 
+(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t: 
+T).(\forall (t6: T).((((eq T t t6) \to (\forall (P: Prop).P))) \to ((pr2 c t 
+t6) \to (sn3 c t6))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t: 
+T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3) 
+\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5)) 
+H9))))))))))) t1 t2 H1 H3)) H2)))))))).
+
+theorem sn3_cast:
+ \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to 
+(sn3 c (THead (Flat Cast) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda 
+(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0))))) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2 
+t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0: 
+T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3: 
+((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 
+t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1 
+t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2: 
+T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def 
+(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c 
+t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0 
+x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12 
+\def (eq_ind T t2 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) t3) \to 
+(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T 
+(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def 
+(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1) 
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14: 
+(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t3: T).((eq T (THead (Flat 
+Cast) t1 t0) (THead (Flat Cast) t3 x1)) \to (\forall (P: Prop).P))) H12 t1 
+H14) in (let H16 \def (eq_ind T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 
+H14) in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) 
+(let H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) 
+((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1)) 
+(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t3: 
+T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t3)) \to (\forall 
+(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t3: 
+T).(pr2 c t0 t3)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead 
+(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c 
+(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to 
+(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14)))) 
+(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda 
+(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda 
+(t3: T).((eq T t3 t1) \to (\forall (P0: Prop).P0))) H14 t1 H15) in (let H17 
+\def (eq_ind_r T x0 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead 
+(Flat Cast) t3 x1)) \to (\forall (P0: Prop).P0))) H12 t1 H15) in (let H18 
+\def (eq_ind_r T x0 (\lambda (t3: T).(pr2 c t1 t3)) H10 t1 H15) in (H16 
+(refl_equal T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec 
+t0 x1) in (let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to 
+(\forall (P: Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def 
+(eq_ind_r T x1 (\lambda (t3: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat 
+Cast) x0 t3)) \to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def 
+(eq_ind_r T x1 (\lambda (t3: T).(pr2 c t0 t3)) H11 t0 H16) in (eq_ind T t0 
+(\lambda (t3: T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: 
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 
+H11))) H15))))) H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 
+t2)).(sn3_pr3_trans c t0 (sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) 
+H7))))))))) t H2)))))) u H))).
+
+lemma sn3_cflat:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u: 
+T).(sn3 (CHead c (Flat f) u) t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f: 
+F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) t0)) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1 
+(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2 
+(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
+
+lemma sn3_shift:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c 
+(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let 
+H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c 
+(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b) 
+v) t)).H2)) H0))))))).
+
+lemma sn3_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1: 
+T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3 
+(CHead c (Bind b) v2) t)))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda 
+(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda 
+(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind 
+b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3 
+(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to 
+(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1 
+(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P: 
+Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3 
+(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4 
+v1)))))))))) t H0))))))).
+
+lemma sn3_gen_def:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef 
+i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v) 
+(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef 
+i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop 
+Abbr c d v i H))))))).
+
+lemma sn3_cdelta:
+ \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T 
+(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d: 
+C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
+\def
+ \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w: 
+T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0 
+\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c: 
+C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to 
+(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind 
+(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall 
+(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1) 
+\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr) 
+v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3))))))) 
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
+(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda 
+(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3 
+c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4 
+H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda 
+(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda 
+(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr) 
+v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) 
+v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0 
+(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0) 
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3 
+c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s 
+(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: 
+C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to 
+(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def 
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 
+(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11: 
+(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b 
+(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4) 
+H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 
+t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0) 
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda 
+(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8) 
+in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda 
+(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3 
+H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c: 
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to 
+(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d: 
+C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d 
+v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d 
+(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def 
+(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1))) 
+H0)))))).
+
+lemma sn3_cpr3_trans:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
+(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2) 
+t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1 
+u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1) 
+t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0)) 
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1) 
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2) 
+t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T 
+t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1 
+t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
+
+theorem sn3_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: 
+T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c 
+u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t) 
+t0) \to (sn3 c (THead (Bind b) t t0))))) (\lambda (t1: T).(\lambda (_: 
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 
+t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (t: T).((sn3 (CHead c 
+(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t: 
+T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b) 
+t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2: 
+T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b) 
+t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b) 
+t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda 
+(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda 
+(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst) 
+in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3) 
+(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c 
+(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b 
+(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P: 
+Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall 
+(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind 
+b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let 
+H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3 
+(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b 
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to 
+(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def 
+(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0 
+x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall 
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3 
+(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P: 
+Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind 
+Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in 
+(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let 
+H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) 
+(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let 
+H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in 
+(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1 
+\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda 
+(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T 
+(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P: 
+Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) 
+H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1 
+t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst) 
+t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P: 
+Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20: 
+(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1) 
+in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let 
+H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda 
+(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans 
+c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1 
+H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20 
+H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst 
+t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b 
+Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind 
+b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
+(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b) 
+t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq 
+T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13: 
+(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0: 
+T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead 
+(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
+T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in 
+(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0 
+(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to 
+(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda 
+(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c 
+(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def 
+H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r 
+T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0)) 
+\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1 
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T 
+t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead 
+(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20: 
+(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0 
+H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2 
+\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2 
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18: 
+(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c 
+(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c 
+(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind 
+b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq 
+T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1 
+x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10)) 
+(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O 
+t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c 
+(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10) 
+c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t 
+H2)))))) u H)))).
+
+theorem sn3_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v 
+t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead 
+(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3 
+c t0)) (\lambda (_: T).(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat 
+Appl) v (THead (Bind Abst) w t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c 
+y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to 
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) 
+w t0))))))) (unintro T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead 
+(Bind Abbr) t0 x)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat 
+Appl) t0 (THead (Bind Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall 
+(x: T).(\forall (x0: T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: 
+T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) w 
+x0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda 
+(H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 
+c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Bind Abbr) x 
+x0)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead 
+(Bind Abst) w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: 
+(eq T t1 (THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c 
+w)).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: 
+T).(\forall (x2: T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: 
+T).((sn3 c w0) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 
+x2)))))))))))) H2 (THead (Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 
+(\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t0 t2) \to (sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in 
+(sn3_ind c (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 
+x0)))) (\lambda (t2: T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: 
+((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 
+t3) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 
+x0)))))))).(sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
+(\lambda (t3: T).(\lambda (H9: (((eq T (THead (Flat Appl) x (THead (Bind 
+Abst) t2 x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H10: (pr2 c (THead 
+(Flat Appl) x (THead (Bind Abst) t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x 
+(THead (Bind Abst) t2 x0) t3 H10) in (or3_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+(THead (Bind Abst) t2 x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H12: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+(THead (Bind Abst) t2 x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) 
+t2 x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq 
+T t3 (THead (Flat Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: 
+(pr2 c (THead (Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda 
+(t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to 
+(\forall (P: Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T 
+(THead (Flat Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def 
+(pr2_gen_abst c t2 x0 x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T x2 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t2 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) 
+(\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) 
+x3 x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind 
+T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) 
+(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst) 
+x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c 
+(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def 
+H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2 
+x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0 
+x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0: 
+T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def 
+(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to 
+(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2 
+x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0: 
+T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) 
+t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let 
+H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind 
+T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2 
+x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T 
+x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
+(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind 
+Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall 
+(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 
+H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead 
+(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind 
+Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4 
+H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead 
+(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind 
+Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x x4) H32) in (let H34 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H31 x0 H33) in (let H35 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H20 x0 H33) in (H34 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H20 
+Abbr x))) x x4 (refl_equal T (THead (Bind Abbr) x x4)) t2 (sn3_sing c t2 
+H7))) H30))) x1 H27)))) (\lambda (H27: (((eq T x x1) \to (\forall (P: 
+Prop).P)))).(H5 (THead (Bind Abbr) x1 x4) (\lambda (H28: (eq T (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x1 x4))).(\lambda (P: Prop).(let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
+_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x1 x4) H28) in (\lambda (H31: (eq T x x1)).(let H32 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H20 x0 H30) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x 
+t0) \to (\forall (P0: Prop).P0))) H27 x H31) in (let H34 \def (eq_ind_r T x1 
+(\lambda (t0: T).(pr2 c x t0)) H14 x H31) in (H33 (refl_equal T x) P)))))) 
+H29)))) (pr3_head_12 c x x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 
+(CHead c (Bind Abbr) x1) x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead 
+(Bind Abbr) x1 x4)) t2 (sn3_sing c t2 H7))) H26))) x3 H23)))) (\lambda (H23: 
+(((eq T t2 x3) \to (\forall (P: Prop).P)))).(let H_x0 \def (term_dec x x1) in 
+(let H24 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda 
+(H25: (eq T x x1)).(let H26 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
+t0)) H14 x H25) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 
+(THead (Bind Abst) x3 x4)))) (let H_x1 \def (term_dec x0 x4) in (let H27 \def 
+H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 
+x4)).(let H29 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (eq_ind T x0 
+(\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) x3 t0)))) (H8 
+x3 H23 (pr3_pr2 c t2 x3 H19)) x4 H28))) (\lambda (H28: (((eq T x0 x4) \to 
+(\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x x4) (\lambda (H29: (eq T 
+(THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4))).(\lambda (P: Prop).(let 
+H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 
+| (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4 
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in 
+(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0) 
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) 
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead 
+(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25))) 
+(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind 
+Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t0 _) 
+\Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in 
+((let H28 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) 
+(THead (Bind Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq 
+T x x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let 
+H31 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: 
+Prop).P0))) H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 
+c x t0)) H14 x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x 
+x1 (pr3_pr2 c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) 
+x0 x4 (H20 Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 
+x3 H23 (pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 
+H13))))))) H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3 
+x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) 
+\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T 
+(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal 
+T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t2 | (TLRef _) 
+\Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) t2 x0) 
+(THead (Bind Abst) x1 x2) H13) in ((let H19 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) 
+x1 x2) H13) in (\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 
+x4)))) H16 x0 H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in 
+(or_ind (eq T x x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Bind Abbr) x3 x4)) (\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: 
+T).(sn3 c (THead (Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let 
+H25 \def H_x0 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let 
+H27 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x0 t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: 
+T).(sn3 c (THead (Bind Abbr) x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) 
+x4 H26))) (\lambda (H26: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 
+(THead (Bind Abbr) x x4) (\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead 
+(Bind Abbr) x x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2 c (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4 (Bind Abbr) (H21 
+Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3) \to (\forall (P: 
+Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq T (THead (Bind 
+Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P: Prop).(let H25 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef 
+_) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) 
+(THead (Bind Abbr) x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr) 
+x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def (eq_ind_r T x4 
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 
+t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T x 
+t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30 \def (eq_ind_r T x3 
+(\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29 (refl_equal T x) P)))))) 
+H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15) (Bind Abbr) x0 x4 (pr3_pr2 
+(CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3))))) H22)))))) H18)) t3 
+H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+Abst) t2 x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 x0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14: 
+(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq 
+T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c 
+(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P: 
+Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
+x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | 
+(THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) 
+in ((let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H22 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) t2 
+x0) (THead (Bind x1) x2 x3) H14) in (\lambda (H23: (eq T t2 x2)).(\lambda 
+(H24: (eq B Abst x1)).(let H25 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 
+(CHead c (Bind x1) x6) t0 x4)) H18 x0 H22) in (let H26 \def (eq_ind_r T x2 
+(\lambda (t0: T).(pr2 c t0 x6)) H17 t2 H23) in (let H27 \def (eq_ind_r B x1 
+(\lambda (b: B).(pr2 (CHead c (Bind b) x6) x0 x4)) H25 Abst H24) in (let H28 
+\def (eq_ind_r B x1 (\lambda (b: B).(not (eq B b Abst))) H13 Abst H24) in 
+(eq_ind B Abst (\lambda (b: B).(sn3 c (THead (Bind b) x6 (THead (Flat Appl) 
+(lift (S O) O x5) x4)))) (let H29 \def (match (H28 (refl_equal B Abst)) in 
+False with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12)) 
+H11))))))))) w H4))))))))))) y H0))))) H)))).
+
+lemma sn3_appl_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: 
+T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead 
+(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef 
+i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2: 
+T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall 
+(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i)) 
+t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
+(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1 
+x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) 
+H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1) 
+(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq 
+T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda 
+(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i) 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1 
+x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T 
+t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat 
+Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P: 
+Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c 
+t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t 
+(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c 
+(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0) 
+\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H 
+x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T 
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8: 
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
+H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) 
+t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r 
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind 
+T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
+(Bind Abst) x0 x1) H7) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H12)) 
+t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H8: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda (H9: 
+(eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P))) H3 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H9) in 
+(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H8) in 
+(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3))) H14)) t2 H9)))))))))))))) H6)) H5))))))))) v H0))))).
+
+lemma sn3_appl_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v 
+(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c 
+(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v 
+(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (\lambda (_: T).(sn3 c (THead 
+(Flat Appl) v (TLRef i)))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro 
+T v (\lambda (t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 
+c (THead (Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: 
+T).((eq T t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat 
+Appl) x (TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
+t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat 
+Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef 
+i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift 
+(S i) O w)))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: 
+T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall 
+(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead 
+(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w)) 
+H4) in (let H6 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t 
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H2 
+(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat 
+Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl) 
+x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead 
+(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8) 
+in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) 
+(sn3 c t2) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))))).(ex3_2_ind T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: 
+T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c 
+x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2 
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: 
+Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat 
+Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i 
+H13) in (or_ind (eq T x1 (TLRef i)) (ex2_2 C T (\lambda (d0: C).(\lambda (u: 
+T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq 
+T x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16: 
+(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead 
+(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x 
+in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def 
+(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead 
+(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21 
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T 
+(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0 
+H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead 
+(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x 
+(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P: 
+Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) H20) in (let H22 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to 
+(\forall (P0: Prop).P0))) H19 x H21) in (let H23 \def (eq_ind_r T x0 (\lambda 
+(t: T).(pr2 c x t)) H12 x H21) in (H22 (refl_equal T x) P)))))) (pr3_pr2 c 
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O 
+w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))) x0 (refl_equal T 
+(THead (Flat Appl) x0 (lift (S i) O w))))) H18))) x1 H16))) (\lambda (H16: 
+(ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) 
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
+u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x1 (lift (S i) O 
+u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr) x3))).(\lambda (H18: (eq T 
+x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1 (\lambda (t: T).((eq T 
+(THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P: 
+Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T (lift (S i) O x3) 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20 \def (eq_ind C 
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x2 (Bind 
+Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) 
+H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) 
+(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 
+(Bind Abbr) x3) H17)) in ((let H22 \def (f_equal C T (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind 
+Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H 
+(CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24 \def 
+(eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20 w 
+H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S i) 
+O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0 
+(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def 
+H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c 
+(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28 
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x 
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c 
+(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x 
+x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w)) 
+(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat 
+Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x | (TLRef _) 
+\Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S 
+i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) H28) in (let H30 \def 
+(eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H27 
+x H29) in (let H31 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x 
+H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift 
+(S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 
+(Flat Appl) (lift (S i) O w))))) H26)))) x3 H22)))) H21))) x1 H18)))))) H16)) 
+H15)) t2 H11))))))) H10)) (\lambda (H10: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) 
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H11: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H12: 
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c x x2)).(\lambda (_: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let 
+H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) 
+t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2 x3) H12) in (eq_ind_r 
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H16 \def (eq_ind 
+T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
+(Bind Abst) x0 x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) 
+t2 H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq 
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(sn3 c t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 
+Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda 
+(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: 
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) in 
+(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) 
+(\lambda (t: T).(sn3 c t)) (let H18 \def (eq_ind T (TLRef i) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H12) in 
+(False_ind (sn3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) 
+x3))) H18)) t2 H13)))))))))))))) H10)) H9))))))))))))) y H1)))) H0))))))).
+
+theorem sn3_appl_cast:
+ \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v 
+u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead 
+(Flat Appl) v (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead 
+(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3 
+c t)) (\lambda (_: T).(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to 
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t0)))))) (\lambda (y: 
+T).(\lambda (H0: (sn3 c y)).(unintro T u (\lambda (t: T).((eq T y (THead 
+(Flat Appl) v t)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to 
+(sn3 c (THead (Flat Appl) v (THead (Flat Cast) t t0))))))) (unintro T v 
+(\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to 
+(\forall (t0: T).((sn3 c (THead (Flat Appl) t t0)) \to (sn3 c (THead (Flat 
+Appl) t (THead (Flat Cast) x t0)))))))) (sn3_ind c (\lambda (t: T).(\forall 
+(x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (t0: 
+T).((sn3 c (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead 
+(Flat Cast) x0 t0))))))))) (\lambda (t1: T).(\lambda (H1: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c 
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 
+(THead (Flat Appl) x x0)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) x 
+t)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 
+t))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead 
+(Flat Appl) x x0))).(\lambda (t: T).(\lambda (H4: (sn3 c (THead (Flat Appl) x 
+t))).(insert_eq T (THead (Flat Appl) x t) (\lambda (t0: T).(sn3 c t0)) 
+(\lambda (_: T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t)))) 
+(\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0: T).((eq 
+T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat 
+Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0 
+(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) 
+x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda 
+(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 
+c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1: 
+T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T 
+t0 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).((eq T t3 (THead (Flat 
+Appl) x x2)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 
+x2))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0 
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H6 (THead (Flat Appl) x x1) H8) in (let 
+H11 \def (eq_ind T t1 (\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x2: T).(\forall (x3: 
+T).((eq T t3 (THead (Flat Appl) x2 x3)) \to (\forall (t4: T).((sn3 c (THead 
+(Flat Appl) x2 t4)) \to (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x3 
+t4)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1 
+(\lambda (t2: T).(\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t2 t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in 
+(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda 
+(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 
+x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat 
+Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x 
+(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T 
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c 
+(THead (Flat Cast) x0 x1) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast) 
+x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq 
+T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19: 
+(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda 
+(t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to 
+(\forall (P: Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T 
+(THead (Flat Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def 
+(pr2_gen_cast c x0 x1 x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c 
+x1 x3) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T x3 (THead (Flat Cast) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c x1 
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead 
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2 
+x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat 
+Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1 
+x5)).(let H26 \def (eq_ind T x3 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall (P: 
+Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat 
+Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x 
+\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let 
+H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
+x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5))) 
+(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 
+x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) \Rightarrow t3])) 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28) in ((let H30 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t3) \Rightarrow t3])) (THead (Flat Appl) x x0) 
+(THead (Flat Appl) x2 x4) H28) in (\lambda (H31: (eq T x x2)).(let H32 \def 
+(eq_ind_r T x4 (\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat 
+Cast) x0 x1)) (THead (Flat Appl) x2 (THead (Flat Cast) t3 x5))) \to (\forall 
+(P: Prop).P))) H26 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t3: 
+T).(pr2 c x0 t3)) H24 x0 H30) in (eq_ind T x0 (\lambda (t3: T).(sn3 c (THead 
+(Flat Appl) x2 (THead (Flat Cast) t3 x5)))) (let H34 \def (eq_ind_r T x2 
+(\lambda (t3: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) 
+(THead (Flat Appl) t3 (THead (Flat Cast) x0 x5))) \to (\forall (P: Prop).P))) 
+H32 x H31) in (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c x t3)) H18 
+x H31) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 (THead 
+(Flat Cast) x0 x5)))) (let H_x0 \def (term_dec (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x x5)) in (let H36 \def H_x0 in (or_ind (eq T (THead (Flat 
+Appl) x x1) (THead (Flat Appl) x x5)) ((eq T (THead (Flat Appl) x x1) (THead 
+(Flat Appl) x x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x x5))).(let H38 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x x5) H37) in 
+(let H39 \def (eq_ind_r T x5 (\lambda (t3: T).((eq T (THead (Flat Appl) x 
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x (THead (Flat Cast) x0 t3))) 
+\to (\forall (P: Prop).P))) H34 x1 H38) in (let H40 \def (eq_ind_r T x5 
+(\lambda (t3: T).(pr2 c x1 t3)) H25 x1 H38) in (eq_ind T x1 (\lambda (t3: 
+T).(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t3)))) (H39 (refl_equal 
+T (THead (Flat Appl) x (THead (Flat Cast) x0 x1))) (sn3 c (THead (Flat Appl) 
+x (THead (Flat Cast) x0 x1)))) x5 H38))))) (\lambda (H37: (((eq T (THead 
+(Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall (P: Prop).P)))).(H9 
+(THead (Flat Appl) x x5) H37 (pr3_pr2 c (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x x5) (pr2_thin_dx c x1 x5 H25 x Appl)) x5 (refl_equal T (THead (Flat 
+Appl) x x5)))) H36))) x2 H31))) x4 H30))))) H29))) (\lambda (H28: (((eq T 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall (P: 
+Prop).P)))).(let H_x0 \def (term_dec (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x2 x5)) in (let H29 \def H_x0 in (or_ind (eq T (THead (Flat Appl) x x1) 
+(THead (Flat Appl) x2 x5)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) 
+x2 x5)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat 
+Cast) x4 x5))) (\lambda (H30: (eq T (THead (Flat Appl) x x1) (THead (Flat 
+Appl) x2 x5))).(let H31 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in 
+((let H32 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in (\lambda (H33: (eq 
+T x x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t3: T).(pr2 c x1 t3)) H25 x1 
+H32) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead 
+(Flat Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x x0) (THead (Flat Appl) t3 x4)) \to (\forall (P: 
+Prop).P))) H28 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda (t3: T).(pr2 c 
+x t3)) H18 x H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead (Flat Appl) 
+t3 (THead (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35 (pr3_pr2 c 
+(THead (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c x0 x4 H24 x 
+Appl)) x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing c (THead 
+(Flat Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30: (((eq T 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P: 
+Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_flat c x x2 (pr3_pr2 c x 
+x2 H18) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x2 x4 (refl_equal T (THead (Flat 
+Appl) x2 x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_flat c x x2 (pr3_pr2 
+c x x2 H18) x1 x5 (pr3_pr2 c x1 x5 H25) Appl)))) H29)))) H27))) x3 H23))))))) 
+H22)) (\lambda (H22: (pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat 
+Appl) x x1) (THead (Flat Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) 
+x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead 
+(Flat Appl) x2 x3)) (\lambda (H24: (eq T (THead (Flat Appl) x x1) (THead 
+(Flat Appl) x2 x3))).(let H25 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t3 _) 
+\Rightarrow t3])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in 
+((let H26 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t3) \Rightarrow t3])) 
+(THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in (\lambda (H27: (eq 
+T x x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t3: T).(pr2 c x1 t3)) H22 x1 
+H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t3: T).((eq T (THead (Flat 
+Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t3)) \to (\forall 
+(P: Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead 
+(Flat Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t3 x1)) 
+\to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def (eq_ind_r T x2 
+(\lambda (t3: T).(pr2 c x t3)) H18 x H27) in (eq_ind T x (\lambda (t3: 
+T).(sn3 c (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat Appl) x x1) 
+H10) x2 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead (Flat Appl) x 
+x1) (THead (Flat Appl) x2 x3)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat 
+Appl) x2 x3) H24 (pr3_flat c x x2 (pr3_pr2 c x x2 H18) x1 x3 (pr3_pr2 c x1 x3 
+H22) Appl))) H23)))) H21)) t2 H17))))))) H16)) (\lambda (H16: (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Flat Cast) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+u0) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: 
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) (sn3 c t2) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2 x3))).(\lambda 
+(H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c x 
+x4)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) 
+u0) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t3: T).((eq T (THead 
+(Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: Prop).P))) H13 
+(THead (Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) 
+(\lambda (t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 
+x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 
+x3) H17) in (False_ind (sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 
+H18)))))))))) H16)) (\lambda (H16: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat 
+Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) 
+(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18: 
+(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq 
+T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda 
+(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c 
+(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t3: T).((eq T 
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t3) \to (\forall (P: 
+Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) 
+H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) 
+x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast) 
+x0 x1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) 
+H18) in (False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) 
+O x6) x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5)))) 
+H4))))))))) y H0))))) H)))).
+
+theorem sn3_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
+T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u) 
+(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v 
+(THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0: 
+T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O) 
+O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t t0))))))) 
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall 
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
+(\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat 
+Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2 
+t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead 
+(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
+t0)) (\lambda (_: T).(sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t)))) 
+(\lambda (y: T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t 
+(\lambda (t0: T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c 
+(THead (Flat Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: 
+T).(\forall (x: T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 
+c (THead (Flat Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) 
+t1) (\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat 
+Appl) (lift (S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) 
+t1 x0))))))) (\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) 
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 
+(CHead c (Bind b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 
+t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to 
+(\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O 
+x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 
+x0))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead 
+(Flat Appl) (lift (S O) O x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: 
+T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 
+(CHead c (Bind b) t1) t0 t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 
+(THead (Flat Appl) (lift (S O) O x1) x2)) \to (sn3 c (THead (Flat Appl) x1 
+(THead (Bind b) t1 x2)))))))))) H6 (THead (Flat Appl) (lift (S O) O x) x0) 
+H7) in (let H9 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T 
+t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to 
+(sn3 (CHead c (Bind b) t1) t3))))) H5 (THead (Flat Appl) (lift (S O) O x) x0) 
+H7) in (sn3_pr2_intro c (THead (Flat Appl) x (THead (Bind b) t1 x0)) (\lambda 
+(t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) 
+t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x 
+(THead (Bind b) t1 x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) 
+t1 x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) 
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3) 
+(\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4))) (sn3 c 
+t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat 
+Appl) x1 x2))).(\lambda (H15: (pr2 c x x1)).(\lambda (H16: (pr2 c (THead 
+(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
+H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) 
+(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in 
+(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind 
+b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 
+(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
+(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3 
+x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1) 
+x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P: 
+Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3 
+x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def 
+(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3) 
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3 
+x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: 
+T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 
+(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27 
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T 
+t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4)))) 
+(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4) 
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead 
+(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
+(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0 
+H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) 
+t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat 
+Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32 
+\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 
+c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x 
+x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) 
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to 
+(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda 
+(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c 
+(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead 
+(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead 
+(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P: 
+Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O 
+x) | (TLRef _) \Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) 
+| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 
+(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x 
+x1 (S O) O H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x 
+t0)) H15 x (lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) 
+(pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift 
+(CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x 
+x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 
+x0 (refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 
+H29)))) (\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead 
+(Flat Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) 
+(lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: 
+Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) 
+\Rightarrow (lref_map (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 
+_) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
+Appl) (lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x4) H30) in (\lambda (H33: (eq T (lift 
+(S O) O x) (lift (S O) O x1))).(let H34 \def (eq_ind_r T x4 (\lambda (t0: 
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H29 x0 H32) in (let H35 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) 
+t1 x0)) (THead (Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P0: 
+Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 
+(CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let H37 \def (eq_ind_r T x1 
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead 
+(Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall (P0: Prop).P0))) H35 x 
+(lift_inj x x1 (S O) O H33)) in (let H38 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H33)) in (H34 (refl_equal T x0) 
+P)))))))) H31)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S 
+O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl) x1 x4 
+(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x4)))) H28))) x3 H25)))) 
+(\lambda (H25: (((eq T t1 x3) \to (\forall (P: Prop).P)))).(H2 x3 H25 H21 x4 
+x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead (Flat Appl) (lift (S O) O x1) 
+x4) (let H_x1 \def (term_dec x0 x4) in (let H26 \def H_x1 in (or_ind (eq T x0 
+x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda (H27: (eq T x0 x4)).(let 
+H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) 
+H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2 \def (term_dec x x1) in 
+(let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: 
+Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) 
+x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0))) (sn3_sing (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9) x1 H30))) (\lambda 
+(H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift 
+(S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x0))).(\lambda (P: Prop).(let H32 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to 
+(\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O H32)) in (let H34 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O 
+H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift 
+(S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O 
+(drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 
+(pr3_refl (CHead c (Bind b) t1) x0) Appl))) H29))) x4 H27))) (\lambda (H27: 
+(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S 
+O) O x1) x4) (\lambda (H28: (eq T (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P: Prop).(let H29 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow (lref_map 
+(\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) 
+x4) H28) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow 
+t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
+O x1) x4) H28) in (\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O 
+x1))).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to 
+(\forall (P0: Prop).P0))) H27 x0 H30) in (let H33 \def (eq_ind_r T x4 
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 
+\def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) 
+O H31)) in (H32 (refl_equal T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) 
+t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S 
+O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) 
+x0 x4 H22 Appl))) H26)))))) H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 
+(CHead c (Bind b) t1) x0 (lift (S O) O x2))).(sn3_gen_lift (CHead c (Bind b) 
+t1) (THead (Flat Appl) x1 x2) (S O) O (eq_ind_r T (THead (Flat Appl) (lift (S 
+O) O x1) (lift (S O) (s (Flat Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c 
+(Bind b) t1) t0)) (sn3_pr3_trans (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O x1) x0) (let H_x0 \def (term_dec x x1) in (let H20 \def H_x0 in 
+(or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0)) (\lambda (H21: (eq T x 
+x1)).(let H22 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H21) 
+in (eq_ind T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O t0) x0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) 
+(lift (S O) O x) x0) H9) x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall 
+(P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: 
+(eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) 
+O x1) x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 
+(S O))) O x) | (TLRef _) \Rightarrow (lref_map (\lambda (x3: nat).(plus x3 (S 
+O))) O x) | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O 
+x) x0) (THead (Flat Appl) (lift (S O) O x1) x0) H22) in (let H24 \def 
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) 
+H21 x (lift_inj x x1 (S O) O H23)) in (let H25 \def (eq_ind_r T x1 (\lambda 
+(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H23)) in (H24 (refl_equal 
+T x) P)))))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O 
+x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind 
+b) t1) x0) Appl))) H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O 
+x2)) (pr3_thin_dx (CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) 
+O x1) Appl)) (lift (S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) 
+x1 x2 (S O) O)) c (drop_drop (Bind b) O c c (drop_refl c) t1))) H18))) t3 
+H14))))))) H13)) (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: 
+T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind 
+b) t1 x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x 
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: 
+T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))) 
+(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1 
+x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c 
+x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind 
+b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead 
+(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10 
+(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) 
+(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
+k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in 
+((let H20 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H21 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
+(THead (Bind Abst) x1 x2) H14) in (\lambda (_: (eq T t1 x1)).(\lambda (H23: 
+(eq B b Abst)).(let H24 \def (eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: 
+B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let 
+H25 \def (eq_ind B b (\lambda (b0: B).((eq T (THead (Flat Appl) x (THead 
+(Bind b0) t1 x0)) (THead (Bind Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 
+Abst H23) in (let H26 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: 
+T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (P: 
+Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl) (lift (S O) O 
+x) x0) t4) \to (sn3 (CHead c (Bind b0) t1) t4))))) H9 Abst H23) in (let H27 
+\def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat 
+Appl) (lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c 
+(Bind b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (\forall (x5: 
+T).(\forall (x6: T).((eq T t4 (THead (Flat Appl) (lift (S O) O x5) x6)) \to 
+(sn3 c (THead (Flat Appl) x5 (THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in 
+(let H28 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) 
+\to (\forall (P: Prop).P))) \to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall 
+(v0: T).((sn3 (CHead c (Bind b0) t4) (THead (Flat Appl) (lift (S O) O v0) 
+t0)) \to (sn3 c (THead (Flat Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 
+Abst H23) in (let H29 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) 
+H Abst H23) in (let H30 \def (match (H29 (refl_equal B Abst)) in False with 
+[]) in H30)))))))))) H20)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 
+B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T (THead (Bind b) t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda 
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
+T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) y2 (THead (Flat Appl) (lift (S 
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) 
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B 
+b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) t1 x0) (THead 
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b0) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: 
+(eq T (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T 
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead 
+c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T 
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) 
+H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in 
+(eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+(\lambda (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead 
+k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in 
+((let H22 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _) \Rightarrow t0])) 
+(THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H23 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef 
+_) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) t1 x0) 
+(THead (Bind x1) x2 x3) H15) in (\lambda (H24: (eq T t1 x2)).(\lambda (H25: 
+(eq B b x1)).(let H26 \def (eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c 
+(Bind x1) x6) t0 x4)) H19 x0 H23) in (let H27 \def (eq_ind_r T x2 (\lambda 
+(t0: T).(pr2 c t0 x6)) H18 t1 H24) in (let H28 \def (eq_ind_r B x1 (\lambda 
+(b0: B).(pr2 (CHead c (Bind b0) x6) x0 x4)) H26 b H25) in (eq_ind B b 
+(\lambda (b0: B).(sn3 c (THead (Bind b0) x6 (THead (Flat Appl) (lift (S O) O 
+x5) x4)))) (sn3_pr3_trans c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
+O x5) x4)) (sn3_bind b c t1 (sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) 
+O x5) x4) (let H_x \def (term_dec x x5) in (let H29 \def H_x in (or_ind (eq T 
+x x5) ((eq T x x5) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let 
+H31 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind 
+T x (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S 
+O) O t0) x4))) (let H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in 
+(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c 
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 
+x4)).(let H34 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) 
+x0 t0)) H28 x0 H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) 
+t1) (THead (Flat Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) 
+(THead (Flat Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T 
+x0 x4) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) 
+x4) (\lambda (H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat 
+Appl) (lift (S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in (let H36 \def 
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) 
+H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c 
+(Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P)))))) 
+(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O 
+x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c 
+(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift 
+(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x) 
+Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P: 
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
+x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O 
+x) | (TLRef _) \Rightarrow (lref_map (\lambda (x7: nat).(plus x7 (S O))) O x) 
+| (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) 
+(THead (Flat Appl) (lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) 
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat Appl) (lift (S 
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in (\lambda (H34: 
+(eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def (eq_ind_r T x5 
+(\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x 
+x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda (t0: T).(pr2 c x 
+t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def (eq_ind_r T x4 
+(\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0 H33) in (H35 
+(refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) 
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) 
+x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x) (lift (S O) O 
+x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind b) O c c 
+(drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c (Bind b) 
+x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat Appl) (lift 
+(S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl) (lift (S O) 
+O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+(pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O x5) x4)))) 
+x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))))) y 
+H4))))) H3))))))) u H0))))).
+
+theorem sn3_appl_appl:
+ \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
+(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to 
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 
+u1)))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in 
+(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T 
+(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\lambda (t: T).(\forall 
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) 
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
+(sn3 c (THead (Flat Appl) v2 t)))))) (\lambda (y: T).(\lambda (H0: (sn3 c 
+y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to 
+(\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso 
+y u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) 
+\to (sn3 c (THead (Flat Appl) v2 y))))))) (unintro T v1 (\lambda (t: 
+T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: 
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c y u2) \to ((((iso y u2) \to 
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c 
+(THead (Flat Appl) v2 y)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: 
+T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: 
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to 
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c 
+(THead (Flat Appl) v2 t))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall 
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to 
+(sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall 
+(P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T 
+t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall 
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to 
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 
+t3))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 
+(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c 
+v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c t2 u2) \to ((((iso 
+t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) 
+\to (sn3 c (THead (Flat Appl) t t2)))) (\lambda (t0: T).(\lambda (H5: 
+((\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 
+t3) \to (sn3 c t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t0 t3) \to 
+(\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to (((\forall (u2: T).((pr3 c t2 
+u2) \to ((((iso t2 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
+Appl) t3 u2)))))) \to (sn3 c (THead (Flat Appl) t3 t2)))))))).(\lambda (H7: 
+((\forall (u2: T).((pr3 c t2 u2) \to ((((iso t2 u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T 
+t2 (\lambda (t: T).(\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to 
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead 
+(Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall 
+(t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t3) \to 
+(((\forall (u2: T).((pr3 c t u2) \to ((((iso t u2) \to (\forall (P: 
+Prop).P))) \to (sn3 c (THead (Flat Appl) t3 u2)))))) \to (sn3 c (THead (Flat 
+Appl) t3 t))))))) H6 (THead (Flat Appl) x x0) H3) in (let H10 \def (eq_ind T 
+t2 (\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: 
+Prop).P))) \to ((pr3 c t t3) \to (\forall (x1: T).(\forall (x2: T).((eq T t3 
+(THead (Flat Appl) x1 x2)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall 
+(u2: T).((pr3 c t3 u2) \to ((((iso t3 u2) \to (\forall (P: Prop).P))) \to 
+(sn3 c (THead (Flat Appl) v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 
+t3)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H11 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (t3: T).((((eq T t t3) \to (\forall (P: Prop).P))) 
+\to ((pr3 c t t3) \to (sn3 c t3))))) H1 (THead (Flat Appl) x x0) H3) in 
+(eq_ind_r T (THead (Flat Appl) x x0) (\lambda (t: T).(sn3 c (THead (Flat 
+Appl) t0 t))) (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x 
+x0)) (\lambda (t3: T).(\lambda (H12: (((eq T (THead (Flat Appl) t0 (THead 
+(Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H13: (pr2 c 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H14 \def 
+(pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H13) in (or3_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
+x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) 
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H15: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c 
+(THead (Flat Appl) x x0) t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) 
+x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H16: (eq T 
+t3 (THead (Flat Appl) x1 x2))).(\lambda (H17: (pr2 c t0 x1)).(\lambda (H18: 
+(pr2 c (THead (Flat Appl) x x0) x2)).(let H19 \def (eq_ind T t3 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
+Prop).P))) H12 (THead (Flat Appl) x1 x2) H16) in (eq_ind_r T (THead (Flat 
+Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H20 \def (pr2_gen_appl c x x0 x2 
+H18) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c 
+(THead (Flat Appl) x1 x2)) (\lambda (H21: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead 
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) 
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 
+x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq T x2 (THead (Flat 
+Appl) x3 x4))).(\lambda (H23: (pr2 c x x3)).(\lambda (H24: (pr2 c x0 
+x4)).(let H25 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
+Prop).P))) H19 (THead (Flat Appl) x3 x4) H22) in (eq_ind_r T (THead (Flat 
+Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def 
+(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H26 
+\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) 
+((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: 
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda 
+(H27: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H28 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x | 
+(TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x 
+x0) (THead (Flat Appl) x3 x4) H27) in ((let H29 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 
+x4) H27) in (\lambda (H30: (eq T x x3)).(let H31 \def (eq_ind_r T x4 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat 
+Appl) x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H25 x0 H29) 
+in (let H32 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H29) in 
+(eq_ind T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) 
+x3 t)))) (let H33 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat 
+Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t 
+x0))) \to (\forall (P: Prop).P))) H31 x H30) in (let H34 \def (eq_ind_r T x3 
+(\lambda (t: T).(pr2 c x t)) H23 x H30) in (eq_ind T x (\lambda (t: T).(sn3 c 
+(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 
+x1) in (let H35 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall 
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) 
+(\lambda (H36: (eq T t0 x1)).(let H37 \def (eq_ind_r T x1 (\lambda (t: 
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) 
+t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H33 t0 H36) in (let 
+H38 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H17 t0 H36) in (eq_ind 
+T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) 
+(H37 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H36)))) (\lambda (H36: 
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H9 x1 H36 (pr3_pr2 c t0 x1 H17) 
+(\lambda (u2: T).(\lambda (H37: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda 
+(H38: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: 
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H8 u2 H37 H38) (THead 
+(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 
+u2) (pr2_head_1 c t0 x1 H17 (Flat Appl) u2)))))))) H35))) x3 H30))) x4 
+H29))))) H28))) (\lambda (H27: (((eq T (THead (Flat Appl) x x0) (THead (Flat 
+Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H10 (THead (Flat Appl) x3 x4) H27 
+(pr3_flat c x x3 (pr3_pr2 c x x3 H23) x0 x4 (pr3_pr2 c x0 x4 H24) Appl) x3 x4 
+(refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c 
+t0 H5) x1 (pr3_pr2 c t0 x1 H17)) (\lambda (u2: T).(\lambda (H28: (pr3 c 
+(THead (Flat Appl) x3 x4) u2)).(\lambda (H29: (((iso (THead (Flat Appl) x3 
+x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 
+u2) (H8 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) 
+(pr2_thin_dx c x0 x4 H24 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) 
+(THead (Flat Appl) x x4) (pr2_head_1 c x x3 H23 (Flat Appl) x4) u2 H28)) 
+(\lambda (H30: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H29 
+(iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head x3 x 
+x4 x0 (Flat Appl)) u2 H30) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead 
+(Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H17 (Flat 
+Appl) u2)))))))) H26))) x2 H22))))))) H21)) (\lambda (H21: (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T T T 
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda 
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat 
+Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(x6: T).(\lambda (H22: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H23: 
+(eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H24: (pr2 c x x5)).(\lambda 
+(H25: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 
+x6))))).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) 
+t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
+Prop).P))) H19 (THead (Bind Abbr) x5 x6) H23) in (eq_ind_r T (THead (Bind 
+Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def 
+(eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) 
+x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: 
+Prop).P))) H26 (THead (Bind Abst) x3 x4) H22) in (let H28 \def (eq_ind T x0 
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (sn3 c 
+t4))))) H11 (THead (Bind Abst) x3 x4) H22) in (let H29 \def (eq_ind T x0 
+(\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat Appl) x t) t4) \to 
+(\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t4) \to (\forall 
+(x7: T).(\forall (x8: T).((eq T t4 (THead (Flat Appl) x7 x8)) \to (\forall 
+(v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) \to ((((iso t4 u2) 
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v3 u2)))))) \to 
+(sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind Abst) x3 x4) 
+H22) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c 
+(THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to 
+(\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 (THead 
+(Bind Abst) x3 x4) H22) in (let H31 \def (eq_ind T x0 (\lambda (t: 
+T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso 
+(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
+t)))))))) H9 (THead (Bind Abst) x3 x4) H22) in (sn3_pr3_trans c (THead (Flat 
+Appl) t0 (THead (Bind Abbr) x5 x6)) (H30 (THead (Bind Abbr) x5 x6) (pr3_sing 
+c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) 
+(pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind 
+Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind 
+Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H24) (Bind Abbr) x4 x6 
+(pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H25 Abbr x5)))) (\lambda (H32: (iso 
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 
+x6))).(\lambda (P: Prop).(let H33 \def (match H32 with [(iso_sort n1 n2) 
+\Rightarrow (\lambda (H33: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
+Abst) x3 x4)))).(\lambda (H34: (eq T (TSort n2) (THead (Bind Abbr) x5 
+x6))).((let H35 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H33) 
+in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H35)) 
+H34))) | (iso_lref i1 i2) \Rightarrow (\lambda (H33: (eq T (TLRef i1) (THead 
+(Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H34: (eq T (TLRef i2) 
+(THead (Bind Abbr) x5 x6))).((let H35 \def (eq_ind T (TLRef i1) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) 
+x3 x4)) H33) in (False_ind ((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to 
+P) H35)) H34))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H33: (eq T 
+(THead k v4 t4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda 
+(H34: (eq T (THead k v5 t5) (THead (Bind Abbr) x5 x6))).((let H35 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t4 | (TLRef 
+_) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k v4 t4) (THead 
+(Flat Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H36 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow v4 | (TLRef _) 
+\Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t4) (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4)) H33) in ((let H37 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v4 t4) (THead (Flat 
+Appl) x (THead (Bind Abst) x3 x4)) H33) in (eq_ind K (Flat Appl) (\lambda 
+(k0: K).((eq T v4 x) \to ((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq T 
+(THead k0 v5 t5) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H38: (eq T v4 
+x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind Abst) x3 x4)) \to ((eq 
+T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda 
+(H39: (eq T t4 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 
+x4) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t5) (THead (Bind Abbr) x5 
+x6)) \to P)) (\lambda (H40: (eq T (THead (Flat Appl) v5 t5) (THead (Bind 
+Abbr) x5 x6))).(let H41 \def (eq_ind T (THead (Flat Appl) v5 t5) (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k0 _ _) \Rightarrow (match k0 with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H40) in (False_ind P 
+H41))) t4 (sym_eq T t4 (THead (Bind Abst) x3 x4) H39))) v4 (sym_eq T v4 x 
+H38))) k (sym_eq K k (Flat Appl) H37))) H36)) H35)) H34)))]) in (H33 
+(refl_equal T (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T 
+(THead (Bind Abbr) x5 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 
+x6)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat 
+Appl) x1 (THead (Bind Abbr) x5 x6)) (pr2_head_1 c t0 x1 H17 (Flat Appl) 
+(THead (Bind Abbr) x5 x6))))))))) x2 H23)))))))))) H21)) (\lambda (H21: 
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda 
+(x3: B).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (x8: T).(\lambda (H22: (not (eq B x3 Abst))).(\lambda (H23: (eq T 
+x0 (THead (Bind x3) x4 x5))).(\lambda (H24: (eq T x2 (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (H25: (pr2 c x 
+x7)).(\lambda (H26: (pr2 c x4 x8)).(\lambda (H27: (pr2 (CHead c (Bind x3) x8) 
+x5 x6)).(let H28 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) 
+t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: 
+Prop).P))) H19 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
+H24) in (eq_ind_r T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H29 \def (eq_ind 
+T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) 
+(THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
+x7) x6)))) \to (\forall (P: Prop).P))) H28 (THead (Bind x3) x4 x5) H23) in 
+(let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead 
+(Flat Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat 
+Appl) x t) t4) \to (sn3 c t4))))) H11 (THead (Bind x3) x4 x5) H23) in (let 
+H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t4: T).((((eq T (THead (Flat 
+Appl) x t) t4) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x 
+t) t4) \to (\forall (x9: T).(\forall (x10: T).((eq T t4 (THead (Flat Appl) x9 
+x10)) \to (\forall (v3: T).((sn3 c v3) \to (((\forall (u2: T).((pr3 c t4 u2) 
+\to ((((iso t4 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) 
+v3 u2)))))) \to (sn3 c (THead (Flat Appl) v3 t4)))))))))))) H10 (THead (Bind 
+x3) x4 x5) H23) in (let H32 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: 
+T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) 
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H8 
+(THead (Bind x3) x4 x5) H23) in (let H33 \def (eq_ind T x0 (\lambda (t: 
+T).(\forall (t4: T).((((eq T t0 t4) \to (\forall (P: Prop).P))) \to ((pr3 c 
+t0 t4) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso 
+(THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead 
+(Flat Appl) t4 u2)))))) \to (sn3 c (THead (Flat Appl) t4 (THead (Flat Appl) x 
+t)))))))) H9 (THead (Bind x3) x4 x5) H23) in (sn3_pr3_trans c (THead (Flat 
+Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H32 
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c 
+(THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat 
+Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead 
+(Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) 
+x5)) (pr0_upsilon x3 H22 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl 
+x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) 
+(pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H26) (Bind x3) (THead (Flat Appl) (lift 
+(S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c 
+(Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) 
+x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x 
+x7 H25)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) 
+(lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H27 Appl 
+(lift (S O) O x7)))))) (\lambda (H34: (iso (THead (Flat Appl) x (THead (Bind 
+x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) 
+x6)))).(\lambda (P: Prop).(let H35 \def (match H34 with [(iso_sort n1 n2) 
+\Rightarrow (\lambda (H35: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind 
+x3) x4 x5)))).(\lambda (H36: (eq T (TSort n2) (THead (Bind x3) x8 (THead 
+(Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def (eq_ind T (TSort n1) 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x 
+(THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T (TSort n2) (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H37)) H36))) | 
+(iso_lref i1 i2) \Rightarrow (\lambda (H35: (eq T (TLRef i1) (THead (Flat 
+Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H36: (eq T (TLRef i2) (THead 
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H37 \def 
+(eq_ind T (TLRef i1) (\lambda (e: T).(match e with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5)) H35) in (False_ind ((eq T 
+(TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to 
+P) H37)) H36))) | (iso_head v4 v5 t4 t5 k) \Rightarrow (\lambda (H35: (eq T 
+(THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda 
+(H36: (eq T (THead k v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S 
+O) O x7) x6)))).((let H37 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) 
+\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H35) in ((let H38 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) 
+\Rightarrow t])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H35) in ((let H39 \def (f_equal T K (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k v4 t4) (THead (Flat Appl) x (THead (Bind x3) x4 
+x5)) H35) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T 
+t4 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t5) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda (H40: (eq T v4 
+x)).(eq_ind T x (\lambda (_: T).((eq T t4 (THead (Bind x3) x4 x5)) \to ((eq T 
+(THead (Flat Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) 
+O x7) x6))) \to P))) (\lambda (H41: (eq T t4 (THead (Bind x3) x4 
+x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead (Flat 
+Appl) v5 t5) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) 
+\to P)) (\lambda (H42: (eq T (THead (Flat Appl) v5 t5) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))).(let H43 \def (eq_ind T (THead 
+(Flat Appl) v5 t5) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False 
+| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind 
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H42) in (False_ind P H43))) 
+t4 (sym_eq T t4 (THead (Bind x3) x4 x5) H41))) v4 (sym_eq T v4 x H40))) k 
+(sym_eq K k (Flat Appl) H39))) H38)) H37)) H36)))]) in (H35 (refl_equal T 
+(THead (Flat Appl) x (THead (Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) 
+x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))) (THead (Flat Appl) x1 
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr3_pr2 c 
+(THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O 
+x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift 
+(S O) O x7) x6))) (pr2_head_1 c t0 x1 H17 (Flat Appl) (THead (Bind x3) x8 
+(THead (Flat Appl) (lift (S O) O x7) x6)))))))))) x2 H24)))))))))))))) H21)) 
+H20)) t3 H16))))))) H15)) (\lambda (H15: (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) 
+x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))))).(ex4_4_ind T 
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind 
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H16: (eq T (THead (Flat Appl) x x0) (THead 
+(Bind Abst) x1 x2))).(\lambda (H17: (eq T t3 (THead (Bind Abbr) x3 
+x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: 
+T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H20 \def (eq_ind T t3 (\lambda 
+(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall 
+(P: Prop).P))) H12 (THead (Bind Abbr) x3 x4) H17) in (eq_ind_r T (THead (Bind 
+Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat 
+Appl) x x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x1 
+x2) H16) in (False_ind (sn3 c (THead (Bind Abbr) x3 x4)) H21)) t3 
+H17)))))))))) H15)) (\lambda (H15: (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat 
+Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) 
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H17: 
+(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H18: (eq T 
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda 
+(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c 
+(Bind x1) x6) x3 x4)).(let H22 \def (eq_ind T t3 (\lambda (t: T).((eq T 
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: 
+Prop).P))) H12 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) 
+H18) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) 
+x4)) (\lambda (t: T).(sn3 c t)) (let H23 \def (eq_ind T (THead (Flat Appl) x 
+x0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) 
+H17) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) 
+O x5) x4))) H23)) t3 H18)))))))))))))) H15)) H14)))))) t2 H3))))))))) v2 
+H4))))))))) y H0))))) H))))).
+
+theorem sn3_appl_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c 
+(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
+\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w 
+t))))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w: 
+T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind 
+Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind 
+Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind 
+Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind 
+Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w 
+H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead 
+(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind 
+Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat 
+Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c 
+(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl)))))))) 
+H1))))))))).
+
+theorem sn3_appl_appls:
+ \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads 
+(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall 
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) 
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to 
+(sn3 c (THead (Flat Appl) v2 u1))))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads 
+(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead 
+(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0: 
+(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 
+(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads 
+(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat 
+Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0 
+H1))))))))).
+
+lemma sn3_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: 
+TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda 
+(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads 
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H)) 
+(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3 
+c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) 
+(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef 
+i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil 
+(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1 
+in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) 
+TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref 
+c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: 
+(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land 
+(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 
+(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads 
+(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3 
+c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1) 
+(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) 
+(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c 
+t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda 
+(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1) 
+(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat 
+Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl) 
+(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9 
+(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t 
+u2))))))))) H5))) H3))))))) t0))) us)))).
+
+theorem sn3_appls_cast:
+ \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat 
+Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall 
+(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads 
+(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u 
+t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda 
+(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0: 
+TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads 
+(Flat Appl) t1 u)) \to (\forall (t2: T).((sn3 c (THeads (Flat Appl) t1 t2)) 
+\to (sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t2)))))))) \to 
+(\forall (u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to 
+(\forall (t2: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) 
+\to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u 
+t2)))))))))) (\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil 
+u)) \to (\forall (t1: T).((sn3 c (THeads (Flat Appl) TNil t1)) \to (sn3 c 
+(THeads (Flat Appl) TNil (THead (Flat Cast) u t1))))))))).(\lambda (u: 
+T).(\lambda (H0: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil 
+u)))).(\lambda (t1: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) TNil t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: 
+T).(\lambda (t2: TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat 
+Appl) t2 u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) t2 t3)) \to 
+(sn3 c (THeads (Flat Appl) t2 (THead (Flat Cast) u t3)))))))) \to (\forall 
+(u: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall 
+(t3: T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2 t3))) \to (sn3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t2 (THead (Flat Cast) u 
+t3))))))))))).(\lambda (H0: ((\forall (u: T).((sn3 c (THeads (Flat Appl) 
+(TCons t1 t2) u)) \to (\forall (t3: T).((sn3 c (THeads (Flat Appl) (TCons t1 
+t2) t3)) \to (sn3 c (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u 
+t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads 
+(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead 
+(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def 
+(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3 
+\def H_x in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads (Flat 
+Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) 
+(THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c 
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 t3)))).(let H6 \def H5 in (let 
+H_x0 \def (sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in 
+(let H7 \def H_x0 in (land_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1 (THeads 
+(Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 
+t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c 
+(THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def H9 in 
+(sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8 
+(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2) 
+(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl) 
+(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P: 
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons 
+t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl) 
+(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t 
+Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
+
+theorem sn3_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: 
+T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind 
+b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat 
+Appl) vs (THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda 
+(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t: 
+TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts 
+(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0)))))) 
+(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u 
+H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t: 
+TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
+(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u 
+t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl) 
+(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0)))))))) 
+(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) 
+(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b) 
+u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead 
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil) 
+t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads 
+(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead 
+(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead 
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to 
+(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u 
+t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u) 
+(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads 
+(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1: 
+T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O 
+v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def 
+(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl) 
+(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3 
+(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THead (Flat 
+Appl) (lift (S O) O t) (THeads (Flat Appl) (lifts (S O) O t0) t1))) (sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u 
+t1)))) (\lambda (H5: (sn3 (CHead c (Bind b) u) (lift (S O) O v))).(\lambda 
+(H6: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O t) (THeads 
+(Flat Appl) (lifts (S O) O t0) t1)))).(let H_y \def (sn3_gen_lift (CHead c 
+(Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t (THead (Bind b) u t1) t0 c 
+(H2 t1 H6) v (H_y (drop_drop (Bind b) O c c (drop_refl c) u)) (\lambda (u2: 
+T).(\lambda (H7: (pr3 c (THeads (Flat Appl) (TCons t t0) (THead (Bind b) u 
+t1)) u2)).(\lambda (H8: (((iso (THeads (Flat Appl) (TCons t t0) (THead (Bind 
+b) u t1)) u2) \to (\forall (P: Prop).P)))).(let H9 \def (pr3_iso_appls_bind b 
+H (TCons t t0) u t1 c u2 H7 H8) in (sn3_pr3_trans c (THead (Flat Appl) v 
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))) 
+(sn3_appl_bind b H c u H0 (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) 
+t1) v H3) (THead (Flat Appl) v u2) (pr3_flat c v v (pr3_refl c v) (THead 
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) u2 H9 
+Appl)))))))))) H4))))))))) vs0))) vs)))))).
+
+theorem sn3_appls_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c 
+(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c 
+w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) 
+w t))))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us: 
+TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead 
+(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H: 
+(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c 
+w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0: 
+TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0 
+(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads 
+(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 
+c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to 
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat 
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_: 
+(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w: 
+T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads 
+(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1: 
+(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1: 
+TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v 
+t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead 
+(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u 
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c 
+w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl) 
+v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat 
+Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) 
+\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead 
+(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads 
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w: 
+T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads 
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in 
+(land_ind (sn3 c u) (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind Abbr) v t)))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) 
+(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: 
+(sn3 c u)).(\lambda (H5: (sn3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(THead (Bind Abbr) v t))))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead 
+(Bind Abst) w t)) t1 c (H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 
+c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w 
+t))) u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat 
+Appl) v (THead (Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 
+\def (pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c 
+(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v 
+t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0 
+t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
+
+lemma sn3_lift:
+ \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: 
+nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda 
+(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) 
+\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2: 
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d 
+t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: 
+Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall 
+(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c: 
+C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c 
+d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T 
+(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i 
+t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T 
+(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3)) 
+(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda 
+(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t0: T).((eq T (lift h 
+i t1) t0) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T 
+(lift h i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 
+x)).(\lambda (P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T 
+(lift h i t1) (lift h i t0)) \to (\forall (P0: Prop).P0))) H8 t1 H9) in (let 
+H11 \def (eq_ind_r T x (\lambda (t0: T).(pr2 d t1 t0)) H7 t1 H9) in (H10 
+(refl_equal T (lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6))))) 
+H5))))))))))))) t H))).
+
+lemma sn3_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d 
+v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef 
+i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let 
+H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T 
+(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) 
+(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2) 
+(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t: 
+T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in 
+(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i)) 
+(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d0: 
+C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda 
+(d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: 
+C).(\lambda (u: T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) 
+x1))).(\lambda (H6: (eq T t2 (lift (S i) O x1))).(let H7 \def (eq_ind T t2 
+(\lambda (t: T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (lift (S 
+i) O x1) H6) in (eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let 
+H8 \def (eq_ind C (CHead d (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H 
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 
+(Bind Abbr) x1) H5)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d 
+(Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) 
+i H (CHead x0 (Bind Abbr) x1) H5)) in ((let H10 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow 
+t])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d 
+(Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in (\lambda (H11: (eq C d 
+x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind 
+Abbr) t))) H8 v H10) in (eq_ind T v (\lambda (t: T).(sn3 c (lift (S i) O t))) 
+(let H13 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) 
+v))) H12 d H11) in (sn3_lift d v H0 c (S i) O (getl_drop Abbr c d v i H13))) 
+x1 H10)))) H9))) t2 H6)))))) H4)) H3))))))))))).
+
+lemma sn3_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl) 
+vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda 
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind 
+(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3 
+c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O 
+w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H)) 
+in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0: 
+TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift 
+(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c 
+(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_: 
+(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat 
+Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads 
+(Flat Appl) TNil (lift (S i) O w))))).(sn3_appl_abbr c d w i H v H1))) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat 
+Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i))))) 
+\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w)))) 
+\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef 
+i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 
+O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda 
+(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i) 
+O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t 
+t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w)))) (sn3 c (THead 
+(Flat Appl) v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: 
+(sn3 c v)).(\lambda (H5: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 
+(lift (S i) O w))))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda 
+(u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) 
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to 
+(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat 
+Appl) (TCons t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) 
+(pr3_thin_dx c (THeads (Flat Appl) (TCons t t0) (lift (S i) O w)) u2 
+(pr3_iso_appls_abbr c d w i H (TCons t t0) u2 H6 H7) v Appl)))))))) 
+H3)))))))) vs0))) vs)))))).
+
+lemma sns3_lifts:
+ \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h 
+i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda 
+(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t: 
+TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c 
+(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def 
+H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c 
+(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj 
+(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0 
+H4)))) H2)))))) ts)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/spare.ma b/matita/matita/contribs/lambdadelta/basic_1A/spare.ma
new file mode 100644 (file)
index 0000000..372b4f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/spare.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/theory.ma".
+
+axiom pc3_gen_appls_sort_abst:
+ \forall (c: C).(\forall (vs: TList).(\forall (w: T).(\forall (u: T).(\forall 
+(n: nat).((pc3 c (THeads (Flat Appl) vs (TSort n)) (THead (Bind Abst) w u)) 
+\to False)))))
+.
+
+axiom pc3_gen_appls_lref_abst:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (w: T).(\forall 
+(u: T).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THead (Bind Abst) w u)) \to 
+False))))))))
+.
+
+axiom pc3_gen_appls_lref_sort:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abst) v)) \to (\forall (vs: TList).(\forall (ws: 
+TList).(\forall (n: nat).((pc3 c (THeads (Flat Appl) vs (TLRef i)) (THeads 
+(Flat Appl) ws (TSort n))) \to False))))))))
+.
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty0/defs.ma
new file mode 100644 (file)
index 0000000..b197b3c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty0/defs.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/G/defs.ma".
+
+include "basic_1A/getl/defs.ma".
+
+inductive sty0 (g: G): C \to (T \to (T \to Prop)) \def
+| sty0_sort: \forall (c: C).(\forall (n: nat).(sty0 g c (TSort n) (TSort 
+(next g n))))
+| sty0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) 
+\to (sty0 g c (TLRef i) (lift (S i) O w))))))))
+| sty0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((sty0 g d v w) 
+\to (sty0 g c (TLRef i) (lift (S i) O v))))))))
+| sty0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
+T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to (sty0 g c (THead 
+(Bind b) v t1) (THead (Bind b) v t2)))))))
+| sty0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: 
+T).((sty0 g c t1 t2) \to (sty0 g c (THead (Flat Appl) v t1) (THead (Flat 
+Appl) v t2))))))
+| sty0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((sty0 g c v1 
+v2) \to (\forall (t1: T).(\forall (t2: T).((sty0 g c t1 t2) \to (sty0 g c 
+(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty0/fwd.ma
new file mode 100644 (file)
index 0000000..951d79a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty0/fwd.ma
@@ -0,0 +1,553 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty0/defs.ma".
+
+implied rec lemma sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
+(\forall (c: C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0: 
+(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
+(CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w) 
+\to (P c (TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall 
+(d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) 
+\to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift 
+(S i) O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v: 
+T).(\forall (t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to 
+((P (CHead c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind 
+b) v t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1: 
+T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
+Appl) v t1) (THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall 
+(v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1: 
+T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
+Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))))))) (c: C) (t: T) (t0: T) (s0: 
+sty0 g c t t0) on s0: P c t t0 \def match s0 with [(sty0_sort c0 n) 
+\Rightarrow (f c0 n) | (sty0_abbr c0 d v i g0 w s1) \Rightarrow (f0 c0 d v i 
+g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w s1)) | (sty0_abst c0 d v i g0 
+w s1) \Rightarrow (f1 c0 d v i g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w 
+s1)) | (sty0_bind b c0 v t1 t2 s1) \Rightarrow (f2 b c0 v t1 t2 s1 ((sty0_ind 
+g P f f0 f1 f2 f3 f4) (CHead c0 (Bind b) v) t1 t2 s1)) | (sty0_appl c0 v t1 
+t2 s1) \Rightarrow (f3 c0 v t1 t2 s1 ((sty0_ind g P f f0 f1 f2 f3 f4) c0 t1 
+t2 s1)) | (sty0_cast c0 v1 v2 s1 t1 t2 s2) \Rightarrow (f4 c0 v1 v2 s1 
+((sty0_ind g P f f0 f1 f2 f3 f4) c0 v1 v2 s1) t1 t2 s2 ((sty0_ind g P f f0 f1 
+f2 f3 f4) c0 t1 t2 s2))].
+
+lemma sty0_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
+(TSort n) x) \to (eq T x (TSort (next g n)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c 
+t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda 
+(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda 
+(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: 
+C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
+(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
+n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq T (TSort (next g n1)) (TSort 
+(next g n)))) (refl_equal T (TSort (next g n))) n0 H2))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
+w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
+n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
+(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
+H4) in (False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(_: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g 
+d v w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
+n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
+(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
+H4) in (False_ind (eq T (lift (S i) O v) (TSort (next g n))) H5))))))))))) 
+(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 t2)).(\lambda (_: (((eq 
+T t1 (TSort n)) \to (eq T t2 (TSort (next g n)))))).(\lambda (H3: (eq T 
+(THead (Bind b) v t1) (TSort n))).(let H4 \def (eq_ind T (THead (Bind b) v 
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
+(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) 
+(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort 
+(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let 
+H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Flat Appl) v 
+t2) (TSort (next g n))) H4))))))))) (\lambda (c0: C).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 
+(TSort n)) \to (eq T v2 (TSort (next g n)))))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
+T t2 (TSort (next g n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) 
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in (False_ind (eq T 
+(THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) c y x H0))) 
+H))))).
+
+lemma sty0_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c 
+t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T 
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u 
+t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O 
+u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda 
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C 
+T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u 
+t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) 
+O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: 
+nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort 
+n0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in 
+(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(eq T (TSort (next g n0)) (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (TSort (next g n0)) (lift (S n) 
+O u))))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda 
+(i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
+T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
+e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
+e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
+(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
+(Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
+T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O w) 
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (lift (S n0) O w) (lift (S n) O u)))))))) (or_introl (ex3_3 C T 
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (lift (S n) O w) (lift (S n) O u)))))) (ex3_3_intro C T T 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
+(lift (S n) O t))))) d v w H6 H2 (refl_equal T (lift (S n) O w)))) i 
+H5)))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: 
+T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
+e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
+e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
+(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
+(Bind Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
+T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) 
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T 
+T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) 
+(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) 
+(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i 
+H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
+u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) 
+O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T 
+(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v 
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in 
+(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(eq T (THead (Bind b) v t2) (lift (S n) O t)))))) (ex3_3 C T T 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
+t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Bind b) v 
+t2) (lift (S n) O u))))))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
+T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda 
+(_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
+(_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
+u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H4 
+\def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Appl) v t2) (lift 
+(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O u))))))) H4))))))))) 
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
+v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 
+C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
+u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T v2 (lift (S n) 
+O u)))))))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 
+C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
+u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) 
+O u)))))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Cast) v2 t2) (lift 
+(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) H6)))))))))))) 
+c y x H0))) H))))).
+
+lemma sty0_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
+T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda 
+(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead 
+(Bind b) u t2))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) 
+x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) 
+(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) 
+(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda 
+(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g 
+(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u 
+t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) 
+(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in (False_ind 
+(ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: 
+T).(eq T (TSort (next g n)) (THead (Bind b) u t2)))) H2))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
+w)).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: 
+T).(sty0 g (CHead d (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind 
+b) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: 
+T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O 
+w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
+v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind 
+b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda 
+(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) 
+H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 
+t2)) (\lambda (t2: T).(eq T (lift (S i) O v) (THead (Bind b) u t2)))) 
+H5))))))))))) (\lambda (b0: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H1: (sty0 g (CHead c0 (Bind b0) v) t0 
+t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: 
+T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) (\lambda (t3: 
+T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Bind b0) 
+v t0) (THead (Bind b) u t1))).(let H4 \def (f_equal T B (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | 
+(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (THead (Bind b0) v t0) (THead (Bind b) u t1) H3) in ((let 
+H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v | 
+(TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v 
+t0) (THead (Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead (Bind b) u t1) 
+H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 b)).(let H9 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T 
+(\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) 
+(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in (let H10 
+\def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t t2)) H1 t1 
+H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead (Bind b) u 
+t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) t) (Bind 
+b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H9 u H7) 
+in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) t) 
+t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
+T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind 
+b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 (\lambda (b1: 
+B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g 
+(CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 
+(THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B b0 (\lambda 
+(b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in (eq_ind_r B b 
+(\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
+t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead (Bind b) u t3))))) 
+(ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda 
+(t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u t3))) t2 H14 
+(refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) H4)))))))))) 
+(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to 
+(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: 
+T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Flat 
+Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T (THead (Flat Appl) 
+v t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
+H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
+t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u t3)))) 
+H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t1)) \to (ex2 T 
+(\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T 
+v2 (THead (Bind b) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T 
+(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T 
+t2 (THead (Bind b) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) 
+(THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
+H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
+t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Bind b) u 
+t3)))) H6)))))))))))) c y x H0))) H))))))).
+
+lemma sty0_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: 
+T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g 
+c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: 
+T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead 
+(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T 
+(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat 
+Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g 
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
+u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T 
+t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: 
+nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def 
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Appl) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 
+t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Appl) u t2)))) 
+H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
+T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Appl) u 
+t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w 
+(THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat 
+Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
+_) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in (False_ind (ex2 T 
+(\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O w) 
+(THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
+Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
+v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) 
+(\lambda (t2: T).(eq T w (THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T 
+(TLRef i) (THead (Flat Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u 
+t1) H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda 
+(t2: T).(eq T (lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) 
+(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq 
+T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 
+(Bind b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
+t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u 
+t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T 
+(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v 
+t2) (THead (Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
+T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 
+t2)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
+(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
+t3))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead (Flat Appl) u 
+t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) 
+(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v t0) 
+(THead (Flat Appl) u t1) H3) in (\lambda (H6: (eq T v u)).(let H7 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u t1)) \to (ex2 T 
+(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat 
+Appl) u t3)))))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: T).(sty0 
+g c0 t t2)) H1 t1 H5) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
+T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t2) (THead 
+(Flat Appl) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) 
+(\lambda (t3: T).(eq T (THead (Flat Appl) u t2) (THead (Flat Appl) u t3))) t2 
+H8 (refl_equal T (THead (Flat Appl) u t2))) v H6))))) H4))))))))) (\lambda 
+(c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
+(t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T v2 (THead (Flat Appl) u 
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
+(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
+t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) u 
+t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+f) \Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
+True])])])) I (THead (Flat Appl) u t1) H5) in (False_ind (ex2 T (\lambda (t3: 
+T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead 
+(Flat Appl) u t3)))) H6)))))))))))) c y x H0))) H)))))).
+
+lemma sty0_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall 
+(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: 
+T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 
+g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 
+t2))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda 
+(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T 
+(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: 
+T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda 
+(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 
+g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq 
+T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: 
+T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) 
+(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) 
+(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat 
+Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
+_) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H1) in (False_ind (ex3_2 
+T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
+T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq 
+T (TSort (next g n)) (THead (Flat Cast) v2 t2))))) H2))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
+w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda 
+(v2: T).(\lambda (_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: 
+T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat 
+Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 
+t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T 
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda 
+(t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S 
+i) O w) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda 
+(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d 
+(Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: 
+(((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda 
+(_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) 
+(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 
+t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 
+\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda 
+(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: 
+T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O 
+v) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (b: B).(\lambda (c0: 
+C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g 
+(CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 
+t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind 
+b) v) v1 v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) 
+t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 
+t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 
+t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T 
+T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
+T (THead (Bind b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda 
+(c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 
+g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T 
+T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) 
+v t0) (THead (Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v 
+t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow True 
+| Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind 
+(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
+T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 t3))))) H4))))))))) (\lambda 
+(c0: C).(\lambda (v0: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c0 v0 
+v2)).(\lambda (H2: (((eq T v0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T 
+(\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda 
+(t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead 
+(Flat Cast) v3 t2)))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
+(sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) v1 t1)) \to 
+(ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: 
+T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) v3 t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) 
+v0 t0) (THead (Flat Cast) v1 t1))).(let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | 
+(THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) 
+v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
+\Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) v1 t1) H5) in 
+(\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T 
+t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: 
+T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) 
+(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 t3))))))) H4 
+t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H3 t1 
+H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t (THead (Flat Cast) 
+v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
+(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
+T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 v1 H8) in 
+(let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 H8) in 
+(ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
+(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 
+t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) 
+H6)))))))))))) c y x H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty0/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty0/props.ma
new file mode 100644 (file)
index 0000000..cb79d69
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty0/props.ma
@@ -0,0 +1,214 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty0/fwd.ma".
+
+include "basic_1A/getl/drop.ma".
+
+lemma sty0_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e 
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
+e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c) 
+\to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda 
+(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: 
+(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d 
+(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g 
+c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort 
+(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v 
+w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0: 
+nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0 
+w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: 
+(drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 
+(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le 
+i d0 (le_S_n i d0 (le_S_n (S i) (S d0) (le_S (S (S i)) (S d0) (le_n_S (S i) 
+d0 H4)))) c0 c h H3 (CHead d (Bind Abbr) v) H0) in (ex3_2_ind C C (\lambda 
+(e0: C).(\lambda (_: C).(drop i O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
+C).(drop h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
+(CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift 
+(S i) O w))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 
+x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 
+(CHead d (Bind Abbr) v))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n: 
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let 
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d v H8) in (ex2_ind C 
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d0 (S i)) 
+v)))) (\lambda (c1: C).(drop h (minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 
+(TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x: C).(\lambda (H11: 
+(clear x0 (CHead x (Bind Abbr) (lift h (minus d0 (S i)) v)))).(\lambda (H12: 
+(drop h (minus d0 (S i)) x d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g 
+c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat (plus (S i) (minus d0 (S i))) 
+(\lambda (n: nat).(sty0 g c0 (TLRef i) (lift h n (lift (S i) O w)))) 
+(eq_ind_r T (lift (S i) O (lift h (minus d0 (S i)) w)) (\lambda (t: T).(sty0 
+g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: nat).(sty0 g c0 (TLRef i) 
+(lift (S i) O (lift h (minus d0 (S i)) w)))) (sty0_abbr g c0 x (lift h (minus 
+d0 (S i)) v) i (getl_intro i c0 (CHead x (Bind Abbr) (lift h (minus d0 (S i)) 
+v)) x0 H6 H11) (lift h (minus d0 (S i)) w) (H2 x h (minus d0 (S i)) H12)) 
+(plus (S i) (minus d0 (S i))) (le_plus_minus (S i) d0 H4)) (lift h (plus (S 
+i) (minus d0 (S i))) (lift (S i) O w)) (lift_d w h (S i) (minus d0 (S i)) O 
+(le_O_n (minus d0 (S i))))) d0 (le_plus_minus_r (S i) d0 H4)) (lift h d0 
+(TLRef i)) (lift_lref_lt i h d0 H4))))) H10)))))))) H5))) (\lambda (H4: (le 
+d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 
+(lift (S i) O w)))) (eq_ind nat (S i) (\lambda (_: nat).(sty0 g c0 (TLRef 
+(plus i h)) (lift h d0 (lift (S i) O w)))) (eq_ind_r T (lift (plus h (S i)) O 
+w) (\lambda (t: T).(sty0 g c0 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S 
+i) h) (\lambda (n: nat).(sty0 g c0 (TLRef (plus i h)) (lift n O w))) 
+(sty0_abbr g c0 d v (plus i h) (drop_getl_trans_ge i c0 c d0 h H3 (CHead d 
+(Bind Abbr) v) H0 H4) w H1) (plus h (S i)) (plus_sym h (S i))) (lift h d0 
+(lift (S i) O w)) (lift_free w (S i) h O d0 (le_S_n d0 (S i) (le_S (S d0) (S 
+i) (le_n_S d0 i H4))) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O) 
+i) (\lambda (n: nat).(eq nat (S i) n)) (le_antisym (S i) (plus (S O) i) (le_n 
+(plus (S O) i)) (le_n (S i))) (plus i (S O)) (plus_sym i (S O)))) (lift h d0 
+(TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda 
+(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d 
+(Bind Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: 
+((\forall (c0: C).(\forall (h: nat).(\forall (d0: nat).((drop h d0 c0 d) \to 
+(sty0 g c0 (lift h d0 v) (lift h d0 w)))))))).(\lambda (c0: C).(\lambda (h: 
+nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g 
+c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (H4: (lt i 
+d0)).(let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S_n (S i) (S d0) 
+(le_S (S (S i)) (S d0) (le_n_S (S i) d0 H4)))) c0 c h H3 (CHead d (Bind Abst) 
+v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0))) 
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_: 
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g c0 (lift h 
+d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0: C).(\lambda (x1: 
+C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0 
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let H9 \def (eq_ind 
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) 
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) 
+H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind 
+Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S 
+i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) 
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift h (minus 
+d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T 
+(TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind 
+nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i) 
+(lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S 
+i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_: 
+nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) v)))) 
+(sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x 
+(Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i)) 
+w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i))) 
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S 
+i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0 
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0 
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i 
+h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat 
+(S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i) 
+O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0 
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g 
+c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h) 
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus 
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i) 
+h O d0 (le_S_n d0 (S i) (le_S (S d0) (S i) (le_n_S d0 i H4))) (le_O_n d0))) 
+(plus i (S O)) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(eq nat (S i) 
+n)) (le_antisym (S i) (plus (S O) i) (le_n (plus (S O) i)) (le_n (S i))) 
+(plus i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h 
+d0 H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: 
+T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) 
+v) t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h 
+d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s 
+(Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v 
+t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4)) 
+(\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b) 
+d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d) 
+t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2 
+b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift 
+h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda 
+(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g 
+c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d 
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat 
+Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v 
+t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d) 
+t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s 
+(Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl) 
+d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h 
+d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d 
+(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d)))))))))))) 
+(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1 
+v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d 
+v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3 
+t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d 
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: 
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s 
+(Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast) 
+v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast) 
+d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h 
+(s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0 
+h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h 
+(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat 
+Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast) 
+v1 t3 h d))))))))))))))) e t1 t2 H))))).
+
+lemma sty0_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2: 
+T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda 
+(n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2)) 
+(TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0: 
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
+w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2 
+in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g 
+c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w 
+x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i) 
+O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0))))) 
+H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: 
+T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g 
+d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex 
+T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda 
+(_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) 
+t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v 
+i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: 
+T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) 
+v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v) 
+t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0 
+(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) 
+t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3 
+x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead 
+(Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0: 
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 
+t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2 
+\def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: 
+T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3: 
+(sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl) 
+v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2)))))))) 
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
+v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2: 
+T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T 
+(\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda 
+(t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat 
+Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6 
+\def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4: 
+T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7: 
+(sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast) 
+v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7)))) 
+H6)))) H4))))))))))) c t1 t H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty1/cnt.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty1/cnt.ma
new file mode 100644 (file)
index 0000000..0e04ba7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty1/cnt.ma
@@ -0,0 +1,86 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty1/props.ma".
+
+include "basic_1A/cnt/props.ma".
+
+lemma sty1_cnt:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c 
+t1 t) \to (ex2 T (\lambda (t2: T).(sty1 g c t1 t2)) (\lambda (t2: T).(cnt 
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
+T).(ex2 T (\lambda (t3: T).(sty1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) 
+(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 
+(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (sty1_sty0 g c0 
+(TSort n) (TSort (next g n)) (sty0_sort g c0 n)) (cnt_sort (next g n))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 
+g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(sty1 g d v t2)) (\lambda 
+(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d 
+v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef 
+i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (sty1 g d v 
+x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) 
+t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (sty1_abbr g c0 d v i H0 x 
+H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abst) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex2 T 
+(\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def 
+H2 in (ex2_ind T (\lambda (t2: T).(sty1 g d v t2)) (\lambda (t2: T).(cnt t2)) 
+(ex2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) 
+(\lambda (x: T).(\lambda (H4: (sty1 g d v x)).(\lambda (H5: (cnt 
+x)).(ex_intro2 T (\lambda (t2: T).(sty1 g c0 (TLRef i) t2)) (\lambda (t2: 
+T).(cnt t2)) (lift (S i) O x) (sty1_trans g c0 (TLRef i) (lift (S i) O v) 
+(sty1_sty0 g c0 (TLRef i) (lift (S i) O v) (sty0_abst g c0 d v i H0 w H1)) 
+(lift (S i) O x) (sty1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i 
+H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: 
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g 
+(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(sty1 g 
+(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in 
+(ex2_ind T (\lambda (t4: T).(sty1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda 
+(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Bind b) v t2) 
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g (CHead 
+c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: 
+T).(sty1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead 
+(Bind b) v x) (sty1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) 
+H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: 
+T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in 
+(ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) 
+(ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda 
+(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (sty1 g c0 t2 x)).(\lambda 
+(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Appl) v 
+t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (sty1_appl g c0 v 
+t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: 
+C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (sty0 g c0 v1 
+v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(sty1 g c0 v1 t2)) (\lambda (t2: 
+T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 
+t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(sty1 g c0 t2 t4)) (\lambda (t4: 
+T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(sty1 g c0 t2 
+t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead 
+(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda 
+(H5: (sty1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (sty1_cast2 g c0 
+t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(sty1 g 
+c0 v1 v3)) (\lambda (v3: T).(sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat 
+Cast) v3 x))) (ex2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) 
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (sty1 g c0 v1 
+x0)).(\lambda (H9: (sty1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 
+x))).(ex_intro2 T (\lambda (t4: T).(sty1 g c0 (THead (Flat Cast) v1 t2) t4)) 
+(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat 
+Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty1/defs.ma
new file mode 100644 (file)
index 0000000..21e78e8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty1/defs.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty0/defs.ma".
+
+inductive sty1 (g: G) (c: C) (t1: T): T \to Prop \def
+| sty1_sty0: \forall (t2: T).((sty0 g c t1 t2) \to (sty1 g c t1 t2))
+| sty1_sing: \forall (t: T).((sty1 g c t1 t) \to (\forall (t2: T).((sty0 g c 
+t t2) \to (sty1 g c t1 t2)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty1/fwd.ma
new file mode 100644 (file)
index 0000000..6f7b14a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty1/fwd.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty1/defs.ma".
+
+implied rec lemma sty1_ind (g: G) (c: C) (t1: T) (P: (T \to Prop)) (f: 
+(\forall (t2: T).((sty0 g c t1 t2) \to (P t2)))) (f0: (\forall (t: T).((sty1 
+g c t1 t) \to ((P t) \to (\forall (t2: T).((sty0 g c t t2) \to (P t2))))))) 
+(t: T) (s0: sty1 g c t1 t) on s0: P t \def match s0 with [(sty1_sty0 t2 s1) 
+\Rightarrow (f t2 s1) | (sty1_sing t0 s1 t2 s2) \Rightarrow (f0 t0 s1 
+((sty1_ind g c t1 P f f0) t0 s1) t2 s2)].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/sty1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/sty1/props.ma
new file mode 100644 (file)
index 0000000..9b8a08e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/sty1/props.ma
@@ -0,0 +1,142 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/sty1/fwd.ma".
+
+include "basic_1A/sty0/props.ma".
+
+theorem sty1_trans:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c 
+t1 t) \to (\forall (t2: T).((sty1 g c t t2) \to (sty1 g c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(sty1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (sty1 g c t t2)).(sty1_ind g 
+c t (\lambda (t0: T).(sty1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (sty0 g 
+c t t3)).(sty1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (sty1 g 
+c t t0)).(\lambda (H2: (sty1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (sty0 
+g c t0 t3)).(sty1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
+
+lemma sty1_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: 
+T).(\forall (t2: T).((sty1 g (CHead c (Bind b) v) t1 t2) \to (sty1 g c (THead 
+(Bind b) v t1) (THead (Bind b) v t2))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H: (sty1 g (CHead c (Bind b) v) t1 
+t2)).(sty1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(sty1 g c (THead 
+(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (sty0 g 
+(CHead c (Bind b) v) t1 t3)).(sty1_sty0 g c (THead (Bind b) v t1) (THead 
+(Bind b) v t3) (sty0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: 
+(sty1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (sty1 g c (THead (Bind b) v 
+t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g (CHead c 
+(Bind b) v) t t3)).(sty1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) 
+H1 (THead (Bind b) v t3) (sty0_bind g b c v t t3 H2))))))) t2 H))))))).
+
+lemma sty1_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
+(t2: T).((sty1 g c t1 t2) \to (sty1 g c (THead (Flat Appl) v t1) (THead (Flat 
+Appl) v t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(sty1 
+g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: 
+T).(\lambda (H0: (sty0 g c t1 t3)).(sty1_sty0 g c (THead (Flat Appl) v t1) 
+(THead (Flat Appl) v t3) (sty0_appl g c v t1 t3 H0)))) (\lambda (t: 
+T).(\lambda (_: (sty1 g c t1 t)).(\lambda (H1: (sty1 g c (THead (Flat Appl) v 
+t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (sty0 g c t 
+t3)).(sty1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 
+(THead (Flat Appl) v t3) (sty0_appl g c v t t3 H2))))))) t2 H)))))).
+
+lemma sty1_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty1 g e 
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c 
+e) \to (sty1 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (sty1 g e t1 t2)).(sty1_ind g e t1 (\lambda (t: T).(\forall (c: 
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sty1 g c (lift h 
+d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g e t1 
+t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop 
+h d c e)).(sty1_sty0 g c (lift h d t1) (lift h d t3) (sty0_lift g e t1 t3 H0 
+c h d H1)))))))) (\lambda (t: T).(\lambda (_: (sty1 g e t1 t)).(\lambda (H1: 
+((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to 
+(sty1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: 
+(sty0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H3: (drop h d c e)).(sty1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) 
+(lift h d t3) (sty0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
+
+lemma sty1_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty1 g c 
+t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(sty1 g c t1 t)).(sty1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: 
+T).(sty0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (sty0 g c t1 
+t2)).(sty0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (sty1 g c t1 
+t0)).(\lambda (_: (ex T (\lambda (t2: T).(sty0 g c t0 t2)))).(\lambda (t2: 
+T).(\lambda (H2: (sty0 g c t0 t2)).(sty0_correct g c t0 t2 H2)))))) t H))))).
+
+lemma sty1_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: 
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty1 g d v w) 
+\to (sty1 g c (TLRef i) (lift (S i) O w)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: 
+T).(\lambda (H0: (sty1 g d v w)).(sty1_ind g d v (\lambda (t: T).(sty1 g c 
+(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (sty0 g d v 
+t2)).(sty1_sty0 g c (TLRef i) (lift (S i) O t2) (sty0_abbr g c d v i H t2 
+H1)))) (\lambda (t: T).(\lambda (_: (sty1 g d v t)).(\lambda (H2: (sty1 g c 
+(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (sty0 g d t 
+t2)).(sty1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) 
+(sty0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w 
+H0)))))))).
+
+lemma sty1_cast2:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((sty1 g c 
+t1 t2) \to (\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T 
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
+Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (sty1 g c t1 t2)).(sty1_ind g c t1 (\lambda (t: T).(\forall (v1: 
+T).(\forall (v2: T).((sty0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(sty1 g c 
+v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
+Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (sty0 g c t1 t3)).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c v1 v2)).(ex_intro2 T 
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
+Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (sty1_sty0 g c v1 v2 H1) 
+(sty1_sty0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (sty0_cast 
+g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (sty1 g c t1 
+t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to 
+(ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead 
+(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda 
+(H2: (sty0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (sty0 g 
+c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T 
+(\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat 
+Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(sty1 g c v1 
+v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) 
+v3 t3)))) (\lambda (x: T).(\lambda (H5: (sty1 g c v1 x)).(\lambda (H6: (sty1 
+g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def 
+(sty1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: 
+T).(sty0 g c x t4)) (ex2 T (\lambda (v3: T).(sty1 g c v1 v3)) (\lambda (v3: 
+T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda 
+(x0: T).(\lambda (H8: (sty0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(sty1 g 
+c v1 v3)) (\lambda (v3: T).(sty1 g c (THead (Flat Cast) v1 t1) (THead (Flat 
+Cast) v3 t3))) x0 (sty1_sing g c v1 x H5 x0 H8) (sty1_sing g c (THead (Flat 
+Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (sty0_cast 
+g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst/defs.ma
new file mode 100644 (file)
index 0000000..b8b69b8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst/defs.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/defs.ma".
+
+rec definition subst (d: nat) (v: T) (t: T) on t: T \def match t with [(TSort 
+n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (match (blt i d) with [true 
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) | (THead k 
+u t0) \Rightarrow (THead k (subst d v u) (subst (s k d) v t0))].
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst/props.ma
new file mode 100644 (file)
index 0000000..cedfac7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst/props.ma
@@ -0,0 +1,157 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst/defs.ma".
+
+include "basic_1A/subst0/fwd.ma".
+
+lemma subst_sort:
+ \forall (v: T).(\forall (d: nat).(\forall (k: nat).(eq T (subst d v (TSort 
+k)) (TSort k))))
+\def
+ \lambda (_: T).(\lambda (_: nat).(\lambda (k: nat).(refl_equal T (TSort 
+k)))).
+
+lemma subst_lref_lt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt i d) \to (eq T 
+(subst d v (TLRef i)) (TLRef i)))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt i 
+d)).(eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true 
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef i))) 
+(refl_equal T (TLRef i)) (blt i d) (lt_blt d i H))))).
+
+lemma subst_lref_eq:
+ \forall (v: T).(\forall (i: nat).(eq T (subst i v (TLRef i)) (lift i O v)))
+\def
+ \lambda (v: T).(\lambda (i: nat).(eq_ind_r bool false (\lambda (b: bool).(eq 
+T (match b with [true \Rightarrow (TLRef i) | false \Rightarrow (match b with 
+[true \Rightarrow (TLRef (pred i)) | false \Rightarrow (lift i O v)])]) (lift 
+i O v))) (refl_equal T (lift i O v)) (blt i i) (le_bge i i (le_n i)))).
+
+lemma subst_lref_gt:
+ \forall (v: T).(\forall (d: nat).(\forall (i: nat).((lt d i) \to (eq T 
+(subst d v (TLRef i)) (TLRef (pred i))))))
+\def
+ \lambda (v: T).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (lt d 
+i)).(eq_ind_r bool false (\lambda (b: bool).(eq T (match b with [true 
+\Rightarrow (TLRef i) | false \Rightarrow (match (blt d i) with [true 
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)])]) (TLRef 
+(pred i)))) (eq_ind_r bool true (\lambda (b: bool).(eq T (match b with [true 
+\Rightarrow (TLRef (pred i)) | false \Rightarrow (lift d O v)]) (TLRef (pred 
+i)))) (refl_equal T (TLRef (pred i))) (blt d i) (lt_blt i d H)) (blt i d) 
+(le_bge d i (lt_le_weak d i H)))))).
+
+lemma subst_head:
+ \forall (k: K).(\forall (w: T).(\forall (u: T).(\forall (t: T).(\forall (d: 
+nat).(eq T (subst d w (THead k u t)) (THead k (subst d w u) (subst (s k d) w 
+t)))))))
+\def
+ \lambda (k: K).(\lambda (w: T).(\lambda (u: T).(\lambda (t: T).(\lambda (d: 
+nat).(refl_equal T (THead k (subst d w u) (subst (s k d) w t))))))).
+
+lemma subst_lift_SO:
+ \forall (v: T).(\forall (t: T).(\forall (d: nat).(eq T (subst d v (lift (S 
+O) d t)) t)))
+\def
+ \lambda (v: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq 
+T (subst d v (lift (S O) d t0)) t0))) (\lambda (n: nat).(\lambda (d: 
+nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (subst d v t0) (TSort n))) 
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T 
+(TSort n)) (subst d v (TSort n)) (subst_sort v d n)) (lift (S O) d (TSort n)) 
+(lift_sort n (S O) d)))) (\lambda (n: nat).(\lambda (d: nat).(lt_le_e n d (eq 
+T (subst d v (lift (S O) d (TLRef n))) (TLRef n)) (\lambda (H: (lt n 
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T 
+(TLRef n)) (subst d v (TLRef n)) (subst_lref_lt v d n H)) (lift (S O) d 
+(TLRef n)) (lift_lref_lt n (S O) d H))) (\lambda (H: (le d n)).(eq_ind_r T 
+(TLRef (plus n (S O))) (\lambda (t0: T).(eq T (subst d v t0) (TLRef n))) 
+(eq_ind nat (S (plus n O)) (\lambda (n0: nat).(eq T (subst d v (TLRef n0)) 
+(TLRef n))) (eq_ind_r T (TLRef (pred (S (plus n O)))) (\lambda (t0: T).(eq T 
+t0 (TLRef n))) (eq_ind nat (plus n O) (\lambda (n0: nat).(eq T (TLRef n0) 
+(TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) 
+(plus_n_O n))) (pred (S (plus n O))) (pred_Sn (plus n O))) (subst d v (TLRef 
+(S (plus n O)))) (subst_lref_gt v d (S (plus n O)) (le_n_S d (plus n O) 
+(le_plus_trans d n O H)))) (plus n (S O)) (plus_n_Sm n O)) (lift (S O) d 
+(TLRef n)) (lift_lref_ge n (S O) d H)))))) (\lambda (k: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (d: nat).(eq T (subst d v (lift (S O) d t0)) 
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (subst d v 
+(lift (S O) d t1)) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift (S O) 
+d t0) (lift (S O) (s k d) t1)) (\lambda (t2: T).(eq T (subst d v t2) (THead k 
+t0 t1))) (eq_ind_r T (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v 
+(lift (S O) (s k d) t1))) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq 
+T (THead k t0 t1) (THead k (subst d v (lift (S O) d t0)) (subst (s k d) v 
+(lift (S O) (s k d) t1))) (sym_eq T (THead k (subst d v (lift (S O) d t0)) 
+(subst (s k d) v (lift (S O) (s k d) t1))) (THead k t0 t1) (f_equal3 K T T T 
+THead k k (subst d v (lift (S O) d t0)) t0 (subst (s k d) v (lift (S O) (s k 
+d) t1)) t1 (refl_equal K k) (H d) (H0 (s k d))))) (subst d v (THead k (lift 
+(S O) d t0) (lift (S O) (s k d) t1))) (subst_head k v (lift (S O) d t0) (lift 
+(S O) (s k d) t1) d)) (lift (S O) d (THead k t0 t1)) (lift_head k t0 t1 (S O) 
+d)))))))) t)).
+
+lemma subst_subst0:
+ \forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall (d: nat).((subst0 
+d v t1 t2) \to (eq T (subst d v t1) (subst d v t2))))))
+\def
+ \lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (d: nat).(\lambda 
+(H: (subst0 d v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(eq T (subst n t t0) (subst n t t3)))))) 
+(\lambda (v0: T).(\lambda (i: nat).(eq_ind_r T (lift i O v0) (\lambda (t: 
+T).(eq T t (subst i v0 (lift (S i) O v0)))) (eq_ind nat (plus (S O) i) 
+(\lambda (n: nat).(eq T (lift i O v0) (subst i v0 (lift n O v0)))) (eq_ind T 
+(lift (S O) i (lift i O v0)) (\lambda (t: T).(eq T (lift i O v0) (subst i v0 
+t))) (eq_ind_r T (lift i O v0) (\lambda (t: T).(eq T (lift i O v0) t)) 
+(refl_equal T (lift i O v0)) (subst i v0 (lift (S O) i (lift i O v0))) 
+(subst_lift_SO v0 (lift i O v0) i)) (lift (plus (S O) i) O v0) (lift_free v0 
+i (S O) O i (le_n (plus O i)) (le_O_n i))) (S i) (refl_equal nat (S i))) 
+(subst i v0 (TLRef i)) (subst_lref_eq v0 i)))) (\lambda (v0: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v0 u1 
+u2)).(\lambda (H1: (eq T (subst i v0 u1) (subst i v0 u2))).(\lambda (t: 
+T).(\lambda (k: K).(eq_ind_r T (THead k (subst i v0 u1) (subst (s k i) v0 t)) 
+(\lambda (t0: T).(eq T t0 (subst i v0 (THead k u2 t)))) (eq_ind_r T (THead k 
+(subst i v0 u2) (subst (s k i) v0 t)) (\lambda (t0: T).(eq T (THead k (subst 
+i v0 u1) (subst (s k i) v0 t)) t0)) (eq_ind_r T (subst i v0 u2) (\lambda (t0: 
+T).(eq T (THead k t0 (subst (s k i) v0 t)) (THead k (subst i v0 u2) (subst (s 
+k i) v0 t)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t))) 
+(subst i v0 u1) H1) (subst i v0 (THead k u2 t)) (subst_head k v0 u2 t i)) 
+(subst i v0 (THead k u1 t)) (subst_head k v0 u1 t i)))))))))) (\lambda (k: 
+K).(\lambda (v0: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i: 
+nat).(\lambda (_: (subst0 (s k i) v0 t4 t3)).(\lambda (H1: (eq T (subst (s k 
+i) v0 t4) (subst (s k i) v0 t3))).(\lambda (u: T).(eq_ind_r T (THead k (subst 
+i v0 u) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T t (subst i v0 (THead k u 
+t3)))) (eq_ind_r T (THead k (subst i v0 u) (subst (s k i) v0 t3)) (\lambda 
+(t: T).(eq T (THead k (subst i v0 u) (subst (s k i) v0 t4)) t)) (eq_ind_r T 
+(subst (s k i) v0 t3) (\lambda (t: T).(eq T (THead k (subst i v0 u) t) (THead 
+k (subst i v0 u) (subst (s k i) v0 t3)))) (refl_equal T (THead k (subst i v0 
+u) (subst (s k i) v0 t3))) (subst (s k i) v0 t4) H1) (subst i v0 (THead k u 
+t3)) (subst_head k v0 u t3 i)) (subst i v0 (THead k u t4)) (subst_head k v0 u 
+t4 i)))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (H1: (eq T (subst i v0 
+u1) (subst i v0 u2))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (subst0 (s k i) v0 t3 t4)).(\lambda (H3: (eq T (subst (s k i) 
+v0 t3) (subst (s k i) v0 t4))).(eq_ind_r T (THead k (subst i v0 u1) (subst (s 
+k i) v0 t3)) (\lambda (t: T).(eq T t (subst i v0 (THead k u2 t4)))) (eq_ind_r 
+T (THead k (subst i v0 u2) (subst (s k i) v0 t4)) (\lambda (t: T).(eq T 
+(THead k (subst i v0 u1) (subst (s k i) v0 t3)) t)) (eq_ind_r T (subst i v0 
+u2) (\lambda (t: T).(eq T (THead k t (subst (s k i) v0 t3)) (THead k (subst i 
+v0 u2) (subst (s k i) v0 t4)))) (eq_ind_r T (subst (s k i) v0 t4) (\lambda 
+(t: T).(eq T (THead k (subst i v0 u2) t) (THead k (subst i v0 u2) (subst (s k 
+i) v0 t4)))) (refl_equal T (THead k (subst i v0 u2) (subst (s k i) v0 t4))) 
+(subst (s k i) v0 t3) H3) (subst i v0 u1) H1) (subst i v0 (THead k u2 t4)) 
+(subst_head k v0 u2 t4 i)) (subst i v0 (THead k u1 t3)) (subst_head k v0 u1 
+t3 i))))))))))))) d v t1 t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/dec.ma
new file mode 100644 (file)
index 0000000..6f4a1bb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/dec.ma
@@ -0,0 +1,176 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/defs.ma".
+
+include "basic_1A/lift/props.ma".
+
+lemma dnf_dec2:
+ \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
+O) d v))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: 
+T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda 
+(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: 
+nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) 
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d 
+v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort 
+n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T 
+(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: 
+nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: 
+T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T 
+(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: 
+T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d 
+(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind 
+nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 
+w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift 
+(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w 
+(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S 
+O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) 
+(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) 
+(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n 
+(lift n O w)) (lift_free w n (S O) O n (le_plus_r O n) (le_O_n n)))))) d H)) 
+(\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
+(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) 
+(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred 
+n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda 
+(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
+t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
+v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: 
+T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda 
+(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) 
+in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 
+d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
+v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) 
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) 
+d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 
+(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in 
+(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S 
+O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) 
+(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift 
+(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d 
+v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w 
+t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
+T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) 
+in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift 
+(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S 
+O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s 
+k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda 
+(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w 
+(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d 
+w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 
+t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) 
+(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) 
+(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) 
+(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
+(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T 
+(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex 
+T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) 
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) 
+(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def 
+H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T 
+(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d 
+v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d 
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) 
+x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) 
+(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) 
+x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 
+H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
+v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: 
+T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex 
+T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: 
+T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in 
+(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s 
+k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S 
+O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
+k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) 
+(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) 
+d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
+(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 
+t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: 
+T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T 
+(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) 
+(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T 
+(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda 
+(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda 
+(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) 
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) 
+(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift 
+(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) 
+t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift 
+(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) 
+(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
+(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T 
+(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) 
+(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
+k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
+T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror 
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) 
+(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) 
+(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k 
+d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d 
+x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) 
+(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift 
+(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) 
+d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
+
+lemma dnf_dec:
+ \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or 
+(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
+\def
+ \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t 
+d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: 
+T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
+O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
+(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: 
+T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 
+\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T 
+(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
+v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d 
+x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
+(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t 
+(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t 
+(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex 
+T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
+v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T 
+(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 
+(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: 
+T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) 
+(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d 
+x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d 
+x)))) t H1))) H0)) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/defs.ma
new file mode 100644 (file)
index 0000000..9c42c70
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/defs.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/lift/defs.ma".
+
+inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
+| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift 
+(S i) O v)))
+| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: 
+nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v 
+(THead k u1 t) (THead k u2 t))))))))
+| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: 
+T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i 
+v (THead k u t1) (THead k u t2))))))))
+| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: 
+nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: 
+T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 
+t2)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/fwd.ma
new file mode 100644 (file)
index 0000000..4886893
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/fwd.ma
@@ -0,0 +1,912 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/defs.ma".
+
+include "basic_1A/lift/fwd.ma".
+
+implied rec lemma subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f: 
+(\forall (v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0: 
+(\forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 
+i v u1 u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v 
+(THead k u1 t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 
+t2) \to ((P (s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead 
+k u t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2: 
+T).(\forall (i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k: 
+K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k 
+i) v t1 t2) \to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat) 
+(t: T) (t0: T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match 
+s0 with [(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2 
+k) \Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2 
+k) | (subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1 
+((subst0_ind P f f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1 
+k t2 t3 s2) \Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 
+s1) k t2 t3 s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
+
+lemma subst0_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
+i v (TSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) 
+(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda 
+(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda 
+(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def 
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(TSort n) H1) in (False_ind P H2))))) (\lambda (v0: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (t: T).(\lambda (k: 
+K).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(let H4 \def (eq_ind T 
+(THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H3) in (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda 
+(t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 
+t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
+(False_ind P H4))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T 
+u1 (TSort n)) \to P))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TSort 
+n)) \to P))).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) 
+H)))))).
+
+lemma subst0_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
+i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 
+i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) 
+(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: 
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) 
+\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef 
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(land (eq nat n n0) (eq T (lift 
+(S n0) O v0) (lift (S n) O v0)))) (conj (eq nat n n) (eq T (lift (S n) O v0) 
+(lift (S n) O v0)) (refl_equal nat n) (refl_equal T (lift (S n) O v0))) i0 
+H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) 
+\to (land (eq nat n i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) (TLRef n))).(let H4 
+\def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
+t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: 
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s 
+k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in (False_ind (land 
+(eq nat n i0) (eq T (THead k u t2) (lift (S n) O v0))) H4))))))))))) (\lambda 
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n 
+i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O 
+v0)))))).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind 
+T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 t2) (lift 
+(S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
+
+lemma subst0_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
+(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T 
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) 
+x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda 
+(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) 
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
+t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda 
+(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k 
+u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 
+t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 
+t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef 
+i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead k u1 t1) H1) in (False_ind (or3 
+(ex2 T (\lambda (u2: T).(eq T (lift (S i0) O v0) (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t2: T).(eq T (lift (S i0) O 
+v0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))) H2))))) (\lambda (v0: 
+T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 
+i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead k u1 t1)) \to (or3 (ex2 T 
+(\lambda (u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 
+u3))) (ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 
+u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 
+t2)))))))).(\lambda (t: T).(\lambda (k0: K).(\lambda (H3: (eq T (THead k0 u0 
+t) (THead k u1 t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
+_) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k 
+u1 t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) 
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T 
+u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda 
+(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead 
+k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind 
+T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda 
+(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
+(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 
+(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) 
+in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 
+H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 
+t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) 
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T 
+(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 
+H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: 
+T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 
+t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 
+t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) 
+(THead k0 u t0) (THead k u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
+(THead _ t _) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let 
+H6 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u t0) 
+(THead k u1 t1) H3) in (\lambda (H7: (eq T u u1)).(\lambda (H8: (eq K k0 
+k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead 
+k0 t t2) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
+(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead k0 t t2) (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3)))))) (let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 
+t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 (s k0 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead 
+k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H2 t1 H6) in (let H10 \def 
+(eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t t2)) H1 t1 H6) in (let 
+H11 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 
+(ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 
+(s k1 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) 
+(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s k1 i0) v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
+k (s k1 i0)) v0 t1 t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda 
+(k1: K).(subst0 (s k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: 
+K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) 
+(\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead 
+k1 u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: 
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) 
+H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead 
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: 
+(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) 
+(THead k u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0) (THead k 
+u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k0 k)).(let 
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 
+T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k0 
+i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda 
+(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(subst0 (s k0 i0) v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
+k (s k0 i0)) v0 t1 t3))))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda 
+(t: T).(subst0 (s k0 i0) v0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 
+(\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
+T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
+T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 
+u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
+t3))))))) H11 k H10) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 
+(s k1 i0) v0 t1 t2)) H12 k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u3: T).(eq T (THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H15 \def (eq_ind T 
+u0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
+T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T 
+(\lambda (t3: T).(eq T u2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 
+\def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in 
+(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead 
+k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k 
+u2 t2)) H16 H14)))) k0 H10)))))))) H7)) H6)))))))))))))) i v y x H0))) 
+H))))))).
+
+lemma subst0_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) 
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d 
+u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: 
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) 
+x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: 
+T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) 
+in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) 
+t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S 
+(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: 
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
+n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h 
+(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H 
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq 
+T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: 
+(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T 
+(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) 
+(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) 
+O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda 
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
+(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T 
+(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O 
+(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) 
+(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) 
+n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: 
+T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S 
+(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n 
+h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat 
+(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d 
+u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) 
+H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: 
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
+n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) 
+(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) 
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift 
+h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 
+t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t 
+t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) 
+(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i 
+d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) 
+in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus 
+i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) 
+t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) 
+t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
+d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S 
+(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h 
+d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
+t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k 
+(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S 
+(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
+(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h 
+(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
+T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) 
+t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) 
+x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T 
+x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T 
+(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
+(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda 
+(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) 
+t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
+t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T 
+(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h 
+(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k 
+(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 
+t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: 
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) 
+(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) 
+t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i 
+d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
+(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
+T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda 
+(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) 
+x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 
+T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: 
+T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i 
+d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S 
+(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus 
+i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) 
+H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
+(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
+(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) 
+x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s 
+k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h 
+(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i 
+u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T 
+(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
+t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda 
+(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind 
+T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda 
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
+(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i 
+d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) 
+(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i 
+d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) 
+(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 
+H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
+(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq 
+T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d 
+u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda 
+(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x 
+(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) 
+t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
+d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda 
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
+(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda 
+(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i 
+d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) 
+(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 
+(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
+(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda 
+(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k 
+i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T 
+(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S 
+(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S 
+(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
+x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
+t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S 
+(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind 
+nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k 
+(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus 
+i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S 
+(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: 
+T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead 
+k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus 
+i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T 
+(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k 
+x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u 
+t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S 
+(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) 
+(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 
+i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k 
+(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
+H2))))))))))))) t1)).
+
+lemma subst0_gen_lift_false:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u 
+(lift h d t) x) \to (\forall (P: Prop).P)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: 
+T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i 
+(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
+Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda 
+(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) 
+x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda 
+(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in 
+(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: 
+T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: 
+nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: 
+(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P 
+(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
+(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind 
+(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda 
+(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: 
+nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n 
+H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) 
+(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d 
+H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P 
+(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n 
+h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d 
+h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h 
+n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall 
+(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) 
+\to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
+Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall 
+(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to 
+((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: 
+Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus 
+d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: 
+Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: 
+T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k 
+u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) 
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: 
+T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: 
+T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u 
+(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
+(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda 
+(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: 
+(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda 
+(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda 
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: 
+T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u 
+(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k 
+(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) 
+x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) 
+(\lambda (n: nat).(lt (s k i) n)) (s_lt k i (plus d h) H2) (plus (s k d) h) 
+(s_plus k d h)) H7 P)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: (subst0 i u (lift h d 
+t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) t1) x1)).(H u x0 h d 
+i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d t0) (lift h (s k d) 
+t1) x i H4))))))))))))))))) t).
+
+lemma subst0_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h 
+d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: 
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus 
+d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 
+i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 
+T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i 
+h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d 
+(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda 
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef 
+n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) 
+(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in 
+(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq 
+T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) 
+(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 
+\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus 
+d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) 
+(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind 
+T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) 
+(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S 
+(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n 
+h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n 
+h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) 
+(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S 
+(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) 
+(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S 
+(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) 
+t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h 
+d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) 
+(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n 
+h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O 
+u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: 
+nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift 
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n 
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d 
+(plus O (S n)) (le_plus_plus O O d (S n) (le_O_n O) (le_S d n H1))) (le_O_n 
+d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i 
+H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: 
+((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
+nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda 
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 
+t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda 
+(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) 
+(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) 
+t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x 
+(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) 
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda 
+(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
+k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
+(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) 
+(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: 
+T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u 
+(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
+(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u 
+(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
+(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda 
+(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: 
+(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r 
+T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
+(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u 
+(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
+(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift 
+h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u 
+(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) 
+(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k 
+d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) 
+(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) 
+(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T 
+(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
+(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: 
+(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) 
+x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) 
+(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T 
+(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 
+(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d 
+t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) 
+t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda 
+(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) 
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) 
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) 
+(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: 
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t 
+t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: 
+nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r 
+d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) 
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) 
+(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 
+(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t 
+x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) 
+(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) 
+(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
+k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
+u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift 
+h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u 
+(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) 
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq 
+T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) 
+t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda 
+(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T 
+(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead 
+k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) 
+x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda 
+(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t 
+t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda 
+(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t 
+x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T 
+(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead 
+k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda 
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d 
+(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d 
+t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 
+\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 
+(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T 
+(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal 
+T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 
+H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) 
+x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind 
+nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i 
+H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u 
+(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
+
+lemma subst0_gen_lift_rev_ge:
+ \forall (t1: T).(\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i v t1 (lift h d u2)) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T t1 (lift h d u1)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (v: T).(\forall (u2: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i v t (lift 
+h d u2)) \to ((le (plus d h) i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T t (lift h d u1)))))))))))) (\lambda (n: 
+nat).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (subst0 i v (TSort n) (lift h d 
+u2))).(\lambda (_: (le (plus d h) i)).(subst0_gen_sort v (lift h d u2) i n H 
+(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
+(TSort n) (lift h d u1))))))))))))) (\lambda (n: nat).(\lambda (v: 
+T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i v (TLRef n) (lift h d u2))).(\lambda (H0: (le 
+(plus d h) i)).(land_ind (eq nat n i) (eq T (lift h d u2) (lift (S n) O v)) 
+(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
+(TLRef n) (lift h d u1)))) (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T 
+(lift h d u2) (lift (S n) O v))).(let H3 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(le (plus d h) n0)) H0 n H1) in (eq_ind nat n (\lambda (n0: nat).(ex2 T 
+(\lambda (u1: T).(subst0 (minus n0 h) v u1 u2)) (\lambda (u1: T).(eq T (TLRef 
+n) (lift h d u1))))) (eq_ind_r nat (plus (minus n h) h) (\lambda (n0: 
+nat).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) (\lambda (u1: 
+T).(eq T (TLRef n0) (lift h d u1))))) (eq_ind T (lift h d (TLRef (minus n 
+h))) (\lambda (t: T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) 
+(\lambda (u1: T).(eq T t (lift h d u1))))) (let H4 \def (eq_ind nat n 
+(\lambda (n0: nat).(eq T (lift h d u2) (lift (S n0) O v))) H2 (plus h (minus 
+n h)) (le_plus_minus h n (le_trans h (plus d h) n (le_plus_r d h) H3))) in 
+(let H5 \def (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(eq T 
+(lift h d u2) (lift n0 O v))) H4 (plus h (S (minus n h))) (plus_n_Sm h (minus 
+n h))) in (let H6 \def (eq_ind_r T (lift (plus h (S (minus n h))) O v) 
+(\lambda (t: T).(eq T (lift h d u2) t)) H5 (lift h d (lift (S (minus n h)) O 
+v)) (lift_free v (S (minus n h)) h O d (le_S d (minus n h) (le_minus d n h 
+H3)) (le_O_n d))) in (eq_ind_r T (lift (S (minus n h)) O v) (\lambda (t: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 t)) (\lambda (u1: T).(eq 
+T (lift h d (TLRef (minus n h))) (lift h d u1))))) (ex_intro2 T (\lambda (u1: 
+T).(subst0 (minus n h) v u1 (lift (S (minus n h)) O v))) (\lambda (u1: T).(eq 
+T (lift h d (TLRef (minus n h))) (lift h d u1))) (TLRef (minus n h)) 
+(subst0_lref v (minus n h)) (refl_equal T (lift h d (TLRef (minus n h))))) u2 
+(lift_inj u2 (lift (S (minus n h)) O v) h d H6))))) (TLRef (plus (minus n h) 
+h)) (lift_lref_ge (minus n h) h d (le_minus d n h H3))) n (le_plus_minus_sym 
+h n (le_trans h (plus d h) n (le_plus_r d h) H3))) i H1)))) (subst0_gen_lref 
+v (lift h d u2) i n H)))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: 
+((\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: 
+nat).(\forall (d: nat).((subst0 i v t (lift h d u2)) \to ((le (plus d h) i) 
+\to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T t (lift h d u1))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
+nat).((subst0 i v t0 (lift h d u2)) \to ((le (plus d h) i) \to (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T t0 
+(lift h d u1))))))))))))).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (subst0 i v (THead k t 
+t0) (lift h d u2))).(\lambda (H2: (le (plus d h) i)).(or3_ind (ex2 T (\lambda 
+(u3: T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
+u3))) (ex2 T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) (\lambda 
+(t2: T).(subst0 (s k i) v t0 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T (lift h d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(subst0 i v t u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 
+t2)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (H3: (ex2 T (\lambda (u3: 
+T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
+u3)))).(ex2_ind T (\lambda (u3: T).(eq T (lift h d u2) (THead k u3 t0))) 
+(\lambda (u3: T).(subst0 i v t u3)) (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x: T).(\lambda (H4: (eq T (lift h d u2) (THead k x t0))).(\lambda (H5: 
+(subst0 i v t x)).(let H6 \def (sym_eq T (lift h d u2) (THead k x t0) H4) in 
+(let H_x \def (lift_gen_head k x t0 u2 h d H6) in (let H7 \def H_x in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T x (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t0 (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
+h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
+x0 x1))).(\lambda (H9: (eq T x (lift h d x0))).(\lambda (H10: (eq T t0 (lift 
+h (s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 i v t t2)) 
+H5 (lift h d x0) H9) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq T (THead 
+k t t0) (lift h d u1))))) (eq_ind_r T (lift h (s k d) x1) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
+(\lambda (u1: T).(eq T (THead k t t2) (lift h d u1))))) (let H_x0 \def (H v 
+x0 i h d H11 H2) in (let H12 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
+(minus i h) v u1 x0)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: 
+T).(eq T (THead k t (lift h (s k d) x1)) (lift h d u1)))) (\lambda (x2: 
+T).(\lambda (H13: (subst0 (minus i h) v x2 x0)).(\lambda (H14: (eq T t (lift 
+h d x2))).(eq_ind_r T (lift h d x2) (\lambda (t2: T).(ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k 
+t2 (lift h (s k d) x1)) (lift h d u1))))) (eq_ind T (lift h d (THead k x2 
+x1)) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead 
+k x0 x1))) (\lambda (u1: T).(eq T t2 (lift h d u1))))) (ex_intro2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T 
+(lift h d (THead k x2 x1)) (lift h d u1))) (THead k x2 x1) (subst0_fst v x0 
+x2 (minus i h) H13 x1 k) (refl_equal T (lift h d (THead k x2 x1)))) (THead k 
+(lift h d x2) (lift h (s k d) x1)) (lift_head k x2 x1 h d)) t H14)))) H12))) 
+t0 H10) u2 H8))))))) H7))))))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq 
+T (lift h d u2) (THead k t t2))) (\lambda (t2: T).(subst0 (s k i) v t0 
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) 
+(\lambda (t2: T).(subst0 (s k i) v t0 t2)) (ex2 T (\lambda (u1: T).(subst0 
+(minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) 
+(\lambda (x: T).(\lambda (H4: (eq T (lift h d u2) (THead k t x))).(\lambda 
+(H5: (subst0 (s k i) v t0 x)).(let H6 \def (sym_eq T (lift h d u2) (THead k t 
+x) H4) in (let H_x \def (lift_gen_head k t x u2 h d H6) in (let H7 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T x (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
+h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
+x0 x1))).(\lambda (H9: (eq T t (lift h d x0))).(\lambda (H10: (eq T x (lift h 
+(s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 (s k i) v t0 
+t2)) H5 (lift h (s k d) x1) H10) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq 
+T (THead k t t0) (lift h d u1))))) (eq_ind_r T (lift h d x0) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
+(\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) (let H_y \def (H0 v 
+x1 (s k i) h (s k d) H11) in (let H12 \def (eq_ind_r nat (plus (s k d) h) 
+(\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: T).(subst0 (minus 
+(s k i) h) v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) u1)))))) H_y 
+(s k (plus d h)) (s_plus k d h)) in (let H13 \def (eq_ind_r nat (minus (s k 
+i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to (ex2 T (\lambda 
+(u1: T).(subst0 n v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) 
+u1)))))) H12 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) i 
+(le_plus_r d h) H2))) in (let H14 \def (H13 (s_le k (plus d h) i H2)) in 
+(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x1)) (\lambda (u1: 
+T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t0) 
+(lift h d u1)))) (\lambda (x2: T).(\lambda (H15: (subst0 (s k (minus i h)) v 
+x2 x1)).(\lambda (H16: (eq T t0 (lift h (s k d) x2))).(eq_ind_r T (lift h (s 
+k d) x2) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t2) (lift h d 
+u1))))) (eq_ind T (lift h d (THead k x0 x2)) (\lambda (t2: T).(ex2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T t2 
+(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x0 x1))) (\lambda (u1: T).(eq T (lift h d (THead k x0 x2)) (lift h d 
+u1))) (THead k x0 x2) (subst0_snd k v x1 x2 (minus i h) H15 x0) (refl_equal T 
+(lift h d (THead k x0 x2)))) (THead k (lift h d x0) (lift h (s k d) x2)) 
+(lift_head k x0 x2 h d)) t0 H16)))) H14))))) t H9) u2 H8))))))) H7))))))) 
+H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (lift h 
+d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 t2))))).(ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (lift h d u2) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) (\lambda (_: T).(\lambda 
+(t2: T).(subst0 (s k i) v t0 t2))) (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T (lift h d u2) (THead k x0 
+x1))).(\lambda (H5: (subst0 i v t x0)).(\lambda (H6: (subst0 (s k i) v t0 
+x1)).(let H7 \def (sym_eq T (lift h d u2) (THead k x0 x1) H4) in (let H_x 
+\def (lift_gen_head k x0 x1 u2 h d H7) in (let H8 \def H_x in (ex3_2_ind T T 
+(\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T x0 (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T x1 (lift h (s k d) z)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T u2 (THead k x2 x3))).(\lambda 
+(H10: (eq T x0 (lift h d x2))).(\lambda (H11: (eq T x1 (lift h (s k d) 
+x3))).(let H12 \def (eq_ind T x1 (\lambda (t2: T).(subst0 (s k i) v t0 t2)) 
+H6 (lift h (s k d) x3) H11) in (let H13 \def (eq_ind T x0 (\lambda (t2: 
+T).(subst0 i v t t2)) H5 (lift h d x2) H10) in (eq_ind_r T (THead k x2 x3) 
+(\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) 
+(\lambda (u1: T).(eq T (THead k t t0) (lift h d u1))))) (let H_x0 \def (H v 
+x2 i h d H13 H2) in (let H14 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
+(minus i h) v u1 x2)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: 
+T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (x: T).(\lambda (H15: 
+(subst0 (minus i h) v x x2)).(\lambda (H16: (eq T t (lift h d x))).(eq_ind_r 
+T (lift h d x) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v 
+u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) 
+(let H_y \def (H0 v x3 (s k i) h (s k d) H12) in (let H17 \def (eq_ind_r nat 
+(plus (s k d) h) (\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: 
+T).(subst0 (minus (s k i) h) v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h (s k 
+d) u1)))))) H_y (s k (plus d h)) (s_plus k d h)) in (let H18 \def (eq_ind_r 
+nat (minus (s k i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to 
+(ex2 T (\lambda (u1: T).(subst0 n v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h 
+(s k d) u1)))))) H17 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) 
+i (le_plus_r d h) H2))) in (let H19 \def (H18 (s_le k (plus d h) i H2)) in 
+(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x3)) (\lambda (u1: 
+T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t0) (lift 
+h d u1)))) (\lambda (x4: T).(\lambda (H20: (subst0 (s k (minus i h)) v x4 
+x3)).(\lambda (H21: (eq T t0 (lift h (s k d) x4))).(eq_ind_r T (lift h (s k 
+d) x4) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t2) (lift h d 
+u1))))) (eq_ind T (lift h d (THead k x x4)) (\lambda (t2: T).(ex2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T t2 
+(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x2 x3))) (\lambda (u1: T).(eq T (lift h d (THead k x x4)) (lift h d 
+u1))) (THead k x x4) (subst0_both v x x2 (minus i h) H15 k x4 x3 H20) 
+(refl_equal T (lift h d (THead k x x4)))) (THead k (lift h d x) (lift h (s k 
+d) x4)) (lift_head k x x4 h d)) t0 H21)))) H19))))) t H16)))) H14))) u2 
+H9)))))))) H8))))))))) H3)) (subst0_gen_head k v t t0 (lift h d u2) i 
+H1)))))))))))))) t1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/props.ma
new file mode 100644 (file)
index 0000000..38ab424
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/props.ma
@@ -0,0 +1,224 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/fwd.ma".
+
+lemma subst0_refl:
+ \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to 
+(\forall (P: Prop).P))))
+\def
+ \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: 
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: 
+nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort 
+n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda 
+(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef 
+n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O 
+u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O 
+u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 
+P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: 
+Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u 
+t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 
+d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead 
+k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T 
+(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) 
+(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T 
+(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 
+\def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) (THead k t0 t1) 
+(THead k x t1) H3) in (let H6 \def (eq_ind_r T x (\lambda (t2: T).(subst0 d u 
+t0 t2)) H4 t0 H5) in (H d H6 P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: 
+T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u 
+t1 t2)))).(ex2_ind T (\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) 
+(\lambda (t2: T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: 
+(eq T (THead k t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 
+x)).(let H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) \Rightarrow t2])) 
+(THead k t0 t1) (THead k t0 x) H3) in (let H6 \def (eq_ind_r T x (\lambda 
+(t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s k d) H6 P)))))) H2)) 
+(\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 x1))).(\lambda (H4: 
+(subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 x1)).(let H6 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k 
+x0 x1) H3) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) 
+\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T 
+t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 
+t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u 
+t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 
+t1 (THead k t0 t1) d H1)))))))))) t)).
+
+lemma subst0_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i 
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall 
+(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d 
+t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda 
+(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: 
+T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) 
+(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) 
+(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O 
+v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: 
+T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S 
+i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d 
+v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r 
+(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda 
+(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall 
+(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d 
+u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt 
+i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) 
+t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d 
+(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) 
+(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d 
+u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h 
+d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k 
+u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h 
+d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: 
+((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) 
+(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda 
+(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 
+\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s 
+k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) 
+(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r 
+T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 
+(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead 
+k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h 
+(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind 
+nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) 
+(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s 
+k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s 
+k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d 
+u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) 
+(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h 
+d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: 
+nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) 
+(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: 
+nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d 
+(S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda 
+(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) 
+(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: 
+nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 
+t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) 
+(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t 
+(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k 
+d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift 
+h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) 
+(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k 
+d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s 
+k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) 
+(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d 
+(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) 
+(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
+
+lemma subst0_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
+(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: 
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le 
+d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: 
+T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T 
+(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift 
+(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 
+(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda 
+(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r 
+nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) 
+O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S 
+i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d 
+(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d 
+H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le 
+d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T 
+(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 
+h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift 
+h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
+(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 
+h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 
+t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda 
+(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: 
+nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d 
+t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 
+\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: 
+nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 
+(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) 
+(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead 
+k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda 
+(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) 
+(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k 
+d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k 
+u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) 
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda 
+(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to 
+(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: 
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 
+t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k 
+i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le 
+d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: 
+nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) 
+(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T 
+(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 
+h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift 
+h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
+(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 
+h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d 
+i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead 
+k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
+
+lemma subst0_lift_ge_S:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d 
+t1) (lift (S O) d t2))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat 
+(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d 
+t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) 
+i) (\lambda (n: nat).(eq nat n (S i))) (le_antisym (plus (S O) i) (S i) (le_n 
+(S i)) (le_n (plus (S O) i))) (plus i (S O)) (plus_sym i (S O)))))))))).
+
+lemma subst0_lift_ge_s:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s 
+(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda 
+(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/subst0.ma
new file mode 100644 (file)
index 0000000..bbd13a6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/subst0.ma
@@ -0,0 +1,1389 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/props.ma".
+
+include "basic_1A/s/fwd.ma".
+
+theorem subst0_subst0:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i 
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: 
+T).(subst0 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: 
+T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n 
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3))))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda 
+(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i)) 
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda 
+(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t: 
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t 
+(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v 
+u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) 
+(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: 
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
+(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: 
+nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i 
+u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda 
+(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 
+i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 
+x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0: 
+T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) 
+u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst 
+u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: 
+K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: 
+nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: 
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda 
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k 
+i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0: 
+T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda 
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k 
+i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t)) 
+(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x: 
+T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0 
+(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: 
+nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in 
+(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n 
+u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T 
+(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S 
+(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 
+u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 
+H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: 
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda 
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t 
+u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: 
+(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: 
+T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 
+(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t 
+t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: 
+(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: 
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u 
+t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 
+x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda 
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0)) 
+(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: 
+T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda 
+(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0 
+u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 
+(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def 
+(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9 
+(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: 
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u 
+t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) 
+(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) 
+(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
+
+theorem subst0_subst0_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i 
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: 
+T).(subst0 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u: 
+T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n 
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4))))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda 
+(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i)) 
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda 
+(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t: 
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift 
+(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0 
+(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 
+(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda 
+(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: 
+((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to 
+(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus 
+i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: 
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v 
+u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: 
+T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3 
+(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) 
+t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0 
+(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k 
+u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0)) 
+(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0 
+i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0 
+(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall 
+(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 
+t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda 
+(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2: 
+(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t)) 
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t: 
+T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 
+(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3 
+x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def 
+(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4 
+(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k 
+(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i))) 
+(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u 
+t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead 
+k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 
+u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: 
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
+u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) 
+(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: 
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 
+t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: 
+nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) 
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: 
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v 
+u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: 
+T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3 
+(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) 
+t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6: 
+(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i 
+u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda 
+(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 
+i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1 
+x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r 
+nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus 
+i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0 
+i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k 
+(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) 
+t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0 
+x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) 
+H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 
+t1 t2 H))))).
+
+theorem subst0_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 
+i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to 
+(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3: 
+T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false 
+v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0 
+(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: 
+T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1: 
+((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda 
+(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k 
+u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) 
+(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3 
+(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
+T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda 
+(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: 
+T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 
+t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) 
+(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 
+i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead 
+k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda 
+(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: 
+T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k 
+u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k 
+u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda 
+(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: 
+T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 
+H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T 
+t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) 
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: 
+T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5: 
+(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T 
+(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) 
+(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3)) 
+(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0 
+(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0 
+t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4: 
+T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2))) 
+(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s 
+k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0 
+(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 
+t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq 
+T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 
+(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x 
+t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda 
+(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) 
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u 
+t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: 
+T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) 
+(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 
+(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k 
+u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 
+i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 
+t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) 
+t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 
+x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0 
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) 
+t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3)) 
+(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 
+u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 
+u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: 
+(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0) 
+v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: 
+(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4 
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda 
+(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 
+t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) 
+(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead 
+k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda 
+(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 
+i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 
+H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) 
+(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: 
+T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) 
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 
+(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead 
+k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 
+u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda 
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda 
+(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 
+i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) 
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda 
+(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: 
+T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 
+x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 
+H4))))))))))))))) i v t1 t2 H))))).
+
+theorem subst0_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: 
+nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall 
+(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda 
+(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: 
+nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n: 
+nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: 
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq 
+nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5: 
+T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda 
+(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2 
+(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq 
+T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v) 
+t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda 
+(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n: 
+nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda 
+(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda 
+(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in 
+False with []) in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) 
+(\lambda (v: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: 
+T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 
+T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 
+t)))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: 
+T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) 
+t2)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq 
+T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t 
+t3))) (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
+t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: 
+T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) 
+(\lambda (H4: (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
+T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 
+t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 
+i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
+T).(\lambda (H5: (eq T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 
+x)).(eq_ind_r T (THead k x t) (\lambda (t3: T).(ex2 T (\lambda (t4: 
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
+(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i 
+v x t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i v (THead k x t) t3))) (\lambda (x0: T).(\lambda (H7: 
+(subst0 i2 u3 u2 x0)).(\lambda (H8: (subst0 i v x x0)).(ex_intro2 T (\lambda 
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead 
+k x t) t3)) (THead k x0 t) (subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x 
+i H8 t k))))) (H1 x u3 i2 H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T 
+(\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) 
+u3 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda 
+(t3: T).(subst0 (s k i2) u3 t t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 
+(THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
+T).(\lambda (H5: (eq T t2 (THead k u0 x))).(\lambda (H6: (subst0 (s k i2) u3 
+t x)).(eq_ind_r T (THead k u0 x) (\lambda (t3: T).(ex2 T (\lambda (t4: 
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
+(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i v (THead k u0 x) t3)) (THead k u2 x) (subst0_snd k u3 x t i2 H6 
+u2) (subst0_fst v u2 u0 i H0 x k)) t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T 
+(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: 
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i2) u3 t t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda 
+(t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 
+i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) 
+(ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i v t2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
+t2 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: 
+(subst0 (s k i2) u3 t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 
+T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 
+i v t3 t4)))) (ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: 
+T).(subst0 i v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) 
+t3)) (\lambda (t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: 
+T).(\lambda (H8: (subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 
+x)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i v (THead k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x 
+i2 H8 k t x1 H7) (subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 
+H5)))))) H4)) (subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: 
+K).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: 
+nat).(\lambda (H0: (subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: 
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq 
+nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: 
+T).(subst0 (s k i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda 
+(u2: T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) 
+t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq 
+T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda 
+(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: 
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) 
+(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda 
+(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k 
+u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 
+i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
+T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u 
+x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5: 
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
+(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: 
+T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 
+k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T 
+(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) 
+u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda 
+(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
+T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 
+t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5: 
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) 
+(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7: 
+(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2 
+T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i 
+v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) 
+(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t: 
+T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq 
+nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k 
+i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s 
+k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7: 
+(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4)) 
+(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k 
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda 
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: 
+T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k 
+u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u 
+x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1) 
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5)) 
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k 
+i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: 
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 
+x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda 
+(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k 
+x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0))))) 
+(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) 
+(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to 
+False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_: 
+(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj 
+k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k 
+i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 
+t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: 
+((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) 
+\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) 
+(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: 
+T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: 
+((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) 
+\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 
+t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: 
+T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k 
+u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: 
+T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T 
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) 
+u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 
+t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: 
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
+(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda 
+(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k 
+u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: 
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
+(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 
+i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5: 
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x 
+t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2 
+u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: 
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x 
+t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 
+i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T 
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) 
+u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda 
+(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: 
+T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 
+t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5: 
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) 
+(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda 
+(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x 
+x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda 
+(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 
+i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 
+(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0: 
+T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x 
+x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11)))))) 
+(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i 
+i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: 
+T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: 
+T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) 
+u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 
+(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) 
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda 
+(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0 
+x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3 
+t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5: 
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v 
+x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2 
+u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t: 
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T 
+(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v 
+(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3 
+x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t: 
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 
+x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both 
+v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t: 
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq 
+nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k 
+i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat 
+(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda 
+(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 
+H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 
+H4)))))))))))))))))) i1 u1 t0 t1 H))))).
+
+theorem subst0_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
+i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) 
+(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) 
+(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to 
+(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5: 
+T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda 
+(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef 
+i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T 
+(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) 
+t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) 
+(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda 
+(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2) 
+(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t: 
+T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2 
+(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v 
+t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda 
+(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2: 
+T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0 
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0 
+i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda 
+(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T 
+t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t 
+t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) 
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
+(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 
+t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq 
+T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead 
+k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 
+(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x 
+t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda 
+(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v 
+(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v 
+(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x) 
+(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x 
+t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead 
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k 
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2 
+x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t)) 
+(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4: 
+T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x 
+t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x 
+t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) 
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) 
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead 
+k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) 
+(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 
+v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) 
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: 
+T).(\lambda (H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x 
+x0)).(or4_intro1 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: 
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x 
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x 
+t) (THead k u2 t)) (ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) 
+t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) 
+(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) 
+(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x 
+t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x 
+t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t 
+k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) 
+(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) 
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) 
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x 
+i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: 
+T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda 
+(t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 
+t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 (THead k u2 t))) 
+(\lambda (x: T).(\lambda (H4: (eq T t2 (THead k u1 x))).(\lambda (H5: (subst0 
+(s k i0) v t x)).(eq_ind_r T (THead k u1 x) (\lambda (t3: T).(or4 (eq T 
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+u1 u2 i0 H0 k x t3 H8))))) H9)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro3 
+(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v 
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 
+i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 
+t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (\lambda (_: (subst0 i0 v u2 
+u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 
+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 
+x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 
+x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: 
+T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v 
+u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 
+t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: 
+T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) 
+t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 
+(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4 
+(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s 
+k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead 
+k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5)) 
+(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0 
+i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t: 
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t))) 
+(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2 
+t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) 
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda 
+(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t) 
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5)) 
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2 
+t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind 
+(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: 
+T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T 
+(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k 
+u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v 
+(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 
+x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T 
+(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k 
+t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v 
+(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t 
+x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k 
+x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda 
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v 
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: 
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T 
+(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v 
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 
+v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0 
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1) 
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) 
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 
+x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2 
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 
+v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k) 
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 
+x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k 
+x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10: 
+(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 
+i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1 
+x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3 
+t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t: 
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4 
+(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v 
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 
+i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k 
+u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda 
+(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: 
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 
+x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T 
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v 
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T 
+x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda 
+(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead 
+k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v 
+(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k 
+x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: 
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 
+x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t: 
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1 
+i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2 
+t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) 
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) 
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda 
+(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0 
+x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k 
+x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x) 
+(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x 
+H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead 
+k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) 
+t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k 
+u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) 
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0 
+H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0 
+v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda 
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k 
+x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead 
+k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k 
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x) 
+(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11)))) 
+(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T 
+u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 
+v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) 
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) 
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda 
+(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3) 
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5)) 
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t 
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3)))) 
+(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k 
+x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda 
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v 
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: 
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T 
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v 
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0 
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3) 
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) 
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 
+v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9) 
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2 
+x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: 
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k 
+x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda 
+(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) 
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: 
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 
+x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0 
+i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1 
+t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) 
+(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) 
+(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) 
+(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) 
+(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: 
+T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 
+i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) 
+(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) 
+(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T 
+(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v 
+(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 
+v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10)) 
+(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: 
+T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) 
+(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 
+x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: 
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda 
+(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq 
+T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead 
+k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v 
+(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 
+t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda 
+(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0 
+H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10: 
+(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 
+i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead 
+k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0)))) 
+(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead 
+k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda 
+(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead 
+k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 
+i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) 
+(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
+
+theorem subst0_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i 
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0 
+i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i 
+t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: 
+T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S 
+O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) 
+(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def 
+(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i 
+H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) 
+(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: 
+T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) 
+i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i 
+t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) 
+x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S 
+O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) 
+(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift 
+(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1) 
+(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) 
+(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) 
+(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i 
+t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i) 
+(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) 
+i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) 
+(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst0/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst0/tlt.ma
new file mode 100644 (file)
index 0000000..5309cec
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst0/tlt.ma
@@ -0,0 +1,457 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/fwd.ma".
+
+include "basic_1A/lift/tlt.ma".
+
+lemma subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift 
+(S i) O v)) (weight_map g (TLRef i)) (le_S_n (S (weight_map f (lift (S i) O 
+v))) (S (weight_map g (TLRef i))) (le_S (S (S (weight_map f (lift (S i) O 
+v)))) (S (weight_map g (TLRef i))) (le_n_S (S (weight_map f (lift (S i) O 
+v))) (weight_map g (TLRef i)) H1)))))))))) (\lambda (v: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S 
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
+(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S 
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd 
+g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_O_n O) n))))))))) (\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g 
+H2 O O (le_O_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
+u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) 
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g 
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) 
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
+\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
+(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S 
+(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) 
+(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
+g H2 O O (le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g 
+O) (\lambda (m: nat).(wadd_le f g H2 O O (le_O_n O) m)) (eq_ind nat 
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map 
+f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
+(le_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map 
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le 
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2) 
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
+(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) 
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f 
+g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) 
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) 
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat 
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) 
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O 
+(le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
+(\lambda (m: nat).(wadd_le f g H4 O O (le_O_n O) m)) (eq_ind nat (weight_map 
+f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) 
+(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: 
+(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) 
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus 
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 
+f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+lemma subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) 
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: 
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S 
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
+(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
+g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S 
+(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) 
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd 
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_O_n O) n))))))))))) (\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f 
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) 
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n 
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O 
+O (le_O_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
+u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) 
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
+H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
+t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g 
+t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) 
+(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
+(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
+\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
+u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
+nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
+i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f 
+u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
+g H2 O O (le_O_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd 
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_O_n O) m)) (eq_ind nat 
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
+f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
+(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map 
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt 
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) 
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
+(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) 
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
+t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 
+f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) 
+(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S 
+(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) 
+(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O 
+v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f 
+u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) 
+f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
+t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) 
+(wadd g O m) (wadd_le f g H4 O O (le_O_n O) m)))) (eq_ind nat (weight_map f 
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
+(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) 
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) 
+(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_O_n O) 
+m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g 
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v 
+(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) 
+\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) 
+(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
+f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) 
+(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) 
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t 
+z H))))).
+
+lemma subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt 
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus 
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
+(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map 
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map 
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n 
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda 
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n 
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) 
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: 
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda 
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: 
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) 
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda 
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda 
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+lemma subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z 
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx 
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst1/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst1/defs.ma
new file mode 100644 (file)
index 0000000..c11379b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst1/defs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/defs.ma".
+
+inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
+| subst1_refl: subst1 i v t1 t1
+| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst1/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst1/fwd.ma
new file mode 100644 (file)
index 0000000..9552eaa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst1/fwd.ma
@@ -0,0 +1,175 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst1/defs.ma".
+
+include "basic_1A/subst0/fwd.ma".
+
+implied lemma subst1_ind:
+ \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (P: ((T \to 
+Prop))).((P t1) \to (((\forall (t2: T).((subst0 i v t1 t2) \to (P t2)))) \to 
+(\forall (t: T).((subst1 i v t1 t) \to (P t))))))))
+\def
+ \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (P: ((T \to 
+Prop))).(\lambda (f: (P t1)).(\lambda (f0: ((\forall (t2: T).((subst0 i v t1 
+t2) \to (P t2))))).(\lambda (t: T).(\lambda (s0: (subst1 i v t1 t)).(match s0 
+with [subst1_refl \Rightarrow f | (subst1_single x x0) \Rightarrow (f0 x 
+x0)])))))))).
+
+lemma subst1_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
+i v (TSort n) x) \to (eq T x (TSort n))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T 
+t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0 
+i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x 
+H))))).
+
+lemma subst1_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 
+i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift 
+(S n) O v))))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or 
+(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl 
+(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O 
+v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v 
+(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2 
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq 
+nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2 
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i) 
+(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x 
+H))))).
+
+lemma subst1_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
+(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(subst1 (s k i) v t1 t2))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) 
+x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1 
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal 
+T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda 
+(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T 
+(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
+u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2 
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda 
+(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda 
+(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1 
+x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1 
+x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3: 
+T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: 
+T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v 
+t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1) 
+(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i) v t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3: 
+T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda 
+(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 
+i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 
+x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4))))))) 
+H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
+
+lemma subst1_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) 
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst1 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S 
+(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1) 
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
+(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2: 
+T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1)) 
+(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) 
+(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h 
+(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda 
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 
+t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) 
+x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T 
+t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1 
+(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x 
+H))))))).
+
+lemma subst1_gen_lift_eq:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u 
+(lift h d t) x) \to (eq T x (lift h d t))))))))))
+\def
+ \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d 
+h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t) 
+(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda 
+(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t 
+u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
+
+lemma subst1_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst1 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1) 
+x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda 
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2: 
+T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift 
+h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1 
+(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2: 
+T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3: 
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3)) 
+(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 
+(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d 
+x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3: 
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0 
+H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h 
+d H1 H0)))) x H)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst1/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst1/props.ma
new file mode 100644 (file)
index 0000000..7bd1ea2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst1/props.ma
@@ -0,0 +1,165 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst1/fwd.ma".
+
+include "basic_1A/subst0/props.ma".
+
+theorem subst1_head:
+ \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 
+i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s 
+k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
+\def
+ \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k: 
+K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i 
+v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k 
+i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t))) 
+(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k 
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k 
+v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1 
+t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1 
+(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead 
+k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1) 
+(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k 
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both 
+v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
+
+lemma subst1_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i 
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d: 
+nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) 
+(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i 
+d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d 
+t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d: 
+nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h 
+(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d 
+H1 h))))))) t2 H))))).
+
+lemma subst1_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
+(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: 
+T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h 
+d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u 
+(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda 
+(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1) 
+(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
+
+lemma subst1_ex:
+ \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: 
+T).(subst1 d u t1 (lift (S O) d t2))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex 
+T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n: 
+nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n) 
+(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d 
+u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n)) 
+(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d 
+(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda 
+(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) 
+d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n) 
+t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S 
+O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T 
+(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T 
+(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u) 
+(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n) 
+t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S 
+O) n (lift n O u)) (lift_free u n (S O) O n (le_plus_r O n) (le_O_n n)))) d 
+H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) 
+(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t: 
+T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef 
+(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t: 
+T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift 
+(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T 
+(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d: 
+nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2: 
+T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u 
+(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u 
+t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in 
+(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex 
+T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda 
+(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d) 
+x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d 
+t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k 
+d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t 
+(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k 
+x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
+
+lemma subst1_lift_S:
+ \forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i 
+(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
+\def
+ \lambda (u: T).(T_ind (\lambda (t: T).(\forall (i: nat).(\forall (h: 
+nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) (lift (S h) i 
+t)))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (_: 
+(le h i)).(eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef h) t (lift 
+(S h) i (TSort n)))) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 i (TLRef 
+h) (TSort n) t)) (subst1_refl i (TLRef h) (TSort n)) (lift (S h) i (TSort n)) 
+(lift_sort n (S h) i)) (lift (S h) (S i) (TSort n)) (lift_sort n (S h) (S 
+i))))))) (\lambda (n: nat).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H: 
+(le h i)).(lt_eq_gt_e n i (subst1 i (TLRef h) (lift (S h) (S i) (TLRef n)) 
+(lift (S h) i (TLRef n))) (\lambda (H0: (lt n i)).(eq_ind_r T (TLRef n) 
+(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T 
+(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i 
+(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0)) 
+(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S_n (S n) (S i) 
+(le_S (S (S n)) (S i) (le_n_S (S n) i H0)))))) (\lambda (H0: (eq nat n 
+i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: nat).(le h n0)) H n H0) in 
+(eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef h) (lift (S h) (S n0) 
+(TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t: 
+T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) (eq_ind_r T (TLRef (plus 
+n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (eq_ind nat (S 
+(plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) 
+(eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n) 
+(TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: nat).(subst1 n 
+(TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O (TLRef h)) (\lambda 
+(t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n (TLRef h) (TLRef n) 
+(lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) (TLRef (plus h (S n))) 
+(lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) (sym_eq nat (S (plus h 
+n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) (plus_sym n h)) (plus n (S 
+h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) (lift_lref_ge n (S h) n (le_n 
+n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt n (S h) (S n) (le_n (S n)))) 
+i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T (TLRef (plus n (S h))) (\lambda 
+(t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T (TLRef 
+(plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) (TLRef (plus n (S h))) 
+t)) (subst1_refl i (TLRef h) (TLRef (plus n (S h)))) (lift (S h) i (TLRef n)) 
+(lift_lref_ge n (S h) i (le_S_n i n (le_S_n (S i) (S n) (le_S (S (S i)) (S n) 
+(le_n_S (S i) n H0)))))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) 
+(S i) H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i: 
+nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t) 
+(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i: 
+nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) 
+t0) (lift (S h) i t0))))))).(\lambda (i: nat).(\lambda (h: nat).(\lambda (H1: 
+(le h i)).(eq_ind_r T (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) 
+t0)) (\lambda (t1: T).(subst1 i (TLRef h) t1 (lift (S h) i (THead k t t0)))) 
+(eq_ind_r T (THead k (lift (S h) i t) (lift (S h) (s k i) t0)) (\lambda (t1: 
+T).(subst1 i (TLRef h) (THead k (lift (S h) (S i) t) (lift (S h) (s k (S i)) 
+t0)) t1)) (subst1_head (TLRef h) (lift (S h) (S i) t) (lift (S h) i t) i (H i 
+h H1) k (lift (S h) (s k (S i)) t0) (lift (S h) (s k i) t0) (eq_ind_r nat (S 
+(s k i)) (\lambda (n: nat).(subst1 (s k i) (TLRef h) (lift (S h) n t0) (lift 
+(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k 
+(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i)) 
+(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/subst1/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/subst1/subst1.ma
new file mode 100644 (file)
index 0000000..fb466dc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/subst1/subst1.ma
@@ -0,0 +1,196 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst1/fwd.ma".
+
+include "basic_1A/subst0/subst0.ma".
+
+theorem subst1_subst1:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i 
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: 
+T).(subst1 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: 
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda 
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 
+t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: 
+(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda 
+(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl 
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 
+i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (\lambda (_: 
+T).(ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S 
+(plus i j)) u t0 t3)))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 
+y)).(subst1_ind i u u1 (\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: 
+T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) 
+(\lambda (H3: (eq T u1 u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda 
+(t0: T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 
+t3)))) (ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: 
+T).(subst1 (S (plus i j)) u t t3)) t3 (subst1_single j u2 t1 t3 H0) 
+(subst1_refl (S (plus i j)) u t3)) u1 H3)) (\lambda (t0: T).(\lambda (H3: 
+(subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let H5 \def (eq_ind T t0 
+(\lambda (t: T).(subst0 i u u1 t)) H3 u2 H4) in (ex2_ind T (\lambda (t: 
+T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t3)) (ex2 T 
+(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u 
+t t3))) (\lambda (x: T).(\lambda (H6: (subst0 j u1 t1 x)).(\lambda (H7: 
+(subst0 (S (plus i j)) u x t3)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 
+t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t3)) x (subst1_single j u1 t1 
+x H6) (subst1_single (S (plus i j)) u x t3 H7))))) (subst0_subst0 t1 t3 u2 j 
+H0 u1 u i H5)))))) y H2))) H1))))))) t2 H))))).
+
+theorem subst1_subst1_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i 
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: 
+T).(subst1 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda 
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1: 
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda 
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t 
+t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: 
+(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda 
+(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl 
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1 
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1 
+i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0: 
+T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0)))) 
+(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S 
+(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i 
+j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T 
+(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u 
+t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S 
+(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1 
+x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t: 
+T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x 
+(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4))))) 
+(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
+
+theorem subst1_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 
+i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3: 
+T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda 
+(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1 
+t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3 
+(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0: 
+T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans 
+t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
+
+theorem subst1_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: 
+nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall 
+(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda 
+(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1: 
+nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t: 
+T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) 
+\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3)) 
+(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2: 
+T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not 
+(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda 
+(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2: 
+T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2: 
+T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not 
+(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T 
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2 
+(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4: 
+T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1 
+u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1 
+i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda 
+(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T 
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x 
+(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4))))) 
+(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2 
+H2))))) t3 H1)))))))) t1 H))))).
+
+theorem subst1_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 
+i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: 
+T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2: 
+T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3)) 
+(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0: 
+(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda 
+(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2: 
+T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i 
+u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1 
+i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t: 
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u 
+t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u 
+t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t)) 
+(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4) 
+(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t))) 
+(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda 
+(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2 
+T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2 
+(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T 
+(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i 
+u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i 
+u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5: 
+(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda 
+(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4 
+x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t: 
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u 
+t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2 
+t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 
+i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4))) 
+(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
+
+theorem subst1_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i 
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1) 
+(\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(\forall (t2: T).((subst1 
+i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))) (\lambda (y: T).(\lambda (H0: 
+(subst1 i u t0 y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i 
+t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 
+t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda 
+(H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda 
+(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 
+\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u 
+(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda 
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) 
+H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: 
+(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: 
+T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 
+(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T 
+(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (\lambda (_: T).(eq T t1 
+t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 
+(\lambda (t: T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: 
+(eq T t0 (lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: 
+T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in 
+(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r 
+nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i 
+(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: 
+(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def 
+(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in 
+(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) 
+H3))))))) y H0))) H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/theory.ma b/matita/matita/contribs/lambdadelta/basic_1A/theory.ma
new file mode 100644 (file)
index 0000000..5ad07a5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/theory.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/subst0/tlt.ma".
+
+include "basic_1A/subst/props.ma".
+
+include "basic_1A/sty1/cnt.ma".
+
+include "basic_1A/ex0/props.ma".
+
+include "basic_1A/pr3/wcpr0.ma".
+
+include "basic_1A/ex2/props.ma".
+
+include "basic_1A/ex1/props.ma".
+
+include "basic_1A/ty3/sty0.ma".
+
+include "basic_1A/csubt/csuba.ma".
+
+include "basic_1A/ty3/fwd_nf2.ma".
+
+include "basic_1A/ty3/nf2.ma".
+
+include "basic_1A/wf3/props.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlist/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlist/defs.ma
new file mode 100644 (file)
index 0000000..7d892e9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlist/defs.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+inductive TList: Type[0] \def
+| TNil: TList
+| TCons: T \to (TList \to TList).
+
+rec definition THeads (k: K) (us: TList) on us: T \to T \def \lambda (t: 
+T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u 
+(THeads k ul t))]).
+
+rec definition TApp (ts: TList) on ts: T \to TList \def \lambda (v: T).(match 
+ts with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t 
+(TApp ts0 v))]).
+
+rec definition tslen (ts: TList) on ts: nat \def match ts with [TNil 
+\Rightarrow O | (TCons _ ts0) \Rightarrow (S (tslen ts0))].
+
+definition tslt:
+ TList \to (TList \to Prop)
+\def
+ \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlist/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlist/fwd.ma
new file mode 100644 (file)
index 0000000..cced526
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlist/fwd.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/tlist/props.ma".
+
+fact tslt_wf__q_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList 
+\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0) 
+\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen 
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat 
+(tslen ts)))))).
+
+lemma tslt_wf_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1: 
+TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts: 
+TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts: 
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to 
+Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt 
+(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts: 
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n: 
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda 
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t)) 
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2 
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to 
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0) 
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen 
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
+
+lemma tlist_ind_rev:
+ \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts: 
+TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts: 
+TList).(P ts))))
+\def
+ \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0: 
+((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts 
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t)) 
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1: 
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1: 
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0: 
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1)))) 
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0)) 
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in 
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t 
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0) 
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1: 
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat 
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P 
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen 
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0) 
+H4))))) H3))))))) ts2)) ts)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlist/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlist/props.ma
new file mode 100644 (file)
index 0000000..b0a0b99
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlist/props.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/tlist/defs.ma".
+
+lemma theads_tapp:
+ \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T 
+(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads 
+k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda 
+(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k 
+v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0 
+(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads 
+k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
+
+lemma tcons_tapp_ex:
+ \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda 
+(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
+\def
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2 
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp 
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen 
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda 
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal 
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t: 
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T 
+(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2 
+t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen 
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in 
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t 
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) 
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq 
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda 
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda 
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq 
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2: 
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons 
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S 
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n: 
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons 
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq 
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2: 
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2)))) 
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2)))) 
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat 
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlt/defs.ma
new file mode 100644 (file)
index 0000000..d40e8fd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlt/defs.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/defs.ma".
+
+definition wadd:
+ ((nat \to nat)) \to (nat \to (nat \to nat))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n 
+with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
+
+rec definition weight_map (f: (nat \to nat)) (t: T) on t: nat \def match t 
+with [(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f 
+u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f 
+O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) 
+t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f 
+t0)))])].
+
+definition weight:
+ T \to nat
+\def
+ weight_map (\lambda (_: nat).O).
+
+definition tlt:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlt/fwd.ma
new file mode 100644 (file)
index 0000000..06c790a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlt/fwd.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/tlt/defs.ma".
+
+fact tlt_wf__q_ind:
+ \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P0: ((T \to 
+Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P0 
+t))))) P n))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: 
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to 
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t) 
+n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight 
+t)))))).
+
+lemma tlt_wf_ind:
+ \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) 
+\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t: 
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to 
+Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v) 
+(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind 
+(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0: 
+T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) 
+\to (Q (\lambda (t0: T).(P t0)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat 
+(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall 
+(m: nat).((lt m n1) \to (\forall (t1: T).((eq nat (weight t1) m) \to (P 
+t1)))))) H0 (weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt 
+(weight v) (weight t0))).(H2 (weight v) H3 v (refl_equal nat (weight 
+v))))))))))))) t)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/tlt/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/tlt/props.ma
new file mode 100644 (file)
index 0000000..bb9621f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/tlt/props.ma
@@ -0,0 +1,238 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/T/fwd.ma".
+
+include "basic_1A/tlt/defs.ma".
+
+lemma wadd_le:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to 
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: 
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: 
+nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le 
+(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
+
+lemma wadd_lt:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: 
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to 
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: 
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w: 
+nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S_n (S v) (S w) (le_S 
+(S (S v)) (S w) (le_n_S (S v) w H0)))) (\lambda (n0: nat).(\lambda (_: (le 
+(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
+
+lemma wadd_O:
+ \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_: 
+nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat 
+(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
+
+lemma weight_le:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) 
+(weight_map g t)))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
+\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda 
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall 
+(n0: nat).(le (f n0) (g n0))))).(le_O_n (weight_map g (TSort n))))))) 
+(\lambda (n: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H: ((\forall (n0: nat).(le (f n0) (g n0))))).(H n))))) 
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g 
+n)))) \to (le (weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: 
+T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) 
+\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f (THead k0 t0 t1)) 
+(weight_map g (THead k0 t0 t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: 
+B).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) 
+(weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
+\to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
+\to (le (match b0 with [Abbr \Rightarrow (S (plus (weight_map f t0) 
+(weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S (plus 
+(weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus 
+(weight_map f t0) (weight_map (wadd f O) t1)))]) (match b0 with [Abbr 
+\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g (S (weight_map g 
+t0))) t1))) | Abst \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g 
+O) t1))) | Void \Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) 
+t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) 
+\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda 
+(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g 
+t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus 
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus 
+(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1)) 
+(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S 
+(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g 
+H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0))) 
+(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0)) 
+(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n)))))))))))) 
+(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f 
+t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) 
+(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le 
+(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1)) 
+(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map 
+f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1) 
+(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O 
+(le_O_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g 
+n)))) \to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: 
+T).(\lambda (H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) 
+(weight_map g t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus 
+(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0) 
+(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0) 
+(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f 
+O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_O_n O) n)))))))))))) 
+b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n)))) 
+\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda 
+(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g 
+t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus 
+(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g 
+t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1) 
+(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
+
+lemma weight_eq:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f 
+t) (weight_map g t)))))
+\def
+ \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym 
+(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n: 
+nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n) 
+(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0: 
+nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
+
+lemma weight_add_O:
+ \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) 
+(weight_map (\lambda (_: nat).O) t))
+\def
+ \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_: 
+nat).O) (\lambda (n: nat).(wadd_O n))).
+
+lemma weight_add_S:
+ \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) 
+O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
+\def
+ \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O) 
+(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(wadd_le (\lambda (_: 
+nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_O_n O)) O (S m) (le_O_n (S 
+m)) n)))).
+
+theorem tlt_trans:
+ \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to 
+(tlt u t)))))
+\def
+ \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u) 
+(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u) 
+(weight v) (weight t) H H0))))).
+
+lemma tlt_head_sx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead 
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall 
+(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr 
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst 
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda 
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda 
+(u: T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus 
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
+(weight_map (\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: 
+nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: 
+nat).O) u))) t))))) (\lambda (u: T).(\lambda (t: T).(le_n_S (weight_map 
+(\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) (weight_map 
+(wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) 
+u) (weight_map (wadd (\lambda (_: nat).O) O) t))))) (\lambda (u: T).(\lambda 
+(t: T).(le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda 
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l 
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) 
+t))))) b)) (\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_n_S 
+(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u) 
+(weight_map (\lambda (_: nat).O) t)) (le_plus_l (weight_map (\lambda (_: 
+nat).O) u) (weight_map (\lambda (_: nat).O) t)))))) k).
+
+lemma tlt_head_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt 
+(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead 
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall 
+(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr 
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst 
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda 
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda 
+(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S 
+(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_: 
+nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: 
+nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S 
+(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) 
+(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) 
+u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd 
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus 
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
+(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd 
+(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda 
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t 
+(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t) 
+(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map 
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))))))) 
+(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_: 
+nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus 
+(weight_map (\lambda (_: nat).O) u) n)))) (le_n_S (weight_map (\lambda (_: 
+nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: 
+nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map 
+(\lambda (_: nat).O) t))) (weight_map (wadd (\lambda (_: nat).O) O) t) 
+(weight_add_O t)))) (\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map 
+(\lambda (_: nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: 
+nat).O) t) (S (plus (weight_map (\lambda (_: nat).O) u) n)))) (le_n_S 
+(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) 
+(weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: 
+nat).O) u) (weight_map (\lambda (_: nat).O) t))) (weight_map (wadd (\lambda 
+(_: nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u: 
+T).(\lambda (t: T).(le_n_S (weight_map (\lambda (_: nat).O) t) (plus 
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)) 
+(le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: 
+nat).O) t)))))) k).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity.ma
new file mode 100644 (file)
index 0000000..116241d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity.ma
@@ -0,0 +1,182 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/pr3_props.ma".
+
+include "basic_1A/arity/pr3.ma".
+
+include "basic_1A/asucc/fwd.ma".
+
+lemma ty3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity 
+g c t2 (asucc g a1))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g 
+c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity 
+g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
+a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda 
+(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) 
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 
+(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: 
+(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A 
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g 
+a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: 
+(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: 
+T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: 
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) 
+(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 
+x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity 
+g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g 
+(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 
+g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) 
+H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A 
+(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 
+(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort 
+g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g 
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g 
+d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) 
+(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g 
+c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr 
+c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A 
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g 
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) 
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d 
+t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind 
+A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g 
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g 
+a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A 
+(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 
+(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g 
+d u x H4 (asucc g x0) H7)) (arity_repl g c0 (lift (S n) O u) x (arity_lift g 
+d u x H4 c0 (S n) O (getl_drop Abst c0 d u n H0)) (asucc g x0) H7)))) 
+H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity 
+g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: 
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
+u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) 
+u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g 
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) 
+(\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
+g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) 
+u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 u 
+x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H7 \def H3 in (ex2_ind A 
+(\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: 
+A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: 
+A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead 
+(Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g (CHead 
+c0 (Bind b) u) t3 x0)).(\lambda (H9: (arity g (CHead c0 (Bind b) u) t4 (asucc 
+g x0))).(let H_x \def (leq_asucc g x) in (let H10 \def H_x in (ex_ind A 
+(\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 
+(THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) 
+(asucc g a1)))) (\lambda (x1: A).(\lambda (H11: (leq g x (asucc g 
+x1))).(B_ind (\lambda (b0: B).((arity g (CHead c0 (Bind b0) u) t3 x0) \to 
+((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: 
+A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 
+(THead (Bind b0) u t4) (asucc g a1))))))) (\lambda (H12: (arity g (CHead c0 
+(Bind Abbr) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind Abbr) u) t4 
+(asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u 
+t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) 
+x0 (arity_bind g Abbr not_abbr_abst c0 u x H5 t3 x0 H12) (arity_bind g Abbr 
+not_abbr_abst c0 u x H5 t4 (asucc g x0) H13)))) (\lambda (H12: (arity g 
+(CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H13: (arity g (CHead c0 (Bind 
+Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead 
+(Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) 
+(asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H5 
+(asucc g x1) H11) t3 x0 H12) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead 
+x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H5 (asucc g x1) 
+H11) t4 (asucc g x0) H13) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead 
+x1 x0))))))) (\lambda (H12: (arity g (CHead c0 (Bind Void) u) t3 
+x0)).(\lambda (H13: (arity g (CHead c0 (Bind Void) u) t4 (asucc g 
+x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 
+(arity_bind g Void not_void_abst c0 u x H5 t3 x0 H12) (arity_bind g Void 
+not_void_abst c0 u x H5 t4 (asucc g x0) H13)))) b H8 H9))) H10)))))) H7))))) 
+H4)))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: 
+(ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) 
+(\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A 
+(\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind 
+Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: 
+A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A 
+(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: 
+A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) 
+(\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u 
+(asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 
+A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: 
+A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) 
+(\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 
+(THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t 
+(asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A 
+(asucc g x0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u 
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind 
+Abst) u) t a2))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) 
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A 
+(asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g 
+x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def 
+(sym_eq A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g 
+x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) 
+(\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 
+(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) 
+w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: 
+(eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def 
+(eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 
+(asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v 
+a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead 
+(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w 
+(THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl 
+g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g 
+x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind 
+Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) 
+H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A 
+(\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g 
+a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: (ex2 A 
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) 
+(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity 
+g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat 
+Cast) t0 t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 
+x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(let H7 \def H3 in (ex2_ind A 
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g 
+a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) 
+(\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) (asucc g a1)))) 
+(\lambda (x0: A).(\lambda (H8: (arity g c0 t4 x0)).(\lambda (H9: (arity g c0 
+t0 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat 
+Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t0 t4) 
+(asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) (arity_cast g c0 t0 (asucc g 
+x) (arity_repl g c0 t0 (asucc g x0) H9 (asucc g (asucc g x)) (asucc_repl g x0 
+(asucc g x) (arity_mono g c0 t4 x0 H8 (asucc g x) H6))) t4 H6))))) H7))))) 
+H4)))))))))) c t1 t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity_props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity_props.ma
new file mode 100644 (file)
index 0000000..846d98c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/arity_props.ma
@@ -0,0 +1,105 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/arity.ma".
+
+include "basic_1A/sc3/arity.ma".
+
+lemma ty3_predicative:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: 
+T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: 
+Prop).P)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u: 
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u 
+v)).(\lambda (P: Prop).(let H1 \def H in (ex3_2_ind T T (\lambda (t2: 
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))) (\lambda (_: 
+T).(\lambda (t0: T).(ty3 g c v t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) v) t t2))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v 
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(let H_y \def 
+(ty3_conv g c v x1 H3 (THead (Bind Abst) v t) u H H0) in (let H_x \def 
+(ty3_arity g c (THead (Bind Abst) v t) v H_y) in (let H5 \def H_x in (ex2_ind 
+A (\lambda (a1: A).(arity g c (THead (Bind Abst) v t) a1)) (\lambda (a1: 
+A).(arity g c v (asucc g a1))) P (\lambda (x: A).(\lambda (H6: (arity g c 
+(THead (Bind Abst) v t) x)).(\lambda (H7: (arity g c v (asucc g x))).(let H8 
+\def (arity_gen_abst g c v t x H6) in (ex3_2_ind A A (\lambda (a1: 
+A).(\lambda (a2: A).(eq A x (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: 
+A).(arity g c v (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g 
+(CHead c (Bind Abst) v) t a2))) P (\lambda (x2: A).(\lambda (x3: A).(\lambda 
+(H9: (eq A x (AHead x2 x3))).(\lambda (H10: (arity g c v (asucc g 
+x2))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t x3)).(let H12 \def 
+(eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H7 (AHead x2 x3) H9) 
+in (leq_ahead_asucc_false g x2 (asucc g x3) (arity_mono g c v (asucc g (AHead 
+x2 x3)) H12 (asucc g x2) H10) P))))))) H8))))) H5))))))))) (ty3_gen_bind g 
+Abst c v t u H1)))))))))).
+
+theorem ty3_repellent:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (t: T).(\forall (u1: 
+T).((ty3 g c (THead (Bind Abst) w t) u1) \to (\forall (u2: T).((ty3 g (CHead 
+c (Bind Abst) w) t (lift (S O) O u2)) \to ((pc3 c u1 u2) \to (\forall (P: 
+Prop).P)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (t: T).(\lambda (u1: 
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) w t) u1)).(\lambda (u2: 
+T).(\lambda (H0: (ty3 g (CHead c (Bind Abst) w) t (lift (S O) O 
+u2))).(\lambda (H1: (pc3 c u1 u2)).(\lambda (P: Prop).(ex_ind T (\lambda (t0: 
+T).(ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) t0)) P (\lambda (x: 
+T).(\lambda (H2: (ty3 g (CHead c (Bind Abst) w) (lift (S O) O u2) x)).(let H3 
+\def (ty3_gen_lift g (CHead c (Bind Abst) w) u2 x (S O) O H2 c (drop_drop 
+(Bind Abst) O c c (drop_refl c) w)) in (ex2_ind T (\lambda (t2: T).(pc3 
+(CHead c (Bind Abst) w) (lift (S O) O t2) x)) (\lambda (t2: T).(ty3 g c u2 
+t2)) P (\lambda (x0: T).(\lambda (_: (pc3 (CHead c (Bind Abst) w) (lift (S O) 
+O x0) x)).(\lambda (H5: (ty3 g c u2 x0)).(let H_y \def (ty3_conv g c u2 x0 H5 
+(THead (Bind Abst) w t) u1 H H1) in (let H_x \def (ty3_arity g (CHead c (Bind 
+Abst) w) t (lift (S O) O u2) H0) in (let H6 \def H_x in (ex2_ind A (\lambda 
+(a1: A).(arity g (CHead c (Bind Abst) w) t a1)) (\lambda (a1: A).(arity g 
+(CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g a1))) P (\lambda (x1: 
+A).(\lambda (H7: (arity g (CHead c (Bind Abst) w) t x1)).(\lambda (H8: (arity 
+g (CHead c (Bind Abst) w) (lift (S O) O u2) (asucc g x1))).(let H_x0 \def 
+(ty3_arity g c (THead (Bind Abst) w t) u2 H_y) in (let H9 \def H_x0 in 
+(ex2_ind A (\lambda (a1: A).(arity g c (THead (Bind Abst) w t) a1)) (\lambda 
+(a1: A).(arity g c u2 (asucc g a1))) P (\lambda (x2: A).(\lambda (H10: (arity 
+g c (THead (Bind Abst) w t) x2)).(\lambda (H11: (arity g c u2 (asucc g 
+x2))).(arity_repellent g c w t x1 H7 x2 H10 (asucc_inj g x1 x2 (arity_mono g 
+c u2 (asucc g x1) (arity_gen_lift g (CHead c (Bind Abst) w) u2 (asucc g x1) 
+(S O) O H8 c (drop_drop (Bind Abst) O c c (drop_refl c) w)) (asucc g x2) 
+H11)) P)))) H9)))))) H6))))))) H3)))) (ty3_correct g (CHead c (Bind Abst) w) 
+t (lift (S O) O u2) H0))))))))))).
+
+lemma ty3_acyclic:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def 
+(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in 
+(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda 
+(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g 
+c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x 
+(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
+
+lemma ty3_sn3:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in 
+(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u 
+(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t 
+x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t 
+x H1))))) H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/dec.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/dec.ma
new file mode 100644 (file)
index 0000000..8a4f655
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/dec.ma
@@ -0,0 +1,431 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pc3/dec.ma".
+
+include "basic_1A/getl/flt.ma".
+
+include "basic_1A/getl/dec.ma".
+
+include "basic_1A/flt/fwd.ma".
+
+lemma ty3_inference:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: 
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to False)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: 
+C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: 
+T).((ty3 g c0 t t2) \to False))))) (\lambda (c2: C).(\lambda (t2: T).(T_ind 
+(\lambda (t: T).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t) \to (or 
+(ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) 
+\to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall 
+(t3: T).((ty3 g c2 t t3) \to False))))) (\lambda (n: nat).(\lambda (_: 
+((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (TSort n)) \to (or (ex T 
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
+False)))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) 
+(\forall (t3: T).((ty3 g c2 (TSort n) t3) \to False)) (ex_intro T (\lambda 
+(t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n))))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 
+c2 (TLRef n)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: 
+T).((ty3 g c1 t3 t4) \to False)))))))).(let H_x \def (getl_dec c2 n) in (let 
+H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
+(v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) 
+\to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) 
+t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H1: 
+(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead 
+e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda 
+(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) 
+(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 
+(CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) 
+in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g 
+x0 x2 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) 
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (H4: (ex T 
+(\lambda (t3: T).(ty3 g x0 x2 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: 
+T).((ty3 g c2 (TLRef n) t3) \to False))) (\lambda (x: T).(\lambda (H5: (ty3 g 
+x0 x2 x)).(B_ind (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to False))))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) 
+x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall 
+(t3: T).((ty3 g c2 (TLRef n) t3) \to False)) (ex_intro T (\lambda (t3: 
+T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x 
+H5)))) (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T 
+(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef 
+n) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) 
+(lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) (\lambda (H6: (getl n c2 
+(CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 
+(TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) 
+(\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 
+(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) False (\lambda (H8: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind 
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O 
+t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: 
+(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind 
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 
+(Bind Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abbr) x4) 
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) 
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) 
+I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 
+(CHead x3 (Bind Abbr) x4) H10)) in (False_ind False H13))))))))) H8)) 
+(\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: 
+T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False 
+(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift 
+(S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst) 
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind 
+Void) x2) (\lambda (c0: C).(getl n c2 c0)) H6 (CHead x3 (Bind Abst) x4) 
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) 
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | Abst 
+\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) 
+I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 
+(CHead x3 (Bind Abst) x4) H10)) in (False_ind False H13))))))))) H8)) 
+(ty3_gen_lref g c2 t3 n H7)))))) x1 H2))) H4)) (\lambda (H4: ((\forall (t3: 
+T).((ty3 g x0 x2 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g 
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to False)) 
+(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(or_ind (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 
+(lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
+n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) False (\lambda (H6: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind 
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O 
+t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) False (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: 
+(pc3 c2 (lift (S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) 
+x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind 
+x1) x2) (\lambda (c0: C).(getl n c2 c0)) H2 (CHead x3 (Bind Abbr) x4) 
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in 
+(let H11 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) (CHead x0 (Bind x1) x2) 
+(CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead 
+x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k with 
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) 
+x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 
+(CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) 
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 
+(Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda (_: (eq B x1 
+Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: 
+T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 \def 
+(eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def 
+(eq_ind_r C x3 (\lambda (c0: C).(getl n c2 (CHead c0 (Bind Abbr) x2))) H16 x0 
+H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(ty3 g c0 x2 x5)) H17 
+x0 H15) in (H4 x5 H19)))))))) H12)) H11))))))))) H6)) (\lambda (H6: (ex3_3 C 
+T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 
+c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t)))) False (\lambda (x3: C).(\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) t3)).(\lambda 
+(H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 g x3 x4 
+x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c0: C).(getl n 
+c2 c0)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n 
+H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow x0 | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 
+(Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal 
+C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow x1 | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow 
+x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 
+(CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow x2 | (CHead 
+_ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) 
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in 
+(\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def 
+(eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 
+H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 
+H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) 
+t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c0: C).(getl n c2 
+(CHead c0 (Bind Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 
+(\lambda (c0: C).(ty3 g c0 x2 x5)) H17 x0 H15) in (H4 x5 H20))))))))) H12)) 
+H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))) H3)))))) H1)) (\lambda 
+(H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))))).(or_intror 
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 
+(TLRef n) t3) \to False)) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) 
+t3)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) False 
+(\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) False 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift 
+(S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 
+False))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) 
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t)))) False (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: 
+(pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) 
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 
+False))))))) H3)) (ty3_gen_lref g c2 t3 n H2)))))) H0))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 
+t3 c2 t) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: 
+T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g 
+c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)))))).(\lambda (t0: 
+T).(\lambda (_: ((((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 t0) \to 
+(or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 
+t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) 
+(\forall (t3: T).((ty3 g c2 t0 t3) \to False)))))).(\lambda (H1: ((\forall 
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead k t t0)) \to (or (ex T 
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
+False)))))))).(K_ind (\lambda (k0: K).(((\forall (c1: C).(\forall (t3: 
+T).((flt c1 t3 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False))))))) \to (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead k0 t t0) t3) \to False))))) (\lambda (b: B).(\lambda (H2: ((\forall 
+(c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Bind b) t t0)) \to (or (ex T 
+(\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to 
+False)))))))).(let H3 \def (H2 c2 t (flt_thead_sx (Bind b) c2 t t0)) in 
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 
+t t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) 
+(\lambda (H4: (ex T (\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda 
+(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) 
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to 
+False))) (\lambda (x: T).(\lambda (H5: (ty3 g c2 t x)).(let H6 \def (H2 
+(CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T 
+(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 
+g (CHead c2 (Bind b) t) t0 t3) \to False)) (or (ex T (\lambda (t3: T).(ty3 g 
+c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t 
+t0) t3) \to False))) (\lambda (H7: (ex T (\lambda (t3: T).(ty3 g (CHead c2 
+(Bind b) t) t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) 
+t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to False))) (\lambda 
+(x0: T).(\lambda (H8: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(or_introl (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 
+g c2 (THead (Bind b) t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 
+g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H5 b 
+t0 x0 H8))))) H7)) (\lambda (H7: ((\forall (t3: T).((ty3 g (CHead c2 (Bind b) 
+t) t0 t3) \to False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) 
+\to False)) (\lambda (t3: T).(\lambda (H8: (ty3 g c2 (THead (Bind b) t t0) 
+t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) 
+t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c2 t t5))) (\lambda (t4: 
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4))) False (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda 
+(_: (ty3 g c2 t x1)).(\lambda (H11: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(H7 
+x0 H11)))))) (ty3_gen_bind g b c2 t t0 t3 H8)))))) H6)))) H4)) (\lambda (H4: 
+((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Bind b) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H5: (ty3 g 
+c2 (THead (Bind b) t t0) t3)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: 
+T).(pc3 c2 (THead (Bind b) t t4) t3))) (\lambda (_: T).(\lambda (t5: T).(ty3 
+g c2 t t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 
+t4))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead 
+(Bind b) t x0) t3)).(\lambda (H7: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead 
+c2 (Bind b) t) t0 x0)).(H4 x1 H7)))))) (ty3_gen_bind g b c2 t t0 t3 H5)))))) 
+H3)))) (\lambda (f: F).(\lambda (H2: ((\forall (c1: C).(\forall (t3: T).((flt 
+c1 t3 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(F_ind (\lambda 
+(f0: F).(((\forall (c1: C).(\forall (t3: T).((flt c1 t3 c2 (THead (Flat f0) t 
+t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 t4))) (\forall (t4: T).((ty3 
+g c1 t3 t4) \to False))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead 
+(Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) 
+\to False))))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt c1 t3 
+c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 t3 
+t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def (H3 
+c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: 
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H5: (ex T 
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) 
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 
+(flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g 
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: 
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x0: 
+T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) 
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: 
+T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+False))) (\lambda (x2: T).(\lambda (H11: (ty3 g c2 x x2)).(let H12 \def 
+(ty3_sn3 g c2 x x2 H11) in (let H_x \def (nf2_sn3 c2 x H12) in (let H13 \def 
+H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) 
+(or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall 
+(t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (x3: 
+T).(\lambda (H14: (pr3 c2 x x3)).(\lambda (H15: (nf2 c2 x3)).(let H16 \def 
+(ty3_sred_pr3 c2 x x3 H14 g x2 H11) in (let H_x0 \def (pc3_abst_dec g c2 x0 
+x1 H10 x3 x2 H16) in (let H17 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: 
+T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: 
+T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: 
+T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 
+v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to False)) (or 
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False))) (\lambda (H18: (ex4_2 
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) 
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) 
+(\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda 
+(v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 
+c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 
+(THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 
+v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: 
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Appl) t t0) t3) \to False))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H19: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H20: (ty3 
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H21: (pr3 c2 x3 x5)).(\lambda 
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H21 H15) in (let H23 
+\def (eq_ind_r T x5 (\lambda (t3: T).(pr3 c2 x3 t3)) H21 x3 H_y) in (let H24 
+\def (eq_ind_r T x5 (\lambda (t3: T).(ty3 g c2 (THead (Bind Abst) t3 x4) x1)) 
+H20 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) 
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) 
+(THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred 
+g c2 t x H6 x3 H14) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H24 t0 
+x0 H9 H19))))))))))))) H18)) (\lambda (H18: ((\forall (u: T).((pc3 c2 x0 
+(THead (Bind Abst) x3 u)) \to False)))).(or_intror (ex T (\lambda (t3: 
+T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: T).(\lambda (H19: (ty3 
+g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda 
+(t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda 
+(u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c2 t u))) False (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) 
+t3)).(\lambda (H21: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H22: 
+(ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H22 x H6) in (let H_y0 
+\def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H21 x0 H9) in (H18 x5 
+(pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) 
+H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 
+(pc3_pr3_r c2 x x3 H14)) (Bind Abst) x5)))))))))) (ty3_gen_appl g c2 t t0 t3 
+H19)))))) H17))))))) H13)))))) (ty3_correct g c2 t x H6)))) (ty3_correct g c2 
+t0 x0 H9)))) H8)) (\lambda (H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to 
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to 
+False)) (\lambda (t3: T).(\lambda (H9: (ty3 g c2 (THead (Flat Appl) t t0) 
+t3)).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat 
+Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 
+g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 
+t u))) False (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead 
+(Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H11: (ty3 g c2 t0 
+(THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H8 (THead (Bind 
+Abst) x0 x1) H11)))))) (ty3_gen_appl g c2 t t0 t3 H9)))))) H7)))) H5)) 
+(\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to False)))).(or_intror (ex 
+T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to False)) (\lambda (t3: 
+T).(\lambda (H6: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(ex3_2_ind T T 
+(\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind 
+Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind 
+Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) False 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t 
+(THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) 
+x0 x1))).(\lambda (H9: (ty3 g c2 t x0)).(H5 x0 H9)))))) (ty3_gen_appl g c2 t 
+t0 t3 H6)))))) H4))) (\lambda (H3: ((\forall (c1: C).(\forall (t3: T).((flt 
+c1 t3 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t4: T).(ty3 g c1 
+t3 t4))) (\forall (t4: T).((ty3 g c1 t3 t4) \to False)))))))).(let H4 \def 
+(H3 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: 
+T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to False)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (H5: (ex T 
+(\lambda (t3: T).(ty3 g c2 t t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
+(\lambda (x: T).(\lambda (H6: (ty3 g c2 t x)).(let H7 \def (H3 c2 t0 
+(flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g 
+c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to False)) (or (ex T (\lambda 
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 
+(THead (Flat Cast) t t0) t3) \to False))) (\lambda (H8: (ex T (\lambda (t3: 
+T).(ty3 g c2 t0 t3)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) (\lambda (x0: 
+T).(\lambda (H9: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) 
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
+(\lambda (x1: T).(\lambda (H10: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 
+x0 x1 H10 t x H6) in (let H11 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 
+t) \to False) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False))) 
+(\lambda (H12: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) 
+t t0) t3) \to False)) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat 
+Cast) t t0) t3)) (THead (Flat Cast) x t) (ty3_cast g c2 t0 t (ty3_conv g c2 t 
+x H6 t0 x0 H9 H12) x H6)))) (\lambda (H12: (((pc3 c2 x0 t) \to 
+False))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) 
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) 
+(\lambda (t3: T).(\lambda (H13: (ty3 g c2 (THead (Flat Cast) t t0) 
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) 
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False 
+(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x2 t) t3)).(\lambda 
+(H15: (ty3 g c2 t0 t)).(\lambda (H16: (ty3 g c2 t x2)).(let H_y \def 
+(ty3_unique g c2 t x2 H16 x H6) in (let H_y0 \def (ty3_unique g c2 t0 t H15 
+x0 H9) in (H12 (ex2_sym T (pr3 c2 t) (pr3 c2 x0) H_y0)))))))) (ty3_gen_cast g 
+c2 t0 t t3 H13)))))) H11))))) (ty3_correct g c2 t0 x0 H9)))) H8)) (\lambda 
+(H8: ((\forall (t3: T).((ty3 g c2 t0 t3) \to False)))).(or_intror (ex T 
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: 
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to False)) (\lambda (t3: 
+T).(\lambda (H9: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(ex3_ind T (\lambda 
+(t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) (\lambda (_: T).(ty3 g c2 t0 
+t)) (\lambda (t4: T).(ty3 g c2 t t4)) False (\lambda (x0: T).(\lambda (_: 
+(pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda (H11: (ty3 g c2 t0 
+t)).(\lambda (_: (ty3 g c2 t x0)).(H8 t H11))))) (ty3_gen_cast g c2 t0 t t3 
+H9)))))) H7)))) H5)) (\lambda (H5: ((\forall (t3: T).((ty3 g c2 t t3) \to 
+False)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t 
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to 
+False)) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Flat Cast) t t0) 
+t3)).(ex3_ind T (\lambda (t4: T).(pc3 c2 (THead (Flat Cast) t4 t) t3)) 
+(\lambda (_: T).(ty3 g c2 t0 t)) (\lambda (t4: T).(ty3 g c2 t t4)) False 
+(\lambda (x0: T).(\lambda (_: (pc3 c2 (THead (Flat Cast) x0 t) t3)).(\lambda 
+(_: (ty3 g c2 t0 t)).(\lambda (H9: (ty3 g c2 t x0)).(ex_ind T (\lambda (t4: 
+T).(ty3 g c2 x0 t4)) False (\lambda (x: T).(\lambda (_: (ty3 g c2 x0 x)).(H5 
+x0 H9))) (ty3_correct g c2 t x0 H9)))))) (ty3_gen_cast g c2 t0 t t3 H6)))))) 
+H4))) f H2))) k H1))))))) t2))) c t1))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/defs.ma
new file mode 100644 (file)
index 0000000..93ca133
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/defs.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/G/defs.ma".
+
+include "basic_1A/pc3/defs.ma".
+
+inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
+| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) 
+\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to 
+(ty3 g c u t2))))))))
+| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next 
+g m))))
+| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
+T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to 
+(ty3 g c (TLRef n) (lift (S n) O t))))))))
+| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: 
+T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to 
+(ty3 g c (TLRef n) (lift (S n) O u))))))))
+| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to 
+(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) 
+u) t1 t2) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))
+| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to 
+(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 
+g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u 
+t)))))))))
+| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) 
+\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) 
+(THead (Flat Cast) t0 t2))))))).
+
+inductive tys3 (g: G) (c: C): TList \to (T \to Prop) \def
+| tys3_nil: \forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (tys3 g c 
+TNil u)))
+| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
+TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/fsubst0.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fsubst0.ma
new file mode 100644 (file)
index 0000000..582e50f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fsubst0.ma
@@ -0,0 +1,975 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/props.ma".
+
+include "basic_1A/pc3/fsubst0.ma".
+
+include "basic_1A/getl/getl.ma".
+
+lemma ty3_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
+t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind 
+Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
+(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: 
+T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2: 
+T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall 
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 
+c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 
+t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u 
+t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: 
+C).(\forall (t4: T).((fsubst0 i u0 c u c2 t4) \to (\forall (e: C).((getl i c 
+(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (H4: (pc3 c 
+t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4: 
+T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) 
+\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u 
+t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
+u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6) 
+e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda 
+(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2 
+t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u 
+(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3 
+(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6: 
+(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c 
+c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) 
+u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8) 
+t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c 
+t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4 
+H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i: 
+nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0 
+i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0: 
+C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to 
+(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i 
+u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr) 
+u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m)))))))) 
+(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda 
+(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3: 
+T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_: 
+(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind 
+Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g 
+m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda 
+(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: 
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 
+t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
+t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda 
+(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c 
+(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead 
+e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3: 
+T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5: 
+(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S 
+n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda 
+(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: 
+T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(getl n0 c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C 
+(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind 
+Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) 
+H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H8)) in ((let H11 \def (f_equal C T (\lambda (e0: C).(match 
+e0 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
+n H0 (CHead e (Bind Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 
+\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 
+d H12) in (let H14 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d 
+(Bind Abbr) t4))) H13 u H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift 
+(S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop 
+Abbr c d u n H14)) u0 H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n 
+H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: 
+C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 
+(TLRef n) (lift (S n) O t0)) (\lambda (H6: (lt n i)).(let H7 \def 
+(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind 
+(getl n c3 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) 
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T 
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda 
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl 
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) 
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S 
+n) O t0)) (\lambda (H8: (getl n c3 (CHead d (Bind Abbr) u))).(ty3_abbr g n c3 
+d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n) 
+(lift (S n) O t0)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 
+(Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda 
+(H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda 
+(e0: C).(match e0 with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow 
+c0])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in 
+((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr x0)).(\lambda (H16: 
+(eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t3: T).(subst0 (minus i 
+(S n)) u0 t3 x3)) H11 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: 
+C).(getl n c3 (CHead c0 (Bind x0) x3))) H10 d H16) in (let H19 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl n c3 (CHead d (Bind b) x3))) H18 Abbr H15) in (let 
+H20 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead d (Bind 
+Abbr) x3) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) 
+c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) 
+(CHead d (Bind Abbr) x3) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S 
+n)) (S i) (le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in 
+(ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus 
+i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) 
+x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4 
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq 
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: 
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 
+(Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda 
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) 
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: 
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C 
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3 
+(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1 
+x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H13 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
+\Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def 
+(eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead x2 (Bind x0) t3))) H10 u 
+H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S 
+n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: 
+B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abbr H15) in (let H20 \def (eq_ind 
+nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abbr) u) (CHead e 
+(Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 
+(csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead 
+x2 (Bind Abbr) u) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) 
+(le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr 
+g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n)) 
+u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n 
+(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) 
+x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13)) 
+H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind 
+Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) 
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda 
+(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead 
+d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
+\Rightarrow t3])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) 
+in (let H19 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 
+c0 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 
+(CHead x2 (Bind b) x4))) H10 Abbr H16) in (let H21 \def (eq_ind nat (minus i 
+n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) 
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 
+(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr g n c3 x2 x4 H20 t0 (H2 
+(minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) 
+e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S 
+n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) x2 (CHead e 
+(Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) H14)) H13))))))))))) H8)) 
+H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u (csubst0_getl_ge i n H6 c 
+c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda 
+(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 
+c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift 
+(S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O 
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) 
+O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
+(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(csubst0 n0 u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind 
+Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) 
+(getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
+(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) u) 
+(CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H9)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match 
+e0 with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d 
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) 
+n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H14: (eq C d e)).(let H15 
+\def (eq_ind_r C e (\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 
+d H14) in (let H16 \def (eq_ind_r T u0 (\lambda (t4: T).(getl n c (CHead d 
+(Bind Abbr) t4))) H15 u H13) in (let H17 \def (eq_ind_r T u0 (\lambda (t4: 
+T).(csubst0 n t4 c c3)) H10 u H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 
+(lift (S n) O t4) (lift (S n) O t0))) (ty3_lift g d u t0 H1 c3 O (S n) 
+(getl_drop Abbr c3 d u n (csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d 
+(Bind Abbr) u) H16))) u0 H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i 
+n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (n: nat).(\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abst) u))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 d u c2 t2) \to (\forall (e: C).((getl i d (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (i: nat).(\lambda (u0: 
+T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) 
+c2 t2)).(fsubst0_ind i u0 c (TLRef n) (\lambda (c0: C).(\lambda (t3: 
+T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 
+(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) 
+t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) 
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S 
+n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O 
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n) 
+O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e 
+(Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) 
+(\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c 
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (let H10 \def 
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
+Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) 
+H8)) in (False_ind (ty3 g c (lift (S n) O u0) (lift (S n) O u)) H10)))) t3 
+H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda (c3: C).(\lambda (H4: 
+(csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind 
+Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (H6: 
+(lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind 
+Abst) u) H0) in (or4_ind (getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) 
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) 
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i (S n)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) 
+(ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind 
+Abst) u))).(ty3_abst g n c3 d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T 
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) 
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 
+(minus i (S n)) u0 u1 w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead 
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: 
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) 
+u0 u1 w))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda (x0: B).(\lambda 
+(x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind 
+Abst) u) (CHead x1 (Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind 
+x0) x3))).(\lambda (H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow d | 
+(CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x2) H9) in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 with 
+[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k with [(Bind 
+b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) 
+(CHead x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) 
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B 
+Abst x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda 
+(t3: T).(subst0 (minus i (S n)) u0 t3 x3)) H11 u H14) in (let H18 \def 
+(eq_ind_r C x1 (\lambda (c0: C).(getl n c3 (CHead c0 (Bind x0) x3))) H10 d 
+H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead d 
+(Bind b) x3))) H18 Abst H15) in (let H20 \def (eq_ind nat (minus i n) 
+(\lambda (n0: nat).(getl n0 (CHead d (Bind Abst) x3) (CHead e (Bind Abbr) 
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abst) x3) n H19 
+(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u) 
+(lift (S n) O t0) (ty3_lift g d u t0 H1 c3 O (S n) (getl_drop Abst c3 d x3 n 
+H19)) (TLRef n) (lift (S n) O x3) (ty3_abst g n c3 d x3 H19 t0 (H2 (minus i 
+(S n)) u0 d x3 (fsubst0_snd (minus i (S n)) u0 d u x3 H17) e (getl_gen_S 
+(Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20))) (pc3_lift c3 
+d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u (pc3_pr2_x d x3 u (pr2_delta d 
+e u0 (r (Bind Abst) (minus i (S n))) (getl_gen_S (Bind Abst) d (CHead e (Bind 
+Abbr) u0) x3 (minus i (S n)) H20) u u (pr0_refl u) x3 H17))))))))))) H13)) 
+H12))))))))) H8)) (\lambda (H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1)))))) 
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3 
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2))))) (ty3 g c3 (TLRef n) 
+(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x3))).(\lambda (H11: 
+(csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def (f_equal C C (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) 
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def 
+(f_equal C B (\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow Abst | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) 
+\Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
+((let H14 \def (f_equal C T (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow u | (CHead _ _ t3) \Rightarrow t3])) (CHead d (Bind Abst) u) 
+(CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abst x0)).(\lambda (H16: 
+(eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t3: T).(getl n c3 (CHead 
+x2 (Bind x0) t3))) H10 u H14) in (let H18 \def (eq_ind_r C x1 (\lambda (c0: 
+C).(csubst0 (minus i (S n)) u0 c0 x2)) H11 d H16) in (let H19 \def (eq_ind_r 
+B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u))) H17 Abst H15) in (let 
+H20 \def (eq_ind nat (minus i n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind 
+Abst) u) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) 
+c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) 
+(CHead x2 (Bind Abst) u) n H19 (le_S_n n i (le_S_n (S n) (S i) (le_S (S (S 
+n)) (S i) (le_n_S (S n) i H6))))) (S (minus i (S n))) (minus_x_Sy i n H6)) in 
+(ty3_abst g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus 
+i (S n)) u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S 
+n)) (le_n (minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S 
+(Bind Abst) x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) 
+H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: 
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C 
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2 
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_: 
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda 
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0) 
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11: 
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0 
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abst) u) 
+(CHead x1 (Bind x0) x3) H9) in ((let H14 \def (f_equal C B (\lambda (e0: 
+C).(match e0 with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead 
+d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H15 \def (f_equal C T 
+(\lambda (e0: C).(match e0 with [(CSort _) \Rightarrow u | (CHead _ _ t3) 
+\Rightarrow t3])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in 
+(\lambda (H16: (eq B Abst x0)).(\lambda (H17: (eq C d x1)).(let H18 \def 
+(eq_ind_r T x3 (\lambda (t3: T).(subst0 (minus i (S n)) u0 t3 x4)) H11 u H15) 
+in (let H19 \def (eq_ind_r C x1 (\lambda (c0: C).(csubst0 (minus i (S n)) u0 
+c0 x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 
+(CHead x2 (Bind b) x4))) H10 Abst H16) in (let H21 \def (eq_ind nat (minus i 
+n) (\lambda (n0: nat).(getl n0 (CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) 
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n 
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 
+(le_S_n n i (le_S_n (S n) (S i) (le_S (S (S n)) (S i) (le_n_S (S n) i H6))))) 
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u) 
+(lift (S n) O t0) (ty3_lift g x2 u t0 (H2 (minus i (S n)) u0 x2 u 
+(fsubst0_fst (minus i (S n)) u0 d u x2 H19) e (csubst0_getl_ge_back (minus i 
+(S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind 
+Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S 
+n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 n H20)) (TLRef n) (lift (S n) 
+O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4 
+(fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back 
+(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e 
+(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus 
+i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop Abst c3 x2 x4 n H20) x4 u 
+(pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) u0 x2 x4 (fsubst0_both 
+(minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n)) 
+(minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) 
+(getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n)) 
+H21)))))))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i 
+n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind 
+Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef 
+n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: 
+C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) 
+(eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S n) O u)) (\lambda (H7: (eq 
+nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) 
+O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) O u))) (let H9 \def (eq_ind_r 
+nat i (\lambda (n0: nat).(getl n0 c (CHead e (Bind Abbr) u0))) H6 n H7) in 
+(let H10 \def (eq_ind_r nat i (\lambda (n0: nat).(csubst0 n0 u0 c c3)) H5 n 
+H7) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl 
+n c c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n 
+H0 (CHead e (Bind Abbr) u0) H9)) in (let H12 \def (eq_ind C (CHead d (Bind 
+Abst) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with 
+[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | 
+(Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c 
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ty3 
+g c3 (lift (S n) O u0) (lift (S n) O u)) H12))))) t3 H8))) (subst0_gen_lref 
+u0 t3 i n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c: C).(\lambda (u: 
+T).(\lambda (t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda (H1: ((\forall (i: 
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 
+t2) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 
+t0)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: 
+(ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (i: nat).(\forall 
+(u0: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u0 (CHead c (Bind b) u) 
+t2 c2 t4) \to (\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u0: 
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u0 c (THead 
+(Bind b) u t2) c2 t4)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda 
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) 
+\to (ty3 g c0 t5 (THead (Bind b) u t3)))))) (\lambda (t5: T).(\lambda (H5: 
+(subst0 i u0 (THead (Bind b) u t2) t5)).(\lambda (e: C).(\lambda (H6: (getl i 
+c (CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead 
+(Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: 
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) 
+u0 t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c 
+t5 (THead (Bind b) u t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 
+(THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T 
+(\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i 
+u0 u u2)) (ty3 g c t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H8: 
+(eq T t5 (THead (Bind b) x t2))).(\lambda (H9: (subst0 i u0 u x)).(eq_ind_r T 
+(THead (Bind b) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) 
+(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c 
+(THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H10: 
+(ty3 g (CHead c (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead 
+c (Bind b) x) t3 t6)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) 
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g 
+c (THead (Bind b) u t3) (THead (Bind b) u x0) (ty3_bind g c u t0 H0 b t3 x0 
+H10) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 
+c x (fsubst0_snd i u0 c u x H9) e H6) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) 
+x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x) 
+(csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c (CHead e (Bind 
+Abbr) u0) H6 u))) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) 
+(pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3) (fsubst0_snd 
+i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H9 t3 
+(Bind b))) e H6)))) (ty3_correct g (CHead c (Bind b) x) t2 t3 (H3 (S i) u0 
+(CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead 
+c (Bind b) x) (csubst0_snd_bind b i u0 u x H9 c)) e (getl_head (Bind b) i c 
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 
+H2)) t5 H8)))) H7)) (\lambda (H7: (ex2 T (\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) (\lambda 
+(t6: T).(subst0 (s (Bind b) i) u0 t2 t6)) (ty3 g c t5 (THead (Bind b) u t3)) 
+(\lambda (x: T).(\lambda (H8: (eq T t5 (THead (Bind b) u x))).(\lambda (H9: 
+(subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind b) u x) (\lambda 
+(t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g 
+(CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) u x) (THead (Bind b) u 
+t3)) (\lambda (x0: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 
+x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind b) u) x 
+(fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H9) e (getl_head (Bind b) i c 
+(CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g (CHead c (Bind b) u) x t3 
+(H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) 
+u) t2 x H9) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u)))) t5 
+H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T 
+t5 (THead (Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u 
+u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c 
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq 
+T t5 (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i u0 u x0)).(\lambda 
+(H10: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) 
+(\lambda (t6: T).(ty3 g c t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: 
+T).(ty3 g (CHead c (Bind b) u) t3 t6)) (ty3 g c (THead (Bind b) x0 x1) (THead 
+(Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (ty3 g (CHead c (Bind b) u) t3 
+x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c (Bind b) x0) t3 t6)) (ty3 g c 
+(THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: 
+(ty3 g (CHead c (Bind b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) 
+(THead (Bind b) u x) (ty3_bind g c u t0 H0 b t3 x H11) (THead (Bind b) x0 x1) 
+(THead (Bind b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c 
+u x0 H9) e H6) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S 
+i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead c (Bind b) x0) (csubst0_snd_bind 
+b i u0 u x0 H9 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))) 
+(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead 
+(Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c (THead 
+(Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H9 t3 (Bind b))) 
+e H6)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c 
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H10 (CHead 
+c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H9 c)) e (getl_head (Bind b) i c 
+(CHead e (Bind Abbr) u0) H6 u)))))) (ty3_correct g (CHead c (Bind b) u) t2 t3 
+H2)) t5 H8)))))) H7)) (subst0_gen_head (Bind b) u0 u t2 t5 i H5)))))) 
+(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda 
+(H6: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t5: T).(ty3 g 
+(CHead c3 (Bind b) u) t3 t5)) (ty3 g c3 (THead (Bind b) u t2) (THead (Bind b) 
+u t3)) (\lambda (x: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 
+x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H5) e H6) b t2 
+t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind 
+b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H5 u)) e 
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H6 u))))) (ty3_correct g 
+(CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) 
+(csubst0_fst_bind b i c c3 u0 H5 u)) e (getl_head (Bind b) i c (CHead e (Bind 
+Abbr) u0) H6 u)))))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u0 (THead 
+(Bind b) u t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c 
+c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) 
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind b) u2 t2))) 
+(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Bind b) u2 t6)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))) (ty3 g c3 t5 (THead 
+(Bind b) u t3)) (\lambda (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Bind 
+b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: 
+T).(eq T t5 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)) 
+(ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 
+(THead (Bind b) x t2))).(\lambda (H10: (subst0 i u0 u x)).(eq_ind_r T (THead 
+(Bind b) x t2) (\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind 
+T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead 
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g 
+(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 
+(Bind b) u) x0 t6)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3)) 
+(\lambda (x1: T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind T 
+(\lambda (t6: T).(ty3 g (CHead c3 (Bind b) x) t3 t6)) (ty3 g c3 (THead (Bind 
+b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (_: (ty3 g (CHead 
+c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u 
+x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 
+x0 H11) (THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c3 x t0 (H1 i 
+u0 c3 x (fsubst0_both i u0 c u x H10 c3 H6) e H7) b t2 t3 (H3 (S i) u0 (CHead 
+c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 
+(Bind b) x) (csubst0_both_bind b i u0 u x H10 c c3 H6)) e (getl_head (Bind b) 
+i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c (THead (Bind b) u t3) 
+(THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c3 (THead (Bind 
+b) x t3) (fsubst0_both i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) 
+(subst0_fst u0 x u i H10 t3 (Bind b)) c3 H6) e H7)))) (ty3_correct g (CHead 
+c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) 
+u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u 
+x H10 c c3 H6)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) 
+(ty3_correct g (CHead c3 (Bind b) u) t3 x0 H11)))) (ty3_correct g (CHead c3 
+(Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 
+(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 
+H6 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) 
+H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead (Bind b) u t6))) 
+(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6)))).(ex2_ind T (\lambda (t6: 
+T).(eq T t5 (THead (Bind b) u t6))) (\lambda (t6: T).(subst0 (s (Bind b) i) 
+u0 t2 t6)) (ty3 g c3 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: 
+(eq T t5 (THead (Bind b) u x))).(\lambda (H10: (subst0 (s (Bind b) i) u0 t2 
+x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t6: T).(ty3 g c3 t6 (THead 
+(Bind b) u t3))) (ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) t3 
+t6)) (ty3 g c3 (THead (Bind b) u x) (THead (Bind b) u t3)) (\lambda (x0: 
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 
+i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b x t3 (H3 (S i) u0 (CHead c3 
+(Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H10 (CHead c3 
+(Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind b) i c 
+(CHead e (Bind Abbr) u0) H7 u))))) (ty3_correct g (CHead c3 (Bind b) u) x t3 
+(H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) 
+u) t2 x H10 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e 
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))) H8)) 
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Bind b) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Bind b) i) u0 t2 t6))) (ty3 g c3 
+t5 (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq 
+T t5 (THead (Bind b) x0 x1))).(\lambda (H10: (subst0 i u0 u x0)).(\lambda 
+(H11: (subst0 (s (Bind b) i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) 
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Bind b) u t3))) (ex_ind T (\lambda (t6: 
+T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) 
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c3 (Bind 
+b) u) t3 x)).(ex_ind T (\lambda (t6: T).(ty3 g (CHead c3 (Bind b) u) x t6)) 
+(ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: 
+T).(\lambda (_: (ty3 g (CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t6: 
+T).(ty3 g (CHead c3 (Bind b) x0) t3 t6)) (ty3 g c3 (THead (Bind b) x0 x1) 
+(THead (Bind b) u t3)) (\lambda (x3: T).(\lambda (_: (ty3 g (CHead c3 (Bind 
+b) x0) t3 x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) 
+(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H6) e H7) b t3 x 
+H12) (THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i 
+u0 c3 x0 (fsubst0_both i u0 c u x0 H10 c3 H6) e H7) b x1 t3 (H3 (S i) u0 
+(CHead c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 
+H11 (CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e 
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H7 u))) (pc3_fsubst0 c 
+(THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u 
+t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u 
+t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H10 t3 (Bind b)) c3 H6) e 
+H7)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3 
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H11 (CHead 
+c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H10 c c3 H6)) e (getl_head 
+(Bind b) i c (CHead e (Bind Abbr) u0) H7 u)))))) (ty3_correct g (CHead c3 
+(Bind b) u) t3 x H12)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) 
+u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 
+(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H6 u)) e (getl_head (Bind 
+b) i c (CHead e (Bind Abbr) u0) H7 u)))) t5 H9)))))) H8)) (subst0_gen_head 
+(Bind b) u0 u t2 t5 i H5)))))))) c2 t4 H4)))))))))))))))) (\lambda (c: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0: 
+T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: 
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: 
+T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i: 
+nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: 
+(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead 
+(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c 
+(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat 
+Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) 
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
+(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2: 
+T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) 
+(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) 
+x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v) 
+(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) 
+(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c 
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
+(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0) 
+x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c (THead (Bind 
+Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u t5))) (\lambda 
+(t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 t4))) (ty3 g c 
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind Abst) u 
+x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind 
+Abst) u) t0 x1)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead 
+(Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda 
+(x3: T).(\lambda (H14: (ty3 g c u x3)).(ty3_conv g c (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) 
+(ty3_appl g c w u H0 (THead (Bind Abst) u t0) x1 (ty3_bind g c u x3 H14 Abst 
+t0 x1 H13)) (THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u 
+t0)) (ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 
+H2) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead 
+(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u 
+t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+(THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead 
+(Bind Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x 
+(fsubst0_snd i u0 c w x H9) e H6)))))))) (ty3_gen_bind g Abst c u t0 x0 
+H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7)) 
+(\lambda (H7: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) 
+(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat 
+Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w 
+x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead 
+(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x 
+(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) 
+u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c t3 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i 
+u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind 
+Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead 
+(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c 
+(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c u 
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t0 
+t4))) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead 
+(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u x3)).(\lambda (H14: (ty3 g 
+(CHead c (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) 
+(ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) (\lambda (x4: T).(\lambda (H15: (ty3 g c u x4)).(ty3_conv g c (THead 
+(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind 
+Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind g c u 
+x4 H15 Abst t0 x2 H14)) (THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 
+(THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 (fsubst0_snd i u0 
+c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 (fsubst0_snd (s (Flat 
+Appl) i) u0 c v x1 H10) e H6)) (pc3_fsubst0 c (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c 
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) i u0 c (THead (Flat Appl) x0 
+(THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) 
+(subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl))) e H6)))) 
+(ty3_correct g c w u H0))))))) (ty3_gen_bind g Abst c u t0 x H11)))) 
+(ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7)) 
+(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda 
+(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e 
+(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 
+H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda 
+(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\lambda (c3: 
+C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H7: (getl i c 
+(CHead e (Bind Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead 
+(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda 
+(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat 
+Appl) i) u0 v t4))) (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w 
+u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) 
+(ty3 g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: 
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: 
+T).(subst0 i u0 w u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat 
+Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c3 t3 (THead (Flat 
+Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H9: (eq T t3 
+(THead (Flat Appl) x v))).(\lambda (H10: (subst0 i u0 w x)).(eq_ind_r T 
+(THead (Flat Appl) x v) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind 
+Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 (THead 
+(Bind Abst) u t0) x0)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 
+c3 (THead (Bind Abst) u t4) x0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 
+u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 
+t4))) (ty3 g c3 (THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c3 (THead 
+(Bind Abst) u x1) x0)).(\lambda (H13: (ty3 g c3 u x2)).(\lambda (H14: (ty3 g 
+(CHead c3 (Bind Abst) u) t0 x1)).(ty3_conv g c3 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g 
+c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) (THead (Bind Abst) u 
+t0) x1 (ty3_bind g c3 u x2 H13 Abst t0 x1 H14)) (THead (Flat Appl) x v) 
+(THead (Flat Appl) x (THead (Bind Abst) u t0)) (ty3_appl g c3 x u (H1 i u0 c3 
+x (fsubst0_both i u0 c w x H10 c3 H6) e H7) v t0 (H3 i u0 c3 v (fsubst0_fst i 
+u0 c v c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u 
+t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat 
+Appl) w (THead (Bind Abst) u t0))) i u0 c3 (THead (Flat Appl) x (THead (Bind 
+Abst) u t0)) (fsubst0_both i u0 c (THead (Flat Appl) w (THead (Bind Abst) u 
+t0)) (THead (Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H10 
+(THead (Bind Abst) u t0) (Flat Appl)) c3 H6) e H7))))))) (ty3_gen_bind g Abst 
+c3 u t0 x0 H11)))) (ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v 
+(fsubst0_fst i u0 c v c3 H6) e H7))) t3 H9)))) H8)) (\lambda (H8: (ex2 T 
+(\lambda (t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 
+(s (Flat Appl) i) u0 v t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead 
+(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)) (ty3 
+g c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: 
+T).(\lambda (H9: (eq T t3 (THead (Flat Appl) w x))).(\lambda (H10: (subst0 (s 
+(Flat Appl) i) u0 v x)).(eq_ind_r T (THead (Flat Appl) w x) (\lambda (t4: 
+T).(ty3 g c3 t4 (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g 
+c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e H7) x t0 (H3 i u0 c3 x 
+(fsubst0_both i u0 c v x H10 c3 H6) e H7)) t3 H9)))) H8)) (\lambda (H8: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_: 
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c3 t3 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H9: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 
+i u0 w x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T 
+(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind 
+Abst) u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H12: (ty3 g c3 (THead 
+(Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t4: T).(\lambda (_: T).(pc3 c3 
+(THead (Bind Abst) u t4) x))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c3 u 
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c3 (Bind Abst) u) t0 
+t4))) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c3 (THead 
+(Bind Abst) u x2) x)).(\lambda (_: (ty3 g c3 u x3)).(\lambda (H15: (ty3 g 
+(CHead c3 (Bind Abst) u) t0 x2)).(ex_ind T (\lambda (t4: T).(ty3 g c3 u t4)) 
+(ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) (\lambda (x4: T).(\lambda (H16: (ty3 g c3 u x4)).(ty3_conv g c3 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind 
+Abst) u x2)) (ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 H6) e 
+H7) (THead (Bind Abst) u t0) x2 (ty3_bind g c3 u x4 H16 Abst t0 x2 H15)) 
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) 
+(ty3_appl g c3 x0 u (H1 i u0 c3 x0 (fsubst0_both i u0 c w x0 H10 c3 H6) e H7) 
+x1 t0 (H3 i u0 c3 x1 (fsubst0_both i u0 c v x1 H11 c3 H6) e H7)) (pc3_fsubst0 
+c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) i u0 c3 (THead (Flat Appl) x0 (THead (Bind Abst) u t0)) (fsubst0_both i 
+u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 
+(THead (Bind Abst) u t0)) (subst0_fst u0 x0 w i H10 (THead (Bind Abst) u t0) 
+(Flat Appl)) c3 H6) e H7)))) (ty3_correct g c3 w u (H1 i u0 c3 w (fsubst0_fst 
+i u0 c w c3 H6) e H7)))))))) (ty3_gen_bind g Abst c3 u t0 x H12)))) 
+(ty3_correct g c3 v (THead (Bind Abst) u t0) (H3 i u0 c3 v (fsubst0_fst i u0 
+c v c3 H6) e H7))) t3 H9)))))) H8)) (subst0_gen_head (Flat Appl) u0 w v t3 i 
+H5)))))))) c2 t2 H4))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H0: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (i: 
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: T).((fsubst0 i u c t2 c2 
+t4) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2 t4 
+t3)))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c t3 t0)).(\lambda (H3: 
+((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4: 
+T).((fsubst0 i u c t3 c2 t4) \to (\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u)) \to (ty3 g c2 t4 t0)))))))))).(\lambda (i: nat).(\lambda (u: 
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H4: (fsubst0 i u c (THead 
+(Flat Cast) t3 t2) c2 t4)).(fsubst0_ind i u c (THead (Flat Cast) t3 t2) 
+(\lambda (c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind 
+Abbr) u)) \to (ty3 g c0 t5 (THead (Flat Cast) t0 t3)))))) (\lambda (t5: 
+T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda (e: 
+C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda 
+(u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 
+u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda 
+(t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6)))) (ty3 g c t5 (THead (Flat Cast) t0 
+t3)) (\lambda (H7: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 
+t2))) (\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq 
+T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g 
+c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq T t5 (THead 
+(Flat Cast) x t2))).(\lambda (H9: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat 
+Cast) x t2) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) (ex_ind 
+T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x t2) (THead 
+(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (H10: (ty3 g c t0 
+x0)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) 
+(ty3_cast g c t3 t0 H2 x0 H10) (THead (Flat Cast) x t2) (THead (Flat Cast) t0 
+x) (ty3_cast g c t2 x (ty3_conv g c x t0 (H3 i u c x (fsubst0_snd i u c t3 x 
+H9) e H6) t2 t3 H0 (pc3_s c t3 x (pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x 
+(fsubst0_snd i u c t3 x H9) e H6))) t0 (H3 i u c x (fsubst0_snd i u c t3 x 
+H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) 
+(pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead (Flat Cast) t0 x) 
+(fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 x) 
+(subst0_snd (Flat Cast) u x t3 i H9 t0)) e H6)))) (ty3_correct g c x t0 (H3 i 
+u c x (fsubst0_snd i u c t3 x H9) e H6))) t5 H8)))) H7)) (\lambda (H7: (ex2 T 
+(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 
+(THead (Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6)) (ty3 g c t5 (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H8: (eq 
+T t5 (THead (Flat Cast) t3 x))).(\lambda (H9: (subst0 (s (Flat Cast) i) u t2 
+x)).(eq_ind_r T (THead (Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c t6 (THead 
+(Flat Cast) t0 t3))) (ty3_cast g c x t3 (H1 (s (Flat Cast) i) u c x 
+(fsubst0_snd (s (Flat Cast) i) u c t2 x H9) e H6) t0 H2) t5 H8)))) H7)) 
+(\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) 
+(\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g 
+c t5 (THead (Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(H8: (eq T t5 (THead (Flat Cast) x0 x1))).(\lambda (H9: (subst0 i u t3 
+x0)).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead 
+(Flat Cast) x0 x1) (\lambda (t6: T).(ty3 g c t6 (THead (Flat Cast) t0 t3))) 
+(ex_ind T (\lambda (t6: T).(ty3 g c t0 t6)) (ty3 g c (THead (Flat Cast) x0 
+x1) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H11: (ty3 g c t0 
+x)).(ty3_conv g c (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
+(ty3_cast g c t3 t0 H2 x H11) (THead (Flat Cast) x0 x1) (THead (Flat Cast) t0 
+x0) (ty3_cast g c x1 x0 (ty3_conv g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c 
+t3 x0 H9) e H6) x1 t3 (H1 (s (Flat Cast) i) u c x1 (fsubst0_snd (s (Flat 
+Cast) i) u c t2 x1 H10) e H6) (pc3_s c t3 x0 (pc3_fsubst0 c t3 t3 (pc3_refl c 
+t3) i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t0 (H3 i u c x0 
+(fsubst0_snd i u c t3 x0 H9) e H6)) (pc3_fsubst0 c (THead (Flat Cast) t0 t3) 
+(THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) i u c (THead 
+(Flat Cast) t0 x0) (fsubst0_snd i u c (THead (Flat Cast) t0 t3) (THead (Flat 
+Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 i H9 t0)) e H6)))) (ty3_correct 
+g c x0 t0 (H3 i u c x0 (fsubst0_snd i u c t3 x0 H9) e H6))) t5 H8)))))) H7)) 
+(subst0_gen_head (Flat Cast) u t3 t2 t5 i H5)))))) (\lambda (c3: C).(\lambda 
+(H5: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e 
+(Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 
+H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H5) e H6)))))) (\lambda 
+(t5: T).(\lambda (H5: (subst0 i u (THead (Flat Cast) t3 t2) t5)).(\lambda 
+(c3: C).(\lambda (H6: (csubst0 i u c c3)).(\lambda (e: C).(\lambda (H7: (getl 
+i c (CHead e (Bind Abbr) u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 
+(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T 
+(\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) (\lambda (t6: 
+T).(subst0 (s (Flat Cast) i) u t2 t6))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6: T).(subst0 (s (Flat 
+Cast) i) u t2 t6)))) (ty3 g c3 t5 (THead (Flat Cast) t0 t3)) (\lambda (H8: 
+(ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) (\lambda (u2: 
+T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 (THead (Flat 
+Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 (THead (Flat 
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x 
+t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) 
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x t2) (THead (Flat 
+Cast) t0 t3)) (\lambda (x0: T).(\lambda (H11: (ty3 g c3 t0 x0)).(ty3_conv g 
+c3 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x0 t0) (ty3_cast g c3 t3 t0 
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x0 H11) (THead (Flat Cast) x 
+t2) (THead (Flat Cast) t0 x) (ty3_cast g c3 t2 x (ty3_conv g c3 x t0 (H3 i u 
+c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7) t2 t3 (H1 i u c3 t2 
+(fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x (pc3_fsubst0 c t3 t3 
+(pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7))) t0 (H3 i 
+u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) (pc3_fsubst0 c (THead (Flat 
+Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat Cast) t0 t3)) 
+i u c3 (THead (Flat Cast) t0 x) (fsubst0_both i u c (THead (Flat Cast) t0 t3) 
+(THead (Flat Cast) t0 x) (subst0_snd (Flat Cast) u x t3 i H10 t0) c3 H6) e 
+H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e 
+H7))) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t6: T).(eq T t5 (THead 
+(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 
+t6)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) 
+(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 (THead 
+(Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) 
+t3 x))).(\lambda (H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead 
+(Flat Cast) t3 x) (\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) 
+(ty3_cast g c3 x t3 (H1 i u c3 x (fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 
+(H3 i u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda 
+(H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) 
+u2 t6)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: 
+T).(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 (THead 
+(Flat Cast) t0 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t5 
+(THead (Flat Cast) x0 x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: 
+(subst0 (s (Flat Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) 
+(\lambda (t6: T).(ty3 g c3 t6 (THead (Flat Cast) t0 t3))) (ex_ind T (\lambda 
+(t6: T).(ty3 g c3 t0 t6)) (ty3 g c3 (THead (Flat Cast) x0 x1) (THead (Flat 
+Cast) t0 t3)) (\lambda (x: T).(\lambda (H12: (ty3 g c3 t0 x)).(ty3_conv g c3 
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c3 t3 t0 (H3 i 
+u c3 t3 (fsubst0_fst i u c t3 c3 H6) e H7) x H12) (THead (Flat Cast) x0 x1) 
+(THead (Flat Cast) t0 x0) (ty3_cast g c3 x1 x0 (ty3_conv g c3 x0 t0 (H3 i u 
+c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u c3 x1 
+(fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0 c t3 
+t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))) t0 
+(H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7)) (pc3_fsubst0 c 
+(THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 t3) (pc3_refl c (THead (Flat 
+Cast) t0 t3)) i u c3 (THead (Flat Cast) t0 x0) (fsubst0_both i u c (THead 
+(Flat Cast) t0 t3) (THead (Flat Cast) t0 x0) (subst0_snd (Flat Cast) u x0 t3 
+i H10 t0) c3 H6) e H7)))) (ty3_correct g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst 
+i u c t3 c3 H6) e H7))) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 
+t5 i H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
+
+lemma ty3_csubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 
+(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g 
+c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2: 
+C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1 
+(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
+
+lemma ty3_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 
+t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e 
+(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 
+t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: 
+(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1: 
+(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2 
+H1) e H0))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd.ma
new file mode 100644 (file)
index 0000000..560c0df
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd.ma
@@ -0,0 +1,923 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/defs.ma".
+
+include "basic_1A/pc3/props.ma".
+
+implied rec lemma ty3_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
+(\forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to ((P c t2 
+t) \to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((P c u t1) \to 
+((pc3 c t1 t2) \to (P c u t2)))))))))))) (f0: (\forall (c: C).(\forall (m: 
+nat).(P c (TSort m) (TSort (next g m)))))) (f1: (\forall (n: nat).(\forall 
+(c: C).(\forall (d: C).(\forall (u: T).((getl n c (CHead d (Bind Abbr) u)) 
+\to (\forall (t: T).((ty3 g d u t) \to ((P d u t) \to (P c (TLRef n) (lift (S 
+n) O t))))))))))) (f2: (\forall (n: nat).(\forall (c: C).(\forall (d: 
+C).(\forall (u: T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: 
+T).((ty3 g d u t) \to ((P d u t) \to (P c (TLRef n) (lift (S n) O 
+u))))))))))) (f3: (\forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u 
+t) \to ((P c u t) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 
+g (CHead c (Bind b) u) t1 t2) \to ((P (CHead c (Bind b) u) t1 t2) \to (P c 
+(THead (Bind b) u t1) (THead (Bind b) u t2))))))))))))) (f4: (\forall (c: 
+C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to ((P c w u) \to (\forall 
+(v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to ((P c v 
+(THead (Bind Abst) u t)) \to (P c (THead (Flat Appl) w v) (THead (Flat Appl) 
+w (THead (Bind Abst) u t))))))))))))) (f5: (\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).((ty3 g c t1 t2) \to ((P c t1 t2) \to (\forall (t0: 
+T).((ty3 g c t2 t0) \to ((P c t2 t0) \to (P c (THead (Flat Cast) t2 t1) 
+(THead (Flat Cast) t0 t2))))))))))) (c: C) (t: T) (t0: T) (t1: ty3 g c t t0) 
+on t1: P c t t0 \def match t1 with [(ty3_conv c0 t2 t3 t4 u t5 t6 p) 
+\Rightarrow (f c0 t2 t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t2 t3 t4) u 
+t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t5 t6) p) | (ty3_sort c0 m) 
+\Rightarrow (f0 c0 m) | (ty3_abbr n c0 d u g0 t2 t3) \Rightarrow (f1 n c0 d u 
+g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) d u t2 t3)) | (ty3_abst n c0 d u 
+g0 t2 t3) \Rightarrow (f2 n c0 d u g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 
+f5) d u t2 t3)) | (ty3_bind c0 u t2 t3 b t4 t5 t6) \Rightarrow (f3 c0 u t2 t3 
+((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t2 t3) b t4 t5 t6 ((ty3_ind g P f f0 
+f1 f2 f3 f4 f5) (CHead c0 (Bind b) u) t4 t5 t6)) | (ty3_appl c0 w u t2 v t3 
+t4) \Rightarrow (f4 c0 w u t2 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 w u t2) v 
+t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 v (THead (Bind Abst) u t3) t4)) | 
+(ty3_cast c0 t2 t3 t4 t5 t6) \Rightarrow (f5 c0 t2 t3 t4 ((ty3_ind g P f f0 
+f1 f2 f3 f4 f5) c0 t2 t3 t4) t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t3 
+t5 t6))].
+
+implied rec lemma tys3_ind (g: G) (c: C) (P: (TList \to (T \to Prop))) (f: 
+(\forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (P TNil u))))) (f0: 
+(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
+TList).((tys3 g c ts u) \to ((P ts u) \to (P (TCons t ts) u)))))))) (t: 
+TList) (t0: T) (t1: tys3 g c t t0) on t1: P t t0 \def match t1 with 
+[(tys3_nil u u0 t2) \Rightarrow (f u u0 t2) | (tys3_cons t2 u t3 ts t4) 
+\Rightarrow (f0 t2 u t3 ts t4 ((tys3_ind g c P f f0) ts u t4))].
+
+lemma ty3_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
+(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t 
+x)) (\lambda (_: T).(pc3 c (TSort (next g n)) x)) (\lambda (y: T).(\lambda 
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: 
+T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u 
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 
+t2)).(\lambda (H6: (eq T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: 
+T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 
+\def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in 
+(pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 
+H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
+(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e 
+with [(TSort n0) \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) 
+\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0: 
+nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort 
+(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
+(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
+(lift (S n0) O t)) H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
+(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
+(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to 
+(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq 
+T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n)) 
+t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Bind 
+b) u t2)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 
+(TSort (next g n)) u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 
+(TSort (next g n)) (THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead 
+(Flat Appl) w v) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w (THead (Bind Abst) 
+u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 
+c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
+t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) 
+t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) 
+(THead (Flat Cast) t0 t2)) H6))))))))))) c y x H0))) H))))).
+
+lemma ty3_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda 
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
+(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t 
+x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c (lift (S n) O t0) x)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u t0))))))) 
+(\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(pc3 c0 (lift (S n) O t1) 
+t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e 
+u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 
+c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t1: T).(ty3 g e u t1)))))))))) (\lambda (c0: C).(\lambda (t2: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (t0: T).(ty3 g e u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e u 
+t0))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u 
+t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: 
+(eq T u (TLRef n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) 
+H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift 
+(S n) O t3) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0 t3)))))))) H4 (TLRef n) H7) 
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef n) 
+H7) in (let H10 \def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (H12: (pc3 c0 (lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x2) H12 t2 H5) H13 
+H14)))))))) H11)) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0 
+(lift (S n) O x1) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abst) 
+x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 
+(lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) 
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef 
+n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (TSort (next g 
+m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 
+(lift (S n) O u) (TSort (next g m)))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) H2))))) (\lambda (n0: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H1: (getl n0 
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d u 
+t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
+(Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O t))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))) (or_introl (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O t))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
+u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O 
+t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0 H5)))))))))))) 
+(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 
+g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
+(Bind Abst) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O u))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))) (or_intror (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O u))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
+u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O 
+u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O u)) H6 H2)) n0 H5)))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
+t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O u0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
+(Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0)))))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
+n) O t) u)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t: 
+T).(ty3 g e u0 t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t: T).(ty3 g e u0 t))))))))).(\lambda (v: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u 
+t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind 
+Abst) u t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
+t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 
+\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
+(Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
+(THead (Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) H6)))))))))))) 
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T 
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) 
+t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq 
+T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c0 (lift (S n) O t) t0)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TLRef n))).(let H6 
+\def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (THead (Flat 
+Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) 
+H6))))))))))) c y x H0))) H))))).
+
+lemma ty3_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex3_2 T T (\lambda 
+(t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: 
+T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq 
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 
+T T (\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) 
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c u t0))) (\lambda (t2: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))) (\lambda (y: T).(\lambda (H0: 
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq T t (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda 
+(_: T).(pc3 c0 (THead (Bind b) u t2) t0))) (\lambda (_: T).(\lambda (t3: 
+T).(ty3 g c0 u t3))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
+b) u) t1 t2)))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
+u t3) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t3: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (u0: 
+T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0 t0)).(\lambda (H4: (((eq T u0 
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 
+c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u 
+t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t3))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T u0 (THead (Bind 
+b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0 (THead (Bind b) u 
+t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 (THead (Bind 
+b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead 
+(Bind b) u t4) t0))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) 
+(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))))) H4 
+(THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0 (\lambda (t3: T).(ty3 
+g c0 t3 t0)) H3 (THead (Bind b) u t1) H7) in (let H10 \def (H8 (refl_equal T 
+(THead (Bind b) u t1))) in (ex3_2_ind T T (\lambda (t3: T).(\lambda (_: 
+T).(pc3 c0 (THead (Bind b) u t3) t0))) (\lambda (_: T).(\lambda (t4: T).(ty3 
+g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t3))) (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
+t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Bind b) u x0) 
+t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13: (ty3 g (CHead c0 (Bind b) 
+u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead 
+(Bind b) u t3) t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) 
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) x0 x1 
+(pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) 
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
+(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (TSort (next g m))))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u 
+t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
+t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
+T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
+T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
+t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
+t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
+T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
+T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
+t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: 
+(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T 
+T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) 
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda 
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (b0: B).(\lambda 
+(t0: T).(\lambda (t2: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) 
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) 
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind 
+b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind 
+b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match 
+k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind 
+b0) u0 t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
+(THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u 
+t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow 
+t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq 
+T u0 u)).(\lambda (H10: (eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: 
+T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda 
+(_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: 
+T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: 
+T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 
+t4)))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g 
+(CHead c0 (Bind b0) u0) t3 t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 
+(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead (Bind b) u t3) 
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) 
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind 
+b) u) t1 t3)))))) H11 b H10) in (let H14 \def (eq_ind B b0 (\lambda (b1: 
+B).(ty3 g (CHead c0 (Bind b1) u0) t1 t2)) H12 b H10) in (eq_ind_r B b 
+(\lambda (b1: B).(ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead 
+(Bind b) u t3) (THead (Bind b1) u0 t2)))) (\lambda (_: T).(\lambda (t4: 
+T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
+b) u) t1 t3))))) (let H15 \def (eq_ind T u0 (\lambda (t3: T).((eq T t1 (THead 
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead 
+c0 (Bind b) t3) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: 
+T).(ty3 g (CHead c0 (Bind b) t3) u t5))) (\lambda (t4: T).(\lambda (_: 
+T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind b) u) t1 t4)))))) H13 u H9) in 
+(let H16 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b) t3) t1 
+t2)) H14 u H9) in (let H17 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 
+c0 (THead (Bind b) u t4) t))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u 
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g c0 t3 
+t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: T).(ex3_2 T T (\lambda (t4: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind b) t3 t2)))) 
+(\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda 
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) (ex3_2_intro T T (\lambda (t3: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b) u t2)))) 
+(\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda 
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) t2 t (pc3_refl c0 (THead (Bind 
+b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) H7)) H6))))))))))))) (\lambda 
+(c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
+u0)).(\lambda (_: (((eq T w (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
+(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) u0))) (\lambda (_: 
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (_: (((eq T v (THead 
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 
+(THead (Bind b) u t2) (THead (Bind Abst) u0 t)))) (\lambda (_: T).(\lambda 
+(t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 
+(Bind b) u) t1 t2))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead 
+(Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T 
+(\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t))))) (\lambda (_: T).(\lambda (t0: T).(ty3 g 
+c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t2)))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
+u t3) t2))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
+u t4) t3))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: 
+(eq T (THead (Flat Cast) t2 t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind 
+T (THead (Flat Cast) t2 t0) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t4: T).(\lambda 
+(_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 t2)))) (\lambda 
+(_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 
+g (CHead c0 (Bind b) u) t1 t4)))) H6))))))))))) c y x H0))) H))))))).
+
+lemma ty3_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead 
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3_2 T T 
+(\lambda (u: T).(\lambda (t0: T).(pc3 c (THead (Flat Appl) w (THead (Bind 
+Abst) u t0)) x))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c v (THead (Bind 
+Abst) u t0)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))) (\lambda (y: 
+T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
+(u: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t1)) t0))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u 
+t1)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+t))) (\lambda (u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda 
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat 
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u 
+(THead (Flat Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u 
+(THead (Flat Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq 
+T t0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t3: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3)) t1))) (\lambda 
+(u0: T).(\lambda (t3: T).(ty3 g c0 v (THead (Bind Abst) u0 t3)))) (\lambda 
+(u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 (THead (Flat Appl) w v) H7) 
+in (let H9 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (THead 
+(Flat Appl) w v) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat Appl) w 
+v))) in (ex3_2_ind T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0))) (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H11: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
+t1)).(\lambda (H12: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (H13: 
+(ty3 g c0 w x0)).(ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w 
+(THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) 
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat 
+Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) w v) H1) in (False_ind (ex3_2 T T 
+(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (TSort (next g m))))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v 
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) 
+H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
+T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
+(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(lift (S n) O t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
+T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
+(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(lift (S n) O u)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
+H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T 
+T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind 
+Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w 
+u0))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 
+g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w 
+v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) 
+u) (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) v (THead (Bind Abst) u0 
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w 
+u0))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (THead (Flat Appl) w 
+v))).(let H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T 
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t0)) (THead (Bind b) u t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 
+g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g 
+c0 w u0)))) H6))))))))))))) (\lambda (c0: C).(\lambda (w0: T).(\lambda (u: 
+T).(\lambda (H1: (ty3 g c0 w0 u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) 
+w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 
+g c0 v (THead (Bind Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 
+w u0))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 
+(THead (Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) 
+\to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w 
+(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow w0 | (TLRef _) \Rightarrow w0 | 
+(THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) 
+w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow 
+t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: 
+(eq T w0 w)).(let H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead 
+(Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) 
+(\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) 
+(\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 
+\def (eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 
+v H7) in (let H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: 
+T).(ty3 g c0 t0 u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T 
+(\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: 
+T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda 
+(_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T 
+t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: 
+T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g 
+c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T 
+(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind 
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda 
+(H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def 
+(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
+(Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (THead (Flat Cast) 
+t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u 
+t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) c y x 
+H0))) H)))))).
+
+lemma ty3_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall 
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (ex3 T (\lambda (t0: 
+T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
+(\lambda (t0: T).(ty3 g c t2 t0))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T 
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(ex3 
+T (\lambda (t0: T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 
+g c t1 t2)) (\lambda (t0: T).(ty3 g c t2 t0)))) (\lambda (y: T).(\lambda (H0: 
+(ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 
+(THead (Flat Cast) t3 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda 
+(t3: T).(ty3 g c0 t2 t3))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda (_: (((eq T t0 (THead (Flat Cast) 
+t2 t1)) \to (ex3 T (\lambda (t3: T).(pc3 c0 (THead (Flat Cast) t3 t2) t)) 
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t3: T).(ty3 g c0 t2 
+t3)))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u 
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
+(t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)))))).(\lambda (H5: (pc3 c0 t3 
+t0)).(\lambda (H6: (eq T u (THead (Flat Cast) t2 t1))).(let H7 \def (f_equal 
+T T (\lambda (e: T).e) u (THead (Flat Cast) t2 t1) H6) in (let H8 \def 
+(eq_ind T u (\lambda (t4: T).((eq T t4 (THead (Flat Cast) t2 t1)) \to (ex3 T 
+(\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t3)) (\lambda (_: T).(ty3 
+g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 (THead (Flat Cast) t2 
+t1) H7) in (let H9 \def (eq_ind T u (\lambda (t4: T).(ty3 g c0 t4 t3)) H3 
+(THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8 (refl_equal T (THead (Flat 
+Cast) t2 t1))) in (ex3_ind T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 
+t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4)) 
+(ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 t4))) (\lambda (x0: 
+T).(\lambda (H11: (pc3 c0 (THead (Flat Cast) x0 t2) t3)).(\lambda (H12: (ty3 
+g c0 t1 t2)).(\lambda (H13: (ty3 g c0 t2 x0)).(ex3_intro T (\lambda (t4: 
+T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2)) 
+(\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) 
+H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: 
+nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def 
+(eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 
+(THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n: nat).(\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d 
+(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: 
+(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d 
+(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0: 
+T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2 
+t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T 
+(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t))) 
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 
+T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 
+g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef 
+n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in 
+(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S 
+n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to 
+(ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t)) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (b: 
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
+u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T 
+(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Cast) t4 t2) t3)) 
+(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t4: T).(ty3 g 
+(CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T (THead (Bind b) u t0) 
+(THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Bind b) u t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t2 
+t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 
+t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: 
+T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: 
+T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (THead 
+(Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 
+t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead 
+(Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 
+T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u 
+t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 
+t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
+False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda 
+(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind 
+Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) 
+t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 
+t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: (((eq 
+T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead 
+(Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: 
+T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) (THead 
+(Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) 
+\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
+(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq 
+T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
+t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
+H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 
+t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
+t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
+H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 
+t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 
+(THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g 
+c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) 
+(\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T 
+t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 
+(THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) 
+H)))))).
+
+lemma tys3_gen_nil:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
+(\lambda (u0: T).(ty3 g c u u0))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil 
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_: 
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda 
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq 
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0: 
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList 
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda 
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts: 
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to 
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t 
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: 
+TList).(match ee with [TNil \Rightarrow False | (TCons _ _) \Rightarrow 
+True])) I TNil H4) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u0 u1))) 
+H5))))))))) y u H0))) H)))).
+
+lemma tys3_gen_cons:
+ \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
+(u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts 
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda 
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts) 
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u) 
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind 
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to 
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: 
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t 
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee with 
+[TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons t ts) H2) 
+in (False_ind (land (ty3 g c t u0) (tys3 g c ts u0)) H3)))))) (\lambda (t0: 
+T).(\lambda (u0: T).(\lambda (H1: (ty3 g c t0 u0)).(\lambda (ts0: 
+TList).(\lambda (H2: (tys3 g c ts0 u0)).(\lambda (H3: (((eq TList ts0 (TCons 
+t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0))))).(\lambda (H4: (eq TList 
+(TCons t0 ts0) (TCons t ts))).(let H5 \def (f_equal TList T (\lambda (e: 
+TList).(match e with [TNil \Rightarrow t0 | (TCons t1 _) \Rightarrow t1])) 
+(TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal TList TList 
+(\lambda (e: TList).(match e with [TNil \Rightarrow ts0 | (TCons _ t1) 
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 
+t)).(let H8 \def (eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons 
+t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def 
+(eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let 
+H10 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj 
+(ty3 g c t u0) (tys3 g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd_nf2.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd_nf2.ma
new file mode 100644 (file)
index 0000000..7d5c0be
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/fwd_nf2.ma
@@ -0,0 +1,284 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/arity_props.ma".
+
+include "basic_1A/pc3/nf2.ma".
+
+include "basic_1A/nf2/fwd.ma".
+
+lemma ty3_gen_appl_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: 
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x: 
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda 
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
+x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g 
+c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in 
+(let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0 
+x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) 
+w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v 
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) 
+(\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda 
+(x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def 
+(ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t: 
+T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: 
+T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 
+g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) 
+(\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c 
+(THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind 
+Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c 
+(THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda 
+(u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) 
+x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c 
+(THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def 
+(pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u: 
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) 
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) 
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: 
+T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c 
+x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind 
+b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10 
+(THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13 
+(Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda 
+(t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: 
+T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c 
+(THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead 
+(Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6)) 
+(pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w 
+Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead 
+(Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind 
+Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 
+x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2 
+(pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3)))))))) 
+(ty3_gen_appl g c w v x H))))))).
+
+lemma ty3_inv_lref_nf2_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c 
+(TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to 
+((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda 
+(H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t 
+u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c 
+u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda 
+(y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: 
+T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2: 
+T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift 
+(S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2 
+c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T 
+(\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda 
+(t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to 
+((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T 
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0 
+t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda 
+(u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10 
+\def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11 
+\def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to 
+(\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0: 
+T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def 
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y 
+\def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2 
+H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq 
+T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2: 
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m)) 
+u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef i) H1) in (False_ind (ex T (\lambda (u: T).(eq T u2 (lift 
+(S i) O u)))) H5))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(TLRef i)) \to ((nf2 d u) \to (\forall (u2: T).((nf2 d u2) \to ((pc3 d t u2) 
+\to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H4: 
+(eq T (TLRef n) (TLRef i))).(\lambda (H5: (nf2 c0 (TLRef n))).(\lambda (u2: 
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7: (pc3 c0 (lift (S n) O t) u2)).(let 
+H8 \def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n 
+| (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
+i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 
+(TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl 
+n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10 
+(ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))))))))))))))))))))) 
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
+(H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
+d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2: 
+T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S 
+i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5: 
+(nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7: 
+(pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow n | (TLRef n0) \Rightarrow n0 | 
+(THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef i) H4) in (let H9 \def 
+(eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) O u) u2)) H7 i H8) in 
+(let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0 (TLRef n0))) H5 i H8) 
+in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind 
+Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0 (lift (S i) O u) u2 
+H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y d (getl_drop Abst 
+c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2 (lift (S i) O t2))) 
+(\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i) O x))).(\lambda 
+(_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0: T).(ex T (\lambda 
+(u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda (u0: T).(eq T 
+(lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S i) O x))) u2 
+H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 
+c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T (\lambda 
+(u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b: B).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u) t1) 
+\to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0 
+(Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda 
+(_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 
+u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T 
+(THead (Bind b) u t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef i) H5) in (False_ind (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0)))) H9))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef i)) \to ((nf2 
+c0 w) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 u u2) \to (ex T (\lambda 
+(u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (v: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v 
+(TLRef i)) \to ((nf2 c0 v) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 
+(THead (Bind Abst) u t) u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O 
+u0))))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef 
+i))).(\lambda (_: (nf2 c0 (THead (Flat Appl) w v))).(\lambda (u2: T).(\lambda 
+(_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) 
+u t)) u2)).(let H9 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T 
+(\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda 
+(c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: 
+T).((nf2 c0 u2) \to ((pc3 c0 t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift 
+(S i) O u))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda 
+(_: (((eq T t2 (TLRef i)) \to ((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) 
+\to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O 
+u))))))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TLRef 
+i))).(\lambda (_: (nf2 c0 (THead (Flat Cast) t2 t1))).(\lambda (u2: 
+T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead (Flat Cast) t0 t2) 
+u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match 
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ 
+_ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: 
+T).(eq T u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
+
+lemma ty3_inv_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c 
+(TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0: 
+T).(eq T u (lift (S i) O u0))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
+(H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: 
+(nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
+
+lemma ty3_inv_appls_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1: 
+T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to 
+((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S 
+i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u)) 
+u1))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: 
+TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t 
+(TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: 
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t 
+(lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H: 
+(ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c 
+u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in 
+(ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u: 
+T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1))) 
+(\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def 
+(eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r 
+T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i) 
+O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda 
+(u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) 
+(lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2)))))))) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall 
+(i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef 
+i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
+(\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u)) 
+u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead 
+(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c 
+(TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t 
+(THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T 
+T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind 
+Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_: 
+T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst) 
+u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: 
+T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) 
+u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat 
+Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat 
+Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t 
+x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def 
+(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst) 
+x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 
+c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1))) 
+(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0) 
+x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def 
+(H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c 
+(lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) 
+O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O 
+u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift 
+(S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O 
+x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead 
+(Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) 
+(\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S 
+i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c 
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c 
+(THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t 
+Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
+
+lemma ty3_inv_lref_lref_nf2:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c 
+(TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i 
+j)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda 
+(H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda 
+(H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0 
+H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift 
+(S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S 
+i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0 
+in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x 
+(TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x 
+(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt 
+j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda 
+(H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S 
+i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i) 
+j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4))))) 
+H2))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/nf2.ma
new file mode 100644 (file)
index 0000000..1dc0e1a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/nf2.ma
@@ -0,0 +1,455 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/arity.ma".
+
+include "basic_1A/pc3/nf2.ma".
+
+include "basic_1A/nf2/arity.ma".
+
+definition ty3_nf2_inv_abst_premise:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi: 
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs: 
+TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w 
+u)) \to False)))))))).
+
+lemma ty3_nf2_inv_abst_premise_csort:
+ \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise 
+(CSort m) w u)))
+\def
+ \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda 
+(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst) 
+wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs 
+(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d 
+(Bind Abst) wi) H False))))))))).
+
+lemma ty3_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(\lambda (H0: (nf2 c t)).(let H_x \def (ty3_arity g c t u H) 
+in (let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda 
+(a1: A).(arity g c u (asucc g a1))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda 
+(u0: T).(eq T t (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w) 
+u0)))) (ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x: A).(\lambda (H2: 
+(arity g c t x)).(\lambda (_: (arity g c u (asucc g x))).(arity_nf2_inv_all g 
+c t x H2 H0)))) H1)))))))).
+
+lemma ty3_nf2_inv_sort:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (m: nat).((ty3 g c t 
+(TSort m)) \to ((nf2 c t) \to (or (ex2 nat (\lambda (n: nat).(eq T t (TSort 
+n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (m: nat).(\lambda 
+(H: (ty3 g c t (TSort m))).(\lambda (H0: (nf2 c t)).(let H_x \def 
+(ty3_nf2_inv_all g c t (TSort m) H H0) in (let H1 \def H_x in (or3_ind (ex3_2 
+T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind Abst) w u)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: T).(\lambda (u: 
+T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t 
+(TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t 
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) 
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat 
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
+i)))))) (\lambda (H2: (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) 
+u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind 
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u))) (or (ex2 nat (\lambda 
+(n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T t (THead (Bind Abst) x0 
+x1))).(\lambda (_: (nf2 c x0)).(\lambda (_: (nf2 (CHead c (Bind Abst) x0) 
+x1)).(let H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H 
+(THead (Bind Abst) x0 x1) H3) in (eq_ind_r T (THead (Bind Abst) x0 x1) 
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda 
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (ex3_2_ind T T (\lambda (t2: 
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (TSort m)))) (\lambda (_: 
+T).(\lambda (t0: T).(ty3 g c x0 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) x0) x1 t2))) (or (ex2 nat (\lambda (n: nat).(eq T (THead 
+(Bind Abst) x0 x1) (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) 
+(ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind 
+Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c (TLRef i)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: 
+(pc3 c (THead (Bind Abst) x0 x2) (TSort m))).(\lambda (_: (ty3 g c x0 
+x3)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x1 x2)).(pc3_gen_sort_abst 
+c x0 x2 m (pc3_s c (TSort m) (THead (Bind Abst) x0 x2) H7) (or (ex2 nat 
+(\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))) (\lambda (n: 
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
+(i: nat).(eq T (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))) (ty3_gen_bind g Abst c 
+x0 x1 (TSort m) H6)) t H3))))))) H2)) (\lambda (H2: (ex nat (\lambda (n: 
+nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) 
+(or (ex2 nat (\lambda (n: nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat 
+m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
+i)))))) (\lambda (x: nat).(\lambda (H3: (eq T t (TSort x))).(let H4 \def 
+(eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (TSort x) H3) in 
+(eq_ind_r T (TSort x) (\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T 
+t0 (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (eq_ind nat (next g x) 
+(\lambda (n: nat).(or (ex2 nat (\lambda (n0: nat).(eq T (TSort x) (TSort 
+n0))) (\lambda (n0: nat).(eq nat n (next g n0)))) (ex3_2 TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T (TSort x) (THeads (Flat Appl) ws (TLRef 
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_introl (ex2 nat (\lambda 
+(n: nat).(eq T (TSort x) (TSort n))) (\lambda (n: nat).(eq nat (next g x) 
+(next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T 
+(TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda 
+(_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef 
+i))))) (ex_intro2 nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) (\lambda 
+(n: nat).(eq nat (next g x) (next g n))) x (refl_equal T (TSort x)) 
+(refl_equal nat (next g x)))) m (pc3_gen_sort c (next g x) m (ty3_gen_sort g 
+c (TSort m) x H4))) t H3)))) H2)) (\lambda (H2: (ex3_2 TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda 
+(ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (or (ex2 nat (\lambda (n: 
+nat).(eq T t (TSort n))) (\lambda (n: nat).(eq nat m (next g n)))) (ex3_2 
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) 
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) 
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))) (\lambda (x0: 
+TList).(\lambda (x1: nat).(\lambda (H3: (eq T t (THeads (Flat Appl) x0 (TLRef 
+x1)))).(\lambda (H4: (nfs2 c x0)).(\lambda (H5: (nf2 c (TLRef x1))).(let H6 
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (TSort m))) H (THeads (Flat 
+Appl) x0 (TLRef x1)) H3) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) 
+(\lambda (t0: T).(or (ex2 nat (\lambda (n: nat).(eq T t0 (TSort n))) (\lambda 
+(n: nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))))) (or_intror (ex2 nat (\lambda 
+(n: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (TSort n))) (\lambda (n: 
+nat).(eq nat m (next g n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda 
+(i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws 
+(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda 
+(_: TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex3_2_intro TList nat 
+(\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef 
+x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) 
+x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3))))))) 
+H2)) H1)))))))).
+
+fact ty3_nf2_gen__ty3_nf2_inv_abst_aux:
+ \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise 
+c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead 
+(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to 
+(ty3_nf2_inv_abst_premise c w2 u2))))))))
+\def
+ \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d: 
+C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) 
+\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) 
+(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2: 
+T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind 
+Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi: 
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst) 
+wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift 
+(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t 
+(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t 
+(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl) 
+vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst) 
+w1 u1) H0))))))))))))))).
+
+lemma ty3_nf2_inv_abst:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u: 
+T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to 
+((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: 
+T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g 
+c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v 
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) 
+v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c 
+t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w 
+u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in 
+(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq 
+T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c 
+w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0)))) 
+(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws: 
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: 
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v: 
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0: 
+T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 
+c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) 
+u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind 
+Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0: 
+T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda 
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t 
+(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2 
+(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 
+g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r 
+T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst) 
+w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) 
+x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w 
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
+(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex3_2_ind T T 
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w t2) x))) 
+(\lambda (_: T).(\lambda (t0: T).(ty3 g c w t0))) (\lambda (t2: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) w) u t2))) (ex4_2 T T (\lambda (v: 
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w 
+x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T 
+(\lambda (t2: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead 
+(Bind Abst) w u)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c x0 t0))) 
+(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x0) x1 t2))) 
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) 
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) 
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4) 
+(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g 
+(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b: 
+B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda 
+(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w 
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: 
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (H16: (pc3 c 
+x0 w)).(\lambda (H17: ((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind 
+b) u0) x4 u))))).(let H_y \def (pc3_nf2 c x0 w H16 H6 H1) in (let H18 \def 
+(eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0) x1 x4)) H15 w 
+H_y) in (let H19 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead c (Bind Abst) 
+t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
+T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind Abst) w v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T (\lambda (v: 
+T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind Abst) w v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T (THead (Bind Abst) w 
+x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1 x4 H18 (H17 Abst w)) 
+H19) x0 H_y)))))) (pc3_gen_abst c x0 w x4 u H13))))))) (ty3_gen_bind g Abst c 
+x0 x1 (THead (Bind Abst) w u) H8))))))) (ty3_gen_bind g Abst c w u x H9)))) 
+(ty3_correct g c (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u) H8)) t 
+H5))))))) H4)) (\lambda (H4: (ex nat (\lambda (n: nat).(eq T t (TSort 
+n))))).(ex_ind nat (\lambda (n: nat).(eq T t (TSort n))) (ex4_2 T T (\lambda 
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v)))) (\lambda (x: nat).(\lambda (H5: (eq T t (TSort x))).(let 
+H6 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind Abst) w u))) H 
+(TSort x) H5) in (eq_ind_r T (TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda 
+(v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v))))) (pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c 
+(THead (Bind Abst) w u) x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq 
+T (TSort x) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 
+g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v 
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t 
+H5)))) H4)) (\lambda (H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: 
+nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws: TList).(\lambda 
+(i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_: TList).(\lambda (i: 
+nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t 
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) 
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda 
+(x0: TList).(\lambda (x1: nat).(\lambda (H5: (eq T t (THeads (Flat Appl) x0 
+(TLRef x1)))).(\lambda (_: (nfs2 c x0)).(\lambda (H7: (nf2 c (TLRef 
+x1))).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3 g c t0 (THead (Bind 
+Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1)) H5) in (eq_ind_r T (THeads 
+(Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2 T T (\lambda (v: 
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_: 
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c 
+(Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8 in (unintro T u 
+(\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind 
+Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T T (\lambda 
+(v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind 
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0))) (\lambda (v: 
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))))) (unintro T w 
+(\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1)) 
+(THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0 x) \to (ex4_2 T 
+T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1)) 
+(THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c t0 
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) t0) v x))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) t0) v)))))))) 
+(TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2: T).((ty3 g c 
+(THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to 
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda 
+(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2: 
+T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda 
+(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind 
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c 
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) 
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2 
+T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x 
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C 
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O 
+t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
+(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind 
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c 
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef 
+x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x 
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda 
+(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O 
+x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind 
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7 
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind 
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda 
+(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c 
+(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0) 
+(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda 
+(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
+Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15: 
+(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let 
+H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind 
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind 
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: 
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13)) 
+(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0: 
+T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2: 
+T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to 
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda 
+(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2: 
+T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 
+(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13: 
+(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda 
+(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind 
+Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl) 
+t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4)) 
+(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1 
+(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let 
+H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let 
+H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T 
+(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) 
+(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3 
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4))) 
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T 
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat 
+Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
+Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads 
+(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c 
+x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_: 
+(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T 
+(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T 
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat 
+Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda 
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c 
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind 
+Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1)) 
+(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x3 x5) H22) in 
+(False_ind (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat 
+Appl) t0 (THeads (Flat Appl) TNil (TLRef x1))) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v)))) H23))) (\lambda (t2: T).(\lambda (t3: 
+TList).(\lambda (_: (((eq T (THeads (Flat Appl) t3 (TLRef x1)) (THead (Bind 
+Abst) x3 x5)) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead 
+(Flat Appl) t0 (THeads (Flat Appl) t3 (TLRef x1))) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v))))))).(\lambda (H22: (eq T (THeads (Flat 
+Appl) (TCons t2 t3) (TLRef x1)) (THead (Bind Abst) x3 x5))).(let H23 \def 
+(eq_ind T (THead (Flat Appl) t2 (THeads (Flat Appl) t3 (TLRef x1))) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) x3 x5) H22) in (False_ind 
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 
+(THeads (Flat Appl) (TCons t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) 
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda 
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: 
+T).(nf2 (CHead c (Bind Abst) x) v)))) H23)))))) t1 H18))))))) H17))))))))) 
+(ty3_gen_appl g c t0 (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x 
+x2) H12))))))))) x0)) H10)) H9)) t H5))))))) H4)) H3))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3.ma
new file mode 100644 (file)
index 0000000..6a6279f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3.ma
@@ -0,0 +1,676 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/csubt/ty3.ma".
+
+include "basic_1A/ty3/subst1.ma".
+
+include "basic_1A/ty3/fsubst0.ma".
+
+include "basic_1A/pc3/pc1.ma".
+
+include "basic_1A/pc3/wcpr0.ma".
+
+include "basic_1A/pc1/props.ma".
+
+lemma ty3_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 
+t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) 
+\to (ty3 g c2 t2 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda 
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda 
+(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to 
+(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: 
+T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c 
+c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: 
+C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4 
+t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 
+c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 
+H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 
+H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: 
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) 
+t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) 
+(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda 
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind 
+Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: 
+((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g 
+c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: 
+T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda 
+(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 
+(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: 
+(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda 
+(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) 
+(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n 
+H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: 
+T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) 
+\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: 
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) 
+t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind 
+Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind 
+Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g 
+c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u 
+(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S 
+n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) 
+O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 
+H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n 
+H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda 
+(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to 
+(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: 
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) 
+u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) 
+\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2: 
+C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead 
+(Bind b) u t2) t4)).(let H6 \def (match H5 with [(pr0_refl t5) \Rightarrow 
+(\lambda (H6: (eq T t5 (THead (Bind b) u t2))).(\lambda (H7: (eq T t5 
+t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3 
+g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: (eq T (THead (Bind b) u t2) 
+t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).(ty3 g c2 t6 (THead 
+(Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t2 t3 (H3 
+(CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2 
+(pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead (Bind b) u t2) H6) H7))) | 
+(pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 
+t5) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let 
+H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 
+| (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) 
+(THead (Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead (Bind b) u t2) H8) in 
+((let H12 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u1 t5) (THead (Bind b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0: 
+K).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 
+u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) 
+(\lambda (H13: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to 
+((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 
+g c2 t4 (THead (Bind b) u t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t7: T).((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0 
+t7 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) (\lambda (H15: (eq T 
+(THead (Bind b) u2 t6) t4)).(eq_ind T (THead (Bind b) u2 t6) (\lambda (t7: 
+T).((pr0 u u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Bind b) u t3))))) 
+(\lambda (H16: (pr0 u u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda 
+(t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6) 
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H18: (ty3 g (CHead c2 (Bind 
+b) u) t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u2) t3 t7)) 
+(ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x0: 
+T).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead 
+(Bind b) u t3) (THead (Bind b) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl 
+u)) b t3 x H18) (THead (Bind b) u2 t6) (THead (Bind b) u2 t3) (ty3_bind g c2 
+u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 
+H4 u u2 H16 (Bind b)) t6 H17)) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead 
+(Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H16) (Bind b) t3))))) 
+(ty3_correct g (CHead c2 (Bind b) u2) t6 t3 (H3 (CHead c2 (Bind b) u2) 
+(wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))))) (ty3_correct g (CHead c2 
+(Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl 
+u) (Bind b)) t2 (pr0_refl t2)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1 
+(sym_eq T u1 u H13))) k (sym_eq K k (Bind b) H12))) H11)) H10)) H9 H6 H7))) | 
+(pr0_beta u0 v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u0 t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq 
+T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u0 t5)) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) 
+\to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) 
+H10)) H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) 
+\Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 
+t5)) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b0) u1 t5)) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1 
+v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u 
+t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
+\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
+t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
+_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 
+t5) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | 
+(THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u 
+t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(THead (Bind Abbr) u1 t5) (THead (Bind b) u t2) H9) in (eq_ind B Abbr 
+(\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead (Bind Abbr) 
+u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 
+g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T 
+u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind Abbr) u2 w) t4) \to 
+((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead 
+(Bind Abbr) u t3)))))))) (\lambda (H15: (eq T t5 t2)).(eq_ind T t2 (\lambda 
+(t7: T).((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u u2) \to ((pr0 t7 t6) 
+\to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3))))))) 
+(\lambda (H16: (eq T (THead (Bind Abbr) u2 w) t4)).(eq_ind T (THead (Bind 
+Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to ((subst0 O u2 
+t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u t3)))))) (\lambda (H17: (pr0 u 
+u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda (H19: (subst0 O u2 t6 w)).(let H20 
+\def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind 
+b0) u) c3) \to (\forall (t7: T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3 
+Abbr H13) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c 
+(Bind b0) u) t2 t3)) H2 Abbr H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead 
+c2 (Bind Abbr) u) t3 t7)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind 
+Abbr) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) 
+t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7)) 
+(ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: 
+T).(\lambda (_: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 
+H4 u (pr0_refl u)) Abbr t3 x H22) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) 
+u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead 
+c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u 
+u2 H17 (Bind Abbr)) t6 H18) c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x 
+c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 
+(pr2_free c2 u u2 H17) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind 
+Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17 
+(Bind Abbr)) t6 H18))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20 
+(CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2 
+(pr0_refl t2))))))))) t4 H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u 
+H14))) b H13)) H12)) H11)) H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0) 
+\Rightarrow (\lambda (H8: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) (THead 
+(Bind b) u t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow (lref_map (\lambda (x: 
+nat).(plus x (S O))) O t5) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind 
+b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H11 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
+_) \Rightarrow u0 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0 
+(lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) 
+\Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1) 
+\Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) 
+O t5)) (THead (Bind b) u t2) H8) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) 
+\to ((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b1 Abst)) 
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13: 
+(eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t5) t2) \to 
+((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 
+(THead (Bind b) u t3))))))) (\lambda (H14: (eq T (lift (S O) O t5) 
+t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: T).((eq T t6 t4) \to ((not (eq 
+B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) 
+(\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((not (eq B b 
+Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) (\lambda 
+(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t4)).(let H18 \def 
+(eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u) 
+c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 t3)))))) H3 (lift (S 
+O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda (t7: T).(ty3 g (CHead 
+c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in (ex_ind T (\lambda (t7: 
+T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 (THead (Bind b) u t3)) 
+(\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u) t3 x)).(B_ind 
+(\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead c2 (Bind b1) u) t3 
+x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) t3) \to (ty3 g c2 t4 
+(THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda 
+(H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 
+(Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def (ty3_gen_cabbr g (CHead 
+c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O (getl_refl Abbr c2 u) 
+(CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2 
+(drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) (lift (S O) O y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Abbr) u 
+t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: (subst1 O u (lift (S O) 
+O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u t3 (lift (S O) O 
+x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind T x0 (\lambda 
+(t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O (subst1_gen_lift_eq 
+t4 u (lift (S O) O x0) (S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O) 
+(\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O 
+(S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) 
+u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) t4 x1 H28 
+(pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift 
+(S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3) 
+(THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3) 
+(THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3 
+(pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u 
+(lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 
+(pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H24))))) (\lambda (H21: 
+(not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 
+x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t4) t3)).(let 
+H24 \def (match (H21 (refl_equal B Abst)) in False with []) in H24)))) 
+(\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind 
+Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O 
+t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) 
+O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 
+(drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) 
+(ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26: 
+(eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def 
+(eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) u) t7 x)) H22 
+(lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) (\lambda (t7: T).(ty3 
+g c2 t4 (THead (Bind Void) u t7))) (let H29 \def (eq_ind_r T x0 (\lambda (t7: 
+T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S O) O H25)) in (ty3_conv g c2 
+(THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u x) (ty3_bind g 
+c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S O) O x1) x H28) t4 x1 H29 
+(pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pc3_pr2_r c2 (THead 
+(Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Void) u (lift 
+(S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u)))))) t3 H26))))))) 
+H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u 
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O))))) 
+(ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 (H18 (CHead c2 
+(Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4) 
+(pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 H15))) t2 H14)) u0 
+(sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) H9 H6 H7))) | 
+(pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 
+t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def (eq_ind 
+T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 t5 t6) \to 
+(ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 (refl_equal T 
+(THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) (\lambda (c: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g 
+c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v 
+(THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to 
+(\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u 
+t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: 
+T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5 
+with [(pr0_refl t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w 
+v))).(\lambda (H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda 
+(t4: T).((eq T t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))))) (\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T 
+(THead (Flat Appl) w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 
+(H3 c2 H4 v (pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v) 
+H6) H7))) | (pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T 
+(THead k u1 t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4) 
+t2)).((let H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5])) 
+(THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) 
+\Rightarrow u1 | (THead _ t5 _) \Rightarrow t5])) (THead k u1 t3) (THead 
+(Flat Appl) w v) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
+\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K 
+(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead 
+k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat 
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind 
+T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2) 
+\to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v 
+(\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to 
+((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T 
+(THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to 
+(ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda 
+(H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3 
+g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 
+g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda 
+(_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: 
+T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind 
+Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: 
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda 
+(H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat 
+Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u 
+x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 
+(ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead 
+(Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 
+H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead 
+(Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 
+c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0))))))))) 
+(ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst) 
+u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1 
+(sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6 
+H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w 
+v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead 
+(Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead 
+_ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) 
+(THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | 
+(THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 
+t3)) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T 
+(THead (Bind Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to 
+((pr0 t5 v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3) 
+v)).(eq_ind T (THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind 
+Abbr) v2 t4) t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead 
+(Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead 
+(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: 
+T).((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead 
+(Bind Abst) u t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3 
+t4)).(let H16 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c 
+c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u 
+t0))))))) H3 (THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v 
+(\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) 
+u0 t3) H12) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0) 
+t5)) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u 
+t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind 
+Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda 
+(t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 
+(THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u 
+x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind 
+Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 
+(THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_: 
+T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 
+g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) 
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u 
+t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind 
+Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1: 
+T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4) 
+(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0 
+u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b) 
+u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+(THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w 
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0 
+H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u 
+(H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3 
+H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead 
+(Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind 
+Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+(pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+(THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0)))))))) 
+(pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead 
+(Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0 
+(pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x 
+H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0) 
+(H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0 
+t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) | 
+(pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w 
+v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead 
+(Bind b) u1 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in ((let H13 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef 
+_) \Rightarrow v1 | (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 
+(THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda 
+(t5: T).((eq T (THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5 
+v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3) 
+v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to 
+((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat 
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b 
+Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 
+(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq 
+B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda 
+(H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: 
+C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead 
+(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def 
+(eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 
+(THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead 
+(Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift 
+(S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: 
+T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in 
+(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u 
+t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5: 
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: 
+(pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda 
+(H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: 
+T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0)))) 
+(\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda 
+(_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind 
+Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead 
+(Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2 
+x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def 
+(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3 
+(CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27) 
+(THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u 
+t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u 
+t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x))) 
+(ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O) 
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S 
+O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T 
+T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind 
+Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6: 
+T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5: 
+T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S 
+O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u 
+t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b) 
+u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33: 
+(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g 
+(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) 
+t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead 
+(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w 
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0 
+H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S 
+O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat 
+Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind 
+Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) 
+u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17) 
+(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) 
+u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind 
+Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) 
+O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst 
+(lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead 
+(Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) 
+(pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) 
+O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) 
+(pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead 
+(Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+(lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl 
+v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O 
+(THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) 
+(THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w 
+H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind 
+Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u 
+t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0) 
+(THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat 
+Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) 
+(lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) 
+u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31))))))))) 
+(ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind 
+b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g 
+Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind 
+Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 
+(Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7 
+H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq 
+T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T 
+(THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1 
+t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) 
+H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to 
+((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4 
+H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3)) 
+(THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind 
+T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to 
+((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w 
+(THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0) 
+\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) 
+w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast) 
+u0 t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow 
+False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H7) in 
+(False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) 
+w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead 
+(Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g 
+c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: 
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g 
+c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: 
+T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 
+with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat Cast) t3 
+t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3 t2) 
+(\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))) 
+(\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat 
+Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3))) 
+(ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl 
+t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp 
+u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead 
+(Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef 
+_) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead 
+(Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match 
+e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) 
+\Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k | 
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5) 
+(THead (Flat Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq 
+T u1 t3) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2) 
+\to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda 
+(H13: (eq T u1 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T 
+(THead (Flat Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2 
+(\lambda (t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to 
+((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15: 
+(eq T (THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6) 
+(\lambda (t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat 
+Cast) t0 t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2 
+t6)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat 
+Cast) u2 t6) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g 
+c2 t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0) 
+(ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2 
+t6) (THead (Flat Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3 
+c2 H4 u2 H16) t6 t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 
+H16))) t0 (H3 c2 H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat 
+Cast) t0 t3) (pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0 
+u2) (pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct 
+g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) 
+u1 (sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6 
+H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda 
+(H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to 
+((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) 
+H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow 
+(\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (THead 
+(Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t5)) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f 
+with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat 
+Cast) t3 t2) H10) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to 
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8) 
+\Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Flat Cast) 
+t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def 
+(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I 
+(THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w) 
+t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5 
+t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5)) 
+(THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def 
+(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to 
+((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3))))) H10)) H9 H6 H7))) | (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7: 
+(eq T (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq 
+T t6 t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) 
+(THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef 
+_) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t5) 
+(THead (Flat Cast) t3 t2) H7) in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2) 
+\to ((eq T t6 t4) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 
+t3)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6 
+t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda 
+(H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2 
+t4 (THead (Flat Cast) t0 t3)))) (\lambda (H13: (pr0 t2 t4)).(ex_ind T 
+(\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 t4 (THead (Flat Cast) t0 t3)) 
+(\lambda (x: T).(\lambda (H14: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead (Flat 
+Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3 
+(pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 H13) (pc3_pr2_x c2 t3 (THead (Flat 
+Cast) t0 t3) (pr2_free c2 (THead (Flat Cast) t0 t3) t3 (pr0_tau t3 t3 
+(pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3))))) 
+t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10))) 
+H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T 
+t4))))))))))))))) c1 t1 t H))))).
+
+lemma ty3_sred_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall 
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g: 
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1 
+t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
+
+lemma ty3_sred_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall 
+(c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda 
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3: 
+T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c: 
+C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g: 
+G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t 
+(ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
+
+lemma ty3_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g: 
+G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g: 
+G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3 
+t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g: 
+G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0 
+(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t 
+H2)))))))))))))) c t1 t2 H)))).
+
+lemma ty3_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall 
+(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g: 
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda 
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c 
+t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: 
+(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2 
+H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3_props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3_props.ma
new file mode 100644 (file)
index 0000000..f12ad39
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/pr3_props.ma
@@ -0,0 +1,492 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/pr3.ma".
+
+lemma ty3_cred_pr2:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind 
+b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0: 
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b: 
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) 
+t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0 
+(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0 
+(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda 
+(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0 
+t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead 
+c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl 
+c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead 
+c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0) 
+(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1 
+v2 H))))).
+
+lemma ty3_cred_pr3:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c 
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b: 
+B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to 
+(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b: 
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b) 
+t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 
+t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b: 
+B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to 
+(ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b 
+t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
+
+lemma ty3_gen_lift:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop 
+h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: 
+T).(ty3 g e t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T 
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\lambda (_: T).(\forall (e: 
+C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) 
+(\lambda (t2: T).(ty3 g e t1 t2)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c 
+y x)).(unintro nat d (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall 
+(e: C).((drop h n c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) 
+(\lambda (t2: T).(ty3 g e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall 
+(x0: nat).((eq T y (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to 
+(ex2 T (\lambda (t2: T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t 
+t2)))))))) (ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (\forall 
+(e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) 
+t0)) (\lambda (t2: T).(ty3 g e x0 t2))))))))))) (\lambda (c0: C).(\lambda 
+(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (\forall (e: 
+C).((drop h x1 c0 e) \to (ex2 T (\lambda (t3: T).(pc3 c0 (lift h x1 t3) t)) 
+(\lambda (t3: T).(ty3 g e x0 t3)))))))))).(\lambda (u: T).(\lambda (t3: 
+T).(\lambda (H3: (ty3 g c0 u t3)).(\lambda (H4: ((\forall (x0: T).(\forall 
+(x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to 
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e 
+x0 t4)))))))))).(\lambda (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H6: (eq T u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: 
+(drop h x1 c0 e)).(let H8 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: 
+T).(\forall (x3: nat).((eq T t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h 
+x3 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda 
+(t4: T).(ty3 g e0 x2 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def 
+(eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let 
+H10 \def (H8 x0 x1 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda 
+(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 
+t4))) (\lambda (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda 
+(H12: (ty3 g e x0 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) 
+t2)) (\lambda (t4: T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 
+H5) H12)))) H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: 
+nat).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift 
+h x1 x0))).(\lambda (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort 
+m) (\lambda (t: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort 
+(next g m)))) (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e 
+(TSort m) t2)) (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: 
+T).(pc3 c0 t (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 
+(TSort (next g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 
+(lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind 
+Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: 
+((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall 
+(e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) 
+t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda 
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
+x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: 
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda 
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abbr) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: 
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 
+(Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T 
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T 
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u 
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in 
+(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) 
+x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) 
+t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 
+(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) 
+(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4) 
+t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S 
+n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift 
+(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda 
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) 
+(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift 
+(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) 
+O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus 
+x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) 
+(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g 
+n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) 
+(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) 
+H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n 
+h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
+T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0 
+(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: 
+T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) 
+(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift 
+(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda 
+(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h))) 
+(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n 
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0 
+(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans h (plus 
+x1 h) n (le_plus_r x1 h) H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n 
+h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O 
+x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n h))) (le_S_minus x1 h n 
+H8) (le_plus_r O (S (minus n h)))) (le_O_n x1))) (ty3_abbr g (minus n h) e d0 
+u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) x0 
+H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: 
+C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) 
+(\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda 
+(H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 
+\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 
+h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: 
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n)) 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda 
+(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 
+(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 
+t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 
+(S (plus n (minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: 
+T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: 
+(eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 
+(Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 
+\def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T 
+t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0) \to (ex2 T 
+(\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3 g e0 x4 
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u 
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in 
+(eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T (\lambda 
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2: T).(ty3 g e 
+(TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h 
+(minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h 
+(minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda 
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h (minus x1 (S n)) x2)))) 
+(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 
+d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 
+x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T 
+(\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h (minus n0 (S 
+n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda 
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O (lift 
+h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g 
+e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift h (minus 
+x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h (minus (plus 
+(S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 
+c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O (lift h (minus 
+n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1 (S n)) x2))) 
+(plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift h (plus (S 
+n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O 
+(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1 
+(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst 
+c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land 
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h) 
+n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
+t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le 
+(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T 
+(TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h 
+x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T 
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: 
+T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T 
+(lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O 
+u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0 
+O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0) 
+O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h)) 
+(le_plus_minus h n (le_trans h (plus x1 h) n (le_plus_r x1 h) H8))) (plus h 
+(S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) 
+O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus 
+O (S (minus n h))) (le_S_minus x1 h n H8) (le_plus_r O (S (minus n h)))) 
+(le_O_n x1))) (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 
+(Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u 
+t)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
+c0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: 
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) 
+u) t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift 
+h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T 
+(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4: 
+T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: 
+(eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: 
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
+(THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: 
+T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 
+(THead (Bind b) x2 x3))).(\lambda (H8: (eq T u (lift h x1 x2))).(\lambda (H9: 
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda 
+(t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u 
+t3))) (\lambda (t4: T).(ty3 g e t0 t4)))) (let H10 \def (eq_ind T t2 (\lambda 
+(t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to 
+(\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0) \to (ex2 T (\lambda 
+(t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t4) t3)) (\lambda (t4: T).(ty3 
+g e0 x4 t4))))))))) H4 (lift h (S x1) x3) H9) in (let H11 \def (eq_ind T t2 
+(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) t0 t3)) H3 (lift h (S x1) x3) 
+H9) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g (CHead c0 (Bind b) 
+t0) (lift h (S x1) x3) t3)) H11 (lift h x1 x2) H8) in (let H13 \def (eq_ind T 
+u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T (lift h (S x1) 
+x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) t0) 
+e0) \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) t0) (lift h x5 t4) 
+t3)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H10 (lift h x1 x2) H8) in (let 
+H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall (x5: 
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to 
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 
+x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H15 \def (eq_ind T u (\lambda 
+(t0: T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) 
+(\lambda (t0: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind 
+b) t0 t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))) (let H16 
+\def (H14 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda 
+(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) 
+(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4: 
+T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H18: (ty3 g e x2 
+x4)).(let H19 \def (H13 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e 
+(Bind b) x2) (drop_skip_bind h x1 c0 e H6 b x2)) in (ex2_ind T (\lambda (t4: 
+T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda 
+(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e 
+(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H20: (pc3 (CHead c0 
+(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H21: (ty3 g (CHead 
+e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead e (Bind b) 
+x2) x5 t0)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) 
+(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) 
+(\lambda (x6: T).(\lambda (_: (ty3 g (CHead e (Bind b) x2) x5 x6)).(ex_intro2 
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) 
+t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)) (THead (Bind b) 
+x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S x1) x5)) 
+(\lambda (t0: T).(pc3 c0 t0 (THead (Bind b) (lift h x1 x2) t3))) (pc3_head_2 
+c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H20) (lift h x1 (THead (Bind 
+b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H18 b x3 x5 H21)))) 
+(ty3_correct g (CHead e (Bind b) x2) x3 x5 H21))))) H19))))) H16)) u 
+H8))))))) x0 H7)))))) (lift_gen_bind b u t2 x0 h x1 H5))))))))))))))))) 
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w 
+u)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T w (lift h x1 
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
+c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (v: 
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead (Bind Abst) u 
+t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T v (lift h x1 
+x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 
+c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x0 
+t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead 
+(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0 
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat 
+Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) 
+(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1 
+x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) 
+x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead 
+(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2)))) 
+(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x4: T).(\forall (x5: 
+nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to 
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) (THead (Bind Abst) u t))) 
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift h x1 x3) H9) in (let H11 
+\def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 
+(lift h x1 x3) H9) in (let H12 \def (eq_ind T w (\lambda (t0: T).(\forall 
+(x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: 
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2) u)) 
+(\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1 x2) H8) in (let H13 
+\def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1 (lift h x1 x2) H8) in 
+(eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 
+(lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: 
+T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 
+(lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
+(\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4) u)).(\lambda (H16: 
+(ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T (lift h x1 x3)) e H6) 
+in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u 
+t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift 
+h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) 
+(\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x5: 
+T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u t))).(\lambda 
+(H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t2: T).(pr3 
+e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c0 u 
+(lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2)))))) (ex2 T (\lambda 
+(t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5 (THead (Bind Abst) 
+x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1 x6))).(\lambda (H22: ((\forall 
+(b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) 
+x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0)) (ex2 T (\lambda (t2: 
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind 
+Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) 
+(\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(let H_y \def (ty3_sred_pr3 
+e x5 (THead (Bind Abst) x6 x7) H20 g x8 H23) in (ex3_2_ind T T (\lambda (t2: 
+T).(\lambda (_: T).(pc3 e (THead (Bind Abst) x6 t2) x8))) (\lambda (_: 
+T).(\lambda (t0: T).(ty3 g e x6 t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead e (Bind Abst) x6) x7 t2))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 
+t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda 
+(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda 
+(x10: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9) x8)).(\lambda (H25: 
+(ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind Abst) x6) x7 
+x9)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) 
+(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead 
+(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7)) 
+(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst) 
+x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead 
+(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7)) 
+(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h 
+(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t))) 
+(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind 
+Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6)) 
+(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind 
+Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1 
+(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead 
+(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4 
+H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6) 
+(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind 
+Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9 
+H26) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7) H20))))))))) 
+(ty3_gen_bind g Abst e x6 x7 x8 H_y))))) (ty3_correct g e x3 x5 H19))))))) 
+(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0 
+H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda 
+(H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to 
+(\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h 
+x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0: 
+T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall 
+(x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to 
+(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e 
+x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T 
+(THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: 
+(drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
+(THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h 
+x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 t3))) (\lambda 
+(t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq 
+T x0 (THead (Flat Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 
+x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) 
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead 
+(Flat Cast) t0 t3))) (\lambda (t4: T).(ty3 g e t t4)))) (let H10 \def (eq_ind 
+T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 
+x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 
+c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H4 (lift h 
+x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t0)) H3 
+(lift h x1 x2) H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(\forall 
+(x4: T).(\forall (x5: nat).((eq T t2 (lift h x5 x4)) \to (\forall (e0: 
+C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t)) 
+(\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H2 (lift h x1 x2) H8) in (let H13 
+\def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h x1 x2) H8) in 
+(eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) (THead (Flat Cast) t0 t))) (\lambda (t4: T).(ty3 g e (THead 
+(Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g 
+c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t (lift h x5 x4)) 
+\to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t4: T).(pc3 c0 
+(lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4 t4))))))))) H12 
+(lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) 
+e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) 
+(\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 
+t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda (t4: T).(ty3 g e (THead 
+(Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1 
+x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 
+(refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 
+(lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: 
+T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 x2)))) (\lambda 
+(t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda 
+(H20: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 
+T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Flat Cast) t0 (lift h x1 
+x2)))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) (THead (Flat 
+Cast) x5 x2) (eq_ind_r T (THead (Flat Cast) (lift h x1 x5) (lift h x1 x2)) 
+(\lambda (t: T).(pc3 c0 t (THead (Flat Cast) t0 (lift h x1 x2)))) (pc3_head_1 
+c0 (lift h x1 x5) t0 H20 (Flat Cast) (lift h x1 x2)) (lift h x1 (THead (Flat 
+Cast) x5 x2)) (lift_flat Cast x5 x2 h x1)) (ty3_cast g e x3 x2 (ty3_conv g e 
+x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) 
+H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 
+H5))))))))))))))) c y x H0))))) H))))))).
+
+lemma ty3_tred:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T 
+(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1: 
+(ty3 g c t1 x)).(let H_y \def (ty3_sred_pr3 c t1 t2 H0 g x H1) in (ty3_conv g 
+c t2 x H_y u t1 H (pc3_pr3_r c t1 t2 H0))))) (ty3_correct g c u t1 H)))))))).
+
+theorem ty3_sconv_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
+u2) \to (pc3 c t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x: 
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(let H_y \def 
+(ty3_sred_pr3 c u2 x H4 g t2 H0) in (let H_y0 \def (ty3_sred_pr3 c u1 x H3 g 
+t1 H) in (ty3_unique g c x t1 H_y0 t2 H_y)))))) H2)))))))))).
+
+lemma ty3_sred_back:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c 
+t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 
+t) \to (ty3 g c t1 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda 
+(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
+(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t 
+t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g 
+c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t 
+H1)))) (ty3_correct g c t2 t H1)))))))))).
+
+theorem ty3_sconv:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c 
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 
+u2) \to (ty3 g c u1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c 
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda 
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda 
+(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back 
+g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/props.ma
new file mode 100644 (file)
index 0000000..c574309
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/props.ma
@@ -0,0 +1,669 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/fwd.ma".
+
+include "basic_1A/pc3/fwd.ma".
+
+lemma ty3_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e 
+t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c 
+e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to 
+(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h 
+d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 
+g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d 
+t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h 
+d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5) 
+(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: 
+nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop 
+h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort 
+(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 
+(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort 
+(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n: 
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: 
+nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: 
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 
+(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le 
+n d0 (le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) 
+d0 H4)))) c0 c h H3 (CHead d (Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda 
+(e0: C).(\lambda (_: C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
+C).(drop h (minus d0 n) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
+(CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift 
+(S n) O t))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 
+x0)).(\lambda (H7: (drop h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 
+(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
+nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let 
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S n)) H9 Abbr d u H8) in (ex2_ind C 
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d0 (S n)) 
+u)))) (\lambda (c1: C).(drop h (minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 
+(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x: C).(\lambda (H11: 
+(clear x0 (CHead x (Bind Abbr) (lift h (minus d0 (S n)) u)))).(\lambda (H12: 
+(drop h (minus d0 (S n)) x d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g 
+c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat (plus (S n) (minus d0 (S 
+n))) (\lambda (n0: nat).(ty3 g c0 (TLRef n) (lift h n0 (lift (S n) O t)))) 
+(eq_ind_r T (lift (S n) O (lift h (minus d0 (S n)) t)) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) 
+(lift (S n) O (lift h (minus d0 (S n)) t)))) (ty3_abbr g n c0 x (lift h 
+(minus d0 (S n)) u) (getl_intro n c0 (CHead x (Bind Abbr) (lift h (minus d0 
+(S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) t) (H2 x (minus d0 (S n)) h 
+H12)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H4)) (lift h 
+(plus (S n) (minus d0 (S n))) (lift (S n) O t)) (lift_d t h (S n) (minus d0 
+(S n)) O (le_O_n (minus d0 (S n))))) d0 (le_plus_minus_r (S n) d0 H4)) (lift 
+h d0 (TLRef n)) (lift_lref_lt n h d0 H4))))) H10)))))))) H5))) (\lambda (H4: 
+(le d0 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(ty3 g c0 t0 (lift 
+h d0 (lift (S n) O t)))) (eq_ind nat (S n) (\lambda (_: nat).(ty3 g c0 (TLRef 
+(plus n h)) (lift h d0 (lift (S n) O t)))) (eq_ind_r T (lift (plus h (S n)) O 
+t) (\lambda (t0: T).(ty3 g c0 (TLRef (plus n h)) t0)) (eq_ind_r nat (plus (S 
+n) h) (\lambda (n0: nat).(ty3 g c0 (TLRef (plus n h)) (lift n0 O t))) 
+(ty3_abbr g (plus n h) c0 d u (drop_getl_trans_ge n c0 c d0 h H3 (CHead d 
+(Bind Abbr) u) H0 H4) t H1) (plus h (S n)) (plus_sym h (S n))) (lift h d0 
+(lift (S n) O t)) (lift_free t (S n) h O d0 (le_S_n d0 (S n) (le_S (S d0) (S 
+n) (le_n_S d0 n H4))) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (le_antisym (S n) (plus (S O) n) 
+(le_n (plus (S O) n)) (le_n (S n))) (plus n (S O)) (plus_sym n (S O)))) (lift 
+h d0 (TLRef n)) (lift_lref_ge n h d0 H4)))))))))))))))) (\lambda (n: 
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (c0: C).(\forall (d0: nat).(\forall (h: 
+nat).((drop h d0 c0 d) \to (ty3 g c0 (lift h d0 u) (lift h d0 
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3: 
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 
+(lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le 
+n d0 (le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) 
+d0 H4)))) c0 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda 
+(e0: C).(\lambda (_: C).(drop n O c0 e0))) (\lambda (e0: C).(\lambda (e1: 
+C).(drop h (minus d0 n) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
+(CHead d (Bind Abst) u)))) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift 
+(S n) O u))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 
+x0)).(\lambda (H7: (drop h (minus d0 n) x0 x1)).(\lambda (H8: (clear x1 
+(CHead d (Bind Abst) u))).(let H9 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
+nat).(drop h n0 x0 x1)) H7 (S (minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let 
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C 
+(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abst) (lift h (minus d0 (S n)) 
+u)))) (\lambda (c1: C).(drop h (minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 
+(TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (x: C).(\lambda (H11: 
+(clear x0 (CHead x (Bind Abst) (lift h (minus d0 (S n)) u)))).(\lambda (H12: 
+(drop h (minus d0 (S n)) x d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g 
+c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S 
+n))) (\lambda (n0: nat).(ty3 g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) 
+(eq_ind_r T (lift (S n) O (lift h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) 
+(lift (S n) O (lift h (minus d0 (S n)) u)))) (ty3_abst g n c0 x (lift h 
+(minus d0 (S n)) u) (getl_intro n c0 (CHead x (Bind Abst) (lift h (minus d0 
+(S n)) u)) x0 H6 H11) (lift h (minus d0 (S n)) t) (H2 x (minus d0 (S n)) h 
+H12)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H4)) (lift h 
+(plus (S n) (minus d0 (S n))) (lift (S n) O u)) (lift_d u h (S n) (minus d0 
+(S n)) O (le_O_n (minus d0 (S n))))) d0 (le_plus_minus_r (S n) d0 H4)) (lift 
+h d0 (TLRef n)) (lift_lref_lt n h d0 H4))))) H10)))))))) H5))) (\lambda (H4: 
+(le d0 n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(ty3 g c0 t0 (lift 
+h d0 (lift (S n) O u)))) (eq_ind nat (S n) (\lambda (_: nat).(ty3 g c0 (TLRef 
+(plus n h)) (lift h d0 (lift (S n) O u)))) (eq_ind_r T (lift (plus h (S n)) O 
+u) (\lambda (t0: T).(ty3 g c0 (TLRef (plus n h)) t0)) (eq_ind_r nat (plus (S 
+n) h) (\lambda (n0: nat).(ty3 g c0 (TLRef (plus n h)) (lift n0 O u))) 
+(ty3_abst g (plus n h) c0 d u (drop_getl_trans_ge n c0 c d0 h H3 (CHead d 
+(Bind Abst) u) H0 H4) t H1) (plus h (S n)) (plus_sym h (S n))) (lift h d0 
+(lift (S n) O u)) (lift_free u (S n) h O d0 (le_S_n d0 (S n) (le_S (S d0) (S 
+n) (le_n_S d0 n H4))) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O) 
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (le_antisym (S n) (plus (S O) n) 
+(le_n (plus (S O) n)) (le_n (S n))) (plus n (S O)) (plus_sym n (S O)))) (lift 
+h d0 (TLRef n)) (lift_lref_ge n h d0 H4)))))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: 
+((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to 
+(ty3 g c0 (lift h d u) (lift h d t)))))))).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda 
+(H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 
+(CHead c (Bind b) u)) \to (ty3 g c0 (lift h d t0) (lift h d 
+t3)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
+(drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) 
+d) t0)) (\lambda (t4: T).(ty3 g c0 t4 (lift h d (THead (Bind b) u t3)))) 
+(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) t3)) (\lambda 
+(t4: T).(ty3 g c0 (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) 
+t4)) (ty3_bind g c0 (lift h d u) (lift h d t) (H1 c0 d h H4) b (lift h (S d) 
+t0) (lift h (S d) t3) (H3 (CHead c0 (Bind b) (lift h d u)) (S d) h 
+(drop_skip_bind h d c0 c H4 b u))) (lift h d (THead (Bind b) u t3)) 
+(lift_head (Bind b) u t3 h d)) (lift h d (THead (Bind b) u t0)) (lift_head 
+(Bind b) u t0 h d)))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda 
+(u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c0: C).(\forall 
+(d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d w) (lift 
+h d u)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall 
+(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d v) (lift h d (THead (Bind 
+Abst) u t))))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d 
+w) (lift h (s (Flat Appl) d) v)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d 
+(THead (Flat Appl) w (THead (Bind Abst) u t))))) (eq_ind_r T (THead (Flat 
+Appl) (lift h d w) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t))) 
+(\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) (lift h (s (Flat 
+Appl) d) v)) t0)) (eq_ind_r T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) 
+(lift h (s (Bind Abst) (s (Flat Appl) d)) t)) (\lambda (t0: T).(ty3 g c0 
+(THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) (THead (Flat 
+Appl) (lift h d w) t0))) (ty3_appl g c0 (lift h d w) (lift h d u) (H1 c0 d h 
+H4) (lift h d v) (lift h (S d) t) (eq_ind T (lift h d (THead (Bind Abst) u 
+t)) (\lambda (t0: T).(ty3 g c0 (lift h d v) t0)) (H3 c0 d h H4) (THead (Bind 
+Abst) (lift h d u) (lift h (S d) t)) (lift_bind Abst u t h d))) (lift h (s 
+(Flat Appl) d) (THead (Bind Abst) u t)) (lift_head (Bind Abst) u t h (s (Flat 
+Appl) d))) (lift h d (THead (Flat Appl) w (THead (Bind Abst) u t))) 
+(lift_head (Flat Appl) w (THead (Bind Abst) u t) h d)) (lift h d (THead (Flat 
+Appl) w v)) (lift_head (Flat Appl) w v h d))))))))))))))) (\lambda (c: 
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda 
+(H1: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) 
+\to (ty3 g c0 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda 
+(_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d 
+t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4: 
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s 
+(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d (THead (Flat Cast) 
+t4 t3)))) (eq_ind_r T (THead (Flat Cast) (lift h d t4) (lift h (s (Flat Cast) 
+d) t3)) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) (lift h d t3) (lift h (s 
+(Flat Cast) d) t0)) t)) (ty3_cast g c0 (lift h (s (Flat Cast) d) t0) (lift h 
+(s (Flat Cast) d) t3) (H1 c0 (s (Flat Cast) d) h H4) (lift h d t4) (H3 c0 d h 
+H4)) (lift h d (THead (Flat Cast) t4 t3)) (lift_head (Flat Cast) t4 t3 h d)) 
+(lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast) t3 t0 h 
+d)))))))))))))) e t1 t2 H))))).
+
+lemma ty3_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda 
+(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0: 
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda 
+(_: (ex T (\lambda (t3: T).(ty3 g c0 t t3)))).(\lambda (u: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t4: T).(ty3 g 
+c0 t3 t4)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g 
+c0 t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T 
+(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m))) 
+(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda 
+(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3 
+g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda 
+(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0 
+(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n) 
+(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind 
+Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T 
+(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0 
+(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n) 
+(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda 
+(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (H3: (ex T 
+(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)))).(let H4 \def H3 in 
+(ex_ind T (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t3 t4)) (ex T 
+(\lambda (t4: T).(ty3 g c0 (THead (Bind b) u t3) t4))) (\lambda (x: 
+T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 x)).(ex_intro T (\lambda 
+(t4: T).(ty3 g c0 (THead (Bind b) u t3) t4)) (THead (Bind b) u x) (ty3_bind g 
+c0 u t H0 b t3 x H5)))) H4)))))))))))) (\lambda (c0: C).(\lambda (w: 
+T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T (\lambda 
+(t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0: T).(ty3 g c0 
+(THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T (\lambda (t0: 
+T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w 
+(THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_: (ty3 g c0 u 
+x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0 (THead (Bind 
+Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g c0 (THead (Bind 
+Abst) u t) x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 
+(THead (Bind Abst) u t3) x0))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
+t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t 
+t3))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 
+(THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3 g c0 u x2)).(\lambda (H10: 
+(ty3 g (CHead c0 (Bind Abst) u) t x1)).(ex_intro T (\lambda (t0: T).(ty3 g c0 
+(THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat Appl) w 
+(THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u t) x1 
+(ty3_bind g c0 u x2 H9 Abst t x1 H10)))))))) (ty3_gen_bind g Abst c0 u t x0 
+H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g 
+c0 t3 t)))).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 t3 t4)).(\lambda (H3: 
+(ex T (\lambda (t: T).(ty3 g c0 t4 t)))).(let H4 \def H3 in (ex_ind T 
+(\lambda (t: T).(ty3 g c0 t4 t)) (ex T (\lambda (t: T).(ty3 g c0 (THead (Flat 
+Cast) t4 t3) t))) (\lambda (x: T).(\lambda (H5: (ty3 g c0 t4 x)).(ex_intro T 
+(\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t4 t3) t)) (THead (Flat Cast) x 
+t4) (ty3_cast g c0 t3 t4 H2 x H5)))) H4)))))))))) c t1 t2 H))))).
+
+theorem ty3_unique:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0: 
+T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall 
+(t3: T).((ty3 g c0 u0 t3) \to (pc3 c0 t0 t3))))).(\lambda (H4: (pc3 c0 t0 
+t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s 
+c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m: 
+nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g 
+c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0 
+t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) 
+t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) 
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 
+(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda 
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g 
+e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda 
+(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 
+x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n 
+c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n 
+H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) 
+(CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead 
+_ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) 
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: 
+T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H8 u0 H10) in (let H13 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 H10) in (let H14 \def 
+(eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u0))) H12 d 
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d 
+H11) in (pc3_t (lift (S n) O x2) c0 (lift (S n) O t) (pc3_lift c0 d (S n) O 
+(getl_drop Abbr c0 d u0 n H14) t x2 (H2 x2 H15)) t2 H5))))))) H9))))))))) 
+H4)) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T (\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) 
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) 
+(pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: 
+T).(\lambda (_: (pc3 c0 (lift (S n) O x1) t2)).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def 
+(eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead 
+x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 
+(Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u0) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (pc3 c0 
+(lift (S n) O t) t2) H9))))))))) H4)) (ty3_gen_lref g c0 t2 n H3)))))))))))) 
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda 
+(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 
+g d u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t 
+t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind 
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift 
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n 
+c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda 
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) 
+O u0) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))).(ex3_3_ind C T T 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) 
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g 
+e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (x2: T).(\lambda (_: (pc3 c0 (lift (S n) O x2) t2)).(\lambda (H6: 
+(getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 
+\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 
+(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
+x0 (Bind Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u0) 
+(\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) 
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat 
+_) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0 
+(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) 
+(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
+Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 
+t0)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda 
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O 
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7: 
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal 
+C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in 
+((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) 
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
+x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H8 u0 
+H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(ty3 g x0 t0 x2)) H7 u0 
+H10) in (let H14 \def (eq_ind_r T x1 (\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) t2)) H5 u0 H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n 
+c0 (CHead c1 (Bind Abst) u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 
+(\lambda (c1: C).(ty3 g c1 u0 x2)) H13 d H11) in H14))))))) H9))))))))) H4)) 
+(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (c0: C).(\lambda (u0: 
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: 
+T).((ty3 g c0 u0 t2) \to (pc3 c0 t t2))))).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t0 
+t2)).(\lambda (H3: ((\forall (t3: T).((ty3 g (CHead c0 (Bind b) u0) t0 t3) 
+\to (pc3 (CHead c0 (Bind b) u0) t2 t3))))).(\lambda (t3: T).(\lambda (H4: 
+(ty3 g c0 (THead (Bind b) u0 t0) t3)).(ex3_2_ind T T (\lambda (t4: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t4) t3))) (\lambda (_: 
+T).(\lambda (t5: T).(ty3 g c0 u0 t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 
+g (CHead c0 (Bind b) u0) t0 t4))) (pc3 c0 (THead (Bind b) u0 t2) t3) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H5: (pc3 c0 (THead (Bind b) u0 x0) 
+t3)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H7: (ty3 g (CHead c0 (Bind b) 
+u0) t0 x0)).(pc3_t (THead (Bind b) u0 x0) c0 (THead (Bind b) u0 t2) 
+(pc3_head_2 c0 u0 t2 x0 (Bind b) (H3 x0 H7)) t3 H5)))))) (ty3_gen_bind g b c0 
+u0 t0 t3 H4)))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: 
+T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 w 
+t2) \to (pc3 c0 u0 t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 
+g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((ty3 g c0 
+v t2) \to (pc3 c0 (THead (Bind Abst) u0 t) t2))))).(\lambda (t2: T).(\lambda 
+(H4: (ty3 g c0 (THead (Flat Appl) w v) t2)).(ex3_2_ind T T (\lambda (u1: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u1 t0)) 
+t2))) (\lambda (u1: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u1 
+t0)))) (\lambda (u1: T).(\lambda (_: T).(ty3 g c0 w u1))) (pc3 c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u0 t)) t2) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H5: (pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) 
+t2)).(\lambda (H6: (ty3 g c0 v (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 
+g c0 w x0)).(pc3_t (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u0 t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 
+t) (THead (Bind Abst) x0 x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 
+H5)))))) (ty3_gen_appl g c0 w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda 
+(t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: 
+((\forall (t3: T).((ty3 g c0 t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda (H3: ((\forall (t4: T).((ty3 g c0 
+t2 t4) \to (pc3 c0 t3 t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead 
+(Flat Cast) t2 t0) t4)).(ex3_ind T (\lambda (t5: T).(pc3 c0 (THead (Flat 
+Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t0 t2)) (\lambda (t5: T).(ty3 g 
+c0 t2 t5)) (pc3 c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: T).(\lambda 
+(H5: (pc3 c0 (THead (Flat Cast) x0 t2) t4)).(\lambda (_: (ty3 g c0 t0 
+t2)).(\lambda (H7: (ty3 g c0 t2 x0)).(pc3_t (THead (Flat Cast) x0 t2) c0 
+(THead (Flat Cast) t3 t2) (pc3_head_1 c0 t3 x0 (H3 x0 H7) (Flat Cast) t2) t4 
+H5))))) (ty3_gen_cast g c0 t0 t2 t4 H4)))))))))))) c u t1 H))))).
+
+lemma ty3_gen_abst_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
+(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 
+T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) 
+u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u 
+t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T 
+(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) 
+t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) 
+x)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) 
+u t3) x))) (\lambda (_: T).(\lambda (t: T).(ty3 g c u t))) (\lambda (t3: 
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t2 t3))) (ex2 T (\lambda 
+(w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c (THead (Bind Abst) u 
+x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c (Bind 
+Abst) u) t2 x0)).(ex3_2_ind T T (\lambda (t3: T).(\lambda (_: T).(pc3 c 
+(THead (Bind Abst) u t3) (THead (Bind Abst) u t2)))) (\lambda (_: T).(\lambda 
+(t: T).(ty3 g c u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c (Bind 
+Abst) u) t1 t3))) (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 
+g (CHead c (Bind Abst) u) t1 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H4: (pc3 c (THead (Bind Abst) u x2) (THead (Bind Abst) u t2))).(\lambda (H5: 
+(ty3 g c u x3)).(\lambda (H6: (ty3 g (CHead c (Bind Abst) u) t1 x2)).(let H_y 
+\def (pc3_gen_abst_shift c u x2 t2 H4) in (ex_intro2 T (\lambda (w: T).(ty3 g 
+c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x3 H5 
+(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x2 H6 H_y)))))))) 
+(ty3_gen_bind g Abst c u t1 (THead (Bind Abst) u t2) H))))))) (ty3_gen_bind g 
+Abst c u t2 x H0)))) (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind 
+Abst) u t2) H))))))).
+
+lemma ty3_typecheck:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t 
+v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: 
+(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: 
+T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g 
+c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) 
+(THead (Flat Cast) x v) (ty3_cast g c t v H x H0)))) (ty3_correct g c t v 
+H)))))).
+
+lemma ty3_getl_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to (\forall (v0: T).(\forall (t0: T).(\forall (i: nat).((subst0 i v0 t 
+t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c (CHead d 
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(ty3_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
+T).(\forall (v0: T).(\forall (t2: T).(\forall (i: nat).((subst0 i v0 t0 t2) 
+\to (\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d 
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda 
+(c0: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 i v0 t2 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
+w))))))))))))).(\lambda (u0: T).(\lambda (t1: T).(\lambda (_: (ty3 g c0 u0 
+t1)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i: 
+nat).((subst0 i v0 u0 t3) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
+w))))))))))))).(\lambda (_: (pc3 c0 t1 t2)).(\lambda (v0: T).(\lambda (t3: 
+T).(\lambda (i: nat).(\lambda (H5: (subst0 i v0 u0 t3)).(\lambda (b: 
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H6: (getl i c0 (CHead d (Bind b) 
+v))).(H3 v0 t3 i H5 b d v H6))))))))))))))))))) (\lambda (c0: C).(\lambda (m: 
+nat).(\lambda (v0: T).(\lambda (t0: T).(\lambda (i: nat).(\lambda (H0: 
+(subst0 i v0 (TSort m) t0)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: 
+T).(\lambda (_: (getl i c0 (CHead d (Bind b) v))).(subst0_gen_sort v0 t0 i m 
+H0 (ex T (\lambda (w: T).(ty3 g d v w)))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abbr) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 
+t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: 
+nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v: 
+T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v 
+w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda 
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n 
+i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) 
+(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 
+\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) 
+H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: 
+C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind 
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono 
+c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) 
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in 
+((let H11 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abbr) u0) 
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 
+(Bind b) v) H7)) in (\lambda (H12: (eq B Abbr b)).(\lambda (H13: (eq C d 
+d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind 
+b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w: 
+T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n 
+c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T 
+(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0: 
+B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr H12) in (ex_intro T (\lambda 
+(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9)))))) 
+(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (n: nat).(\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d 
+(Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda 
+(_: ((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) 
+\to (\forall (b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 
+(Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda 
+(v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef 
+n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl 
+i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O 
+v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n 
+i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i 
+(\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 
+\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 
+(Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind 
+Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 
+(CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) 
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind 
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) 
+\Rightarrow t2])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono 
+c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: 
+(eq B Abst b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v 
+(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T 
+u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def 
+(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d 
+H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) 
+(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) 
+u0))) H15 Abst H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 
+H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n 
+H3)))))))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: 
+T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1: ((\forall (v0: T).(\forall 
+(t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall 
+(d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda 
+(w: T).(ty3 g d v w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: 
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to 
+(\forall (b0: B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) 
+u0) (CHead d (Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
+w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda 
+(H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: 
+C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind 
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: 
+T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 
+t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: 
+T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) 
+(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) 
+(\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 
+(THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda 
+(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) 
+x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) 
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) 
+(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: 
+T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) 
+v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: 
+(eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 
+x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) 
+H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b) 
+i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 
+(THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 
+u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex 
+T (\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(_: (eq T t3 (THead (Bind b) x0 x1))).(\lambda (H8: (subst0 i v0 u0 
+x0)).(\lambda (_: (subst0 (s (Bind b) i) v0 t1 x1)).(H1 v0 x0 i H8 b0 d v 
+H5)))))) H6)) (subst0_gen_head (Bind b) v0 u0 t1 t3 i H4)))))))))))))))))))) 
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
+u0)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: 
+nat).((subst0 i v0 w t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w0: T).(ty3 g d v 
+w0))))))))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 v 
+(THead (Bind Abst) u0 t0))).(\lambda (H3: ((\forall (v0: T).(\forall (t1: 
+T).(\forall (i: nat).((subst0 i v0 v t1) \to (\forall (b: B).(\forall (d: 
+C).(\forall (v1: T).((getl i c0 (CHead d (Bind b) v1)) \to (ex T (\lambda 
+(w0: T).(ty3 g d v1 w0))))))))))))).(\lambda (v0: T).(\lambda (t1: 
+T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat Appl) w v) 
+t1)).(\lambda (b: B).(\lambda (d: C).(\lambda (v1: T).(\lambda (H5: (getl i 
+c0 (CHead d (Bind b) v1))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t1 (THead 
+(Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w u2))) (ex2 T (\lambda 
+(t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat 
+Appl) i) v0 v t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t1 
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w 
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)))) 
+(ex T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (H6: (ex2 T (\lambda (u2: 
+T).(eq T t1 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i v0 w 
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t1 (THead (Flat Appl) u2 v))) 
+(\lambda (u2: T).(subst0 i v0 w u2)) (ex T (\lambda (w0: T).(ty3 g d v1 w0))) 
+(\lambda (x: T).(\lambda (_: (eq T t1 (THead (Flat Appl) x v))).(\lambda (H8: 
+(subst0 i v0 w x)).(H1 v0 x i H8 b d v1 H5)))) H6)) (\lambda (H6: (ex2 T 
+(\lambda (t2: T).(eq T t1 (THead (Flat Appl) w t2))) (\lambda (t2: T).(subst0 
+(s (Flat Appl) i) v0 v t2)))).(ex2_ind T (\lambda (t2: T).(eq T t1 (THead 
+(Flat Appl) w t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2)) (ex 
+T (\lambda (w0: T).(ty3 g d v1 w0))) (\lambda (x: T).(\lambda (_: (eq T t1 
+(THead (Flat Appl) w x))).(\lambda (H8: (subst0 (s (Flat Appl) i) v0 v 
+x)).(H3 v0 x (s (Flat Appl) i) H8 b d v1 H5)))) H6)) (\lambda (H6: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t1 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v0 w u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) v0 v t2))) (ex T (\lambda (w0: 
+T).(ty3 g d v1 w0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t1 
+(THead (Flat Appl) x0 x1))).(\lambda (_: (subst0 i v0 w x0)).(\lambda (H9: 
+(subst0 (s (Flat Appl) i) v0 v x1)).(H3 v0 x1 (s (Flat Appl) i) H9 b d v1 
+H5)))))) H6)) (subst0_gen_head (Flat Appl) v0 w v t1 i H4))))))))))))))))))) 
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
+t2)).(\lambda (H1: ((\forall (v0: T).(\forall (t0: T).(\forall (i: 
+nat).((subst0 i v0 t1 t0) \to (\forall (b: B).(\forall (d: C).(\forall (v: 
+T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v 
+w))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (H3: 
+((\forall (v0: T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t2 t3) \to 
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b) 
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (v0: 
+T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead (Flat 
+Cast) t2 t1) t3)).(\lambda (b: B).(\lambda (d: C).(\lambda (v: T).(\lambda 
+(H5: (getl i c0 (CHead d (Bind b) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T 
+t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2))) (ex2 T 
+(\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: 
+T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t4: T).(eq T t3 (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v0 t2 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat 
+Cast) i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: 
+(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Cast) u2 t1))) (\lambda (u2: 
+T).(subst0 i v0 t2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat 
+Cast) u2 t1))) (\lambda (u2: T).(subst0 i v0 t2 u2)) (ex T (\lambda (w: 
+T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Flat Cast) x 
+t1))).(\lambda (H8: (subst0 i v0 t2 x)).(H3 v0 x i H8 b d v H5)))) H6)) 
+(\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) 
+(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4)))).(ex2_ind T (\lambda 
+(t4: T).(eq T t3 (THead (Flat Cast) t2 t4))) (\lambda (t4: T).(subst0 (s 
+(Flat Cast) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: 
+T).(\lambda (_: (eq T t3 (THead (Flat Cast) t2 x))).(\lambda (H8: (subst0 (s 
+(Flat Cast) i) v0 t1 x)).(H1 v0 x (s (Flat Cast) i) H8 b d v H5)))) H6)) 
+(\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) 
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 
+t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead 
+(Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 t2 u2))) 
+(\lambda (_: T).(\lambda (t4: T).(subst0 (s (Flat Cast) i) v0 t1 t4))) (ex T 
+(\lambda (w: T).(ty3 g d v w))) (\lambda (x0: T).(\lambda (x1: T).(\lambda 
+(_: (eq T t3 (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i v0 t2 
+x0)).(\lambda (_: (subst0 (s (Flat Cast) i) v0 t1 x1)).(H3 v0 x0 i H8 b d v 
+H5)))))) H6)) (subst0_gen_head (Flat Cast) v0 t2 t1 t3 i H4)))))))))))))))))) 
+c t u H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/sty0.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/sty0.ma
new file mode 100644 (file)
index 0000000..1383143
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/sty0.ma
@@ -0,0 +1,230 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/pr3_props.ma".
+
+include "basic_1A/sty0/fwd.ma".
+
+lemma ty3_sty0:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u 
+t1) \to (\forall (t2: T).((sty0 g c u t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: 
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: 
+T).(\forall (t2: T).((sty0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: 
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda 
+(_: ((\forall (t3: T).((sty0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda 
+(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: 
+((\forall (t4: T).((sty0 g c0 u0 t4) \to (ty3 g c0 u0 t4))))).(\lambda (_: 
+(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (sty0 g c0 u0 t0)).(H3 t0 
+H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda 
+(H0: (sty0 g c0 (TSort m) t2)).(let H_y \def (sty0_gen_sort g c0 t2 m H0) in 
+(let H1 \def (f_equal T T (\lambda (e: T).e) t2 (TSort (next g m)) H_y) in 
+(eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) 
+(ty3_sort g c0 m) t2 H1))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) 
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall 
+(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda 
+(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in 
+(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C 
+T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g 
+e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift 
+(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O 
+t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: 
+(eq T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
+(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (f_equal 
+C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) 
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in 
+((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u0) 
+(CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead 
+x0 (Bind Abbr) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abbr) t0))) H10 u0 
+H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 
+u0 H12) in (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 
+(Bind Abbr) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: 
+C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abbr g n c0 d u0 H16 x2 (H2 x2 
+H17)))))))) H11))) t2 H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) 
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
+(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
+u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abst) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq 
+T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
+(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (eq_ind 
+C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind 
+Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) 
+x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O x1)) H11))) t2 
+H9)))))))) H5)) H4))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 t)).(\lambda (_: ((\forall 
+(t2: T).((sty0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda 
+(H3: (sty0 g c0 (TLRef n) t2)).(let H_x \def (sty0_gen_lref g c0 t2 n H3) in 
+(let H4 \def H_x in (or_ind (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0)))))) (ex3_3 C 
+T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g 
+e u1 t0)))) (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift 
+(S n) O u1)))))) (ty3 g c0 (TLRef n) t2) (\lambda (H5: (ex3_3 C T T (\lambda 
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) 
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(eq T t2 (lift (S n) O 
+t0))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (_: 
+T).(\lambda (t0: T).(eq T t2 (lift (S n) O t0))))) (ty3 g c0 (TLRef n) t2) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abbr) x1))).(\lambda (_: (sty0 g x0 x1 x2)).(\lambda (H8: (eq 
+T t2 (lift (S n) O x2))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
+(lift (S n) O x2) H8) in (eq_ind_r T (lift (S n) O x2) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H11 \def (eq_ind 
+C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind 
+Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) 
+x1) H6)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O x2)) H11))) t2 
+H9)))))))) H5)) (\lambda (H5: (ex3_3 C T T (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: 
+C).(\lambda (u1: T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
+u1))))))).(ex3_3_ind C T T (\lambda (e: C).(\lambda (u1: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: 
+T).(\lambda (t0: T).(sty0 g e u1 t0)))) (\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(eq T t2 (lift (S n) O u1))))) (ty3 g c0 (TLRef n) t2) 
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H6: (getl n c0 
+(CHead x0 (Bind Abst) x1))).(\lambda (H7: (sty0 g x0 x1 x2)).(\lambda (H8: 
+(eq T t2 (lift (S n) O x1))).(let H9 \def (f_equal T T (\lambda (e: T).e) t2 
+(lift (S n) O x1) H8) in (eq_ind_r T (lift (S n) O x1) (\lambda (t0: T).(ty3 
+g c0 (TLRef n) t0)) (let H10 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda 
+(c1: C).(getl n c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d 
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H11 \def (f_equal 
+C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) 
+\Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1) 
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in 
+((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u0) 
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead 
+x0 (Bind Abst) x1) H6)) in (\lambda (H13: (eq C d x0)).(let H14 \def 
+(eq_ind_r T x1 (\lambda (t0: T).(getl n c0 (CHead x0 (Bind Abst) t0))) H10 u0 
+H12) in (let H15 \def (eq_ind_r T x1 (\lambda (t0: T).(sty0 g x0 t0 x2)) H7 
+u0 H12) in (eq_ind T u0 (\lambda (t0: T).(ty3 g c0 (TLRef n) (lift (S n) O 
+t0))) (let H16 \def (eq_ind_r C x0 (\lambda (c1: C).(getl n c0 (CHead c1 
+(Bind Abst) u0))) H14 d H13) in (let H17 \def (eq_ind_r C x0 (\lambda (c1: 
+C).(sty0 g c1 u0 x2)) H15 d H13) in (ty3_abst g n c0 d u0 H16 t H1))) x1 
+H12))))) H11))) t2 H9)))))))) H5)) H4))))))))))))) (\lambda (c0: C).(\lambda 
+(u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall 
+(t2: T).((sty0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 
+t3)).(\lambda (H3: ((\forall (t4: T).((sty0 g (CHead c0 (Bind b) u0) t2 t4) 
+\to (ty3 g (CHead c0 (Bind b) u0) t2 t4))))).(\lambda (t0: T).(\lambda (H4: 
+(sty0 g c0 (THead (Bind b) u0 t2) t0)).(let H_x \def (sty0_gen_bind g b c0 u0 
+t2 t0 H4) in (let H5 \def H_x in (ex2_ind T (\lambda (t4: T).(sty0 g (CHead 
+c0 (Bind b) u0) t2 t4)) (\lambda (t4: T).(eq T t0 (THead (Bind b) u0 t4))) 
+(ty3 g c0 (THead (Bind b) u0 t2) t0) (\lambda (x: T).(\lambda (H6: (sty0 g 
+(CHead c0 (Bind b) u0) t2 x)).(\lambda (H7: (eq T t0 (THead (Bind b) u0 
+x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind b) u0 x) 
+H7) in (eq_ind_r T (THead (Bind b) u0 x) (\lambda (t4: T).(ty3 g c0 (THead 
+(Bind b) u0 t2) t4)) (ty3_bind g c0 u0 t H0 b t2 x (H3 x H6)) t0 H8))))) 
+H5))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda 
+(H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((sty0 g c0 w t2) \to 
+(ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v 
+(THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((sty0 g c0 v t2) 
+\to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (sty0 g c0 (THead 
+(Flat Appl) w v) t2)).(let H_x \def (sty0_gen_appl g c0 w v t2 H4) in (let H5 
+\def H_x in (ex2_ind T (\lambda (t3: T).(sty0 g c0 v t3)) (\lambda (t3: 
+T).(eq T t2 (THead (Flat Appl) w t3))) (ty3 g c0 (THead (Flat Appl) w v) t2) 
+(\lambda (x: T).(\lambda (H6: (sty0 g c0 v x)).(\lambda (H7: (eq T t2 (THead 
+(Flat Appl) w x))).(let H8 \def (f_equal T T (\lambda (e: T).e) t2 (THead 
+(Flat Appl) w x) H7) in (eq_ind_r T (THead (Flat Appl) w x) (\lambda (t0: 
+T).(ty3 g c0 (THead (Flat Appl) w v) t0)) (let H_y \def (H3 x H6) in (let H9 
+\def (ty3_unique g c0 v x H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T 
+(\lambda (t0: T).(ty3 g c0 x t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead 
+(Flat Appl) w x)) (\lambda (x0: T).(\lambda (H10: (ty3 g c0 x x0)).(ex_ind T 
+(\lambda (t0: T).(ty3 g c0 u0 t0)) (ty3 g c0 (THead (Flat Appl) w v) (THead 
+(Flat Appl) w x)) (\lambda (x1: T).(\lambda (_: (ty3 g c0 u0 x1)).(ex_ind T 
+(\lambda (t0: T).(ty3 g c0 (THead (Bind Abst) u0 t) t0)) (ty3 g c0 (THead 
+(Flat Appl) w v) (THead (Flat Appl) w x)) (\lambda (x2: T).(\lambda (H12: 
+(ty3 g c0 (THead (Bind Abst) u0 t) x2)).(ex3_2_ind T T (\lambda (t3: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t3) x2))) (\lambda (_: 
+T).(\lambda (t0: T).(ty3 g c0 u0 t0))) (\lambda (t3: T).(\lambda (_: T).(ty3 
+g (CHead c0 (Bind Abst) u0) t t3))) (ty3 g c0 (THead (Flat Appl) w v) (THead 
+(Flat Appl) w x)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 
+(THead (Bind Abst) u0 x3) x2)).(\lambda (H14: (ty3 g c0 u0 x4)).(\lambda 
+(H15: (ty3 g (CHead c0 (Bind Abst) u0) t x3)).(ty3_conv g c0 (THead (Flat 
+Appl) w x) (THead (Flat Appl) w (THead (Bind Abst) u0 x3)) (ty3_appl g c0 w 
+u0 H0 x x3 (ty3_sconv g c0 x x0 H10 (THead (Bind Abst) u0 t) (THead (Bind 
+Abst) u0 x3) (ty3_bind g c0 u0 x4 H14 Abst t x3 H15) H9)) (THead (Flat Appl) 
+w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v 
+t H2) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) x (ty3_unique g c0 v (THead 
+(Bind Abst) u0 t) H2 x H_y) w Appl))))))) (ty3_gen_bind g Abst c0 u0 t x2 
+H12)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 
+w u0 H0)))) (ty3_correct g c0 v x H_y)))) t2 H8))))) H5)))))))))))))) 
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 
+t3)).(\lambda (H1: ((\forall (t4: T).((sty0 g c0 t2 t4) \to (ty3 g c0 t2 
+t4))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: 
+((\forall (t4: T).((sty0 g c0 t3 t4) \to (ty3 g c0 t3 t4))))).(\lambda (t4: 
+T).(\lambda (H4: (sty0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H_x \def 
+(sty0_gen_cast g c0 t3 t2 t4 H4) in (let H5 \def H_x in (ex3_2_ind T T 
+(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 t3 v2))) (\lambda (_: T).(\lambda 
+(t5: T).(sty0 g c0 t2 t5))) (\lambda (v2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Cast) v2 t5)))) (ty3 g c0 (THead (Flat Cast) t3 t2) t4) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H6: (sty0 g c0 t3 x0)).(\lambda (H7: (sty0 g c0 
+t2 x1)).(\lambda (H8: (eq T t4 (THead (Flat Cast) x0 x1))).(let H9 \def 
+(f_equal T T (\lambda (e: T).e) t4 (THead (Flat Cast) x0 x1) H8) in (eq_ind_r 
+T (THead (Flat Cast) x0 x1) (\lambda (t: T).(ty3 g c0 (THead (Flat Cast) t3 
+t2) t)) (let H_y \def (H1 x1 H7) in (let H_y0 \def (H3 x0 H6) in (let H10 
+\def (ty3_unique g c0 t2 x1 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 
+x0 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 x1)) 
+(\lambda (x: T).(\lambda (H11: (ty3 g c0 x0 x)).(ex_ind T (\lambda (t: 
+T).(ty3 g c0 x1 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) x0 
+x1)) (\lambda (x2: T).(\lambda (H12: (ty3 g c0 x1 x2)).(ty3_conv g c0 (THead 
+(Flat Cast) x0 x1) (THead (Flat Cast) x x0) (ty3_cast g c0 x1 x0 (ty3_sconv g 
+c0 x1 x2 H12 t3 x0 H_y0 H10) x H11) (THead (Flat Cast) t3 t2) (THead (Flat 
+Cast) x0 t3) (ty3_cast g c0 t2 t3 H0 x0 H_y0) (pc3_thin_dx c0 t3 x1 
+(ty3_unique g c0 t2 t3 H0 x1 H_y) x0 Cast)))) (ty3_correct g c0 t2 x1 H_y)))) 
+(ty3_correct g c0 t3 x0 H_y0))))) t4 H9))))))) H5))))))))))))) c u t1 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/ty3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1A/ty3/subst1.ma
new file mode 100644 (file)
index 0000000..bc80f99
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/ty3/subst1.ma
@@ -0,0 +1,1095 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/props.ma".
+
+include "basic_1A/pc3/subst1.ma".
+
+include "basic_1A/getl/getl.ma".
+
+lemma ty3_gen_cabbr:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
+(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
+e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: 
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T 
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u: 
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e: 
+C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) 
+\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d 
+a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0: 
+T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr) 
+u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a: 
+C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d 
+u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e 
+u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d 
+x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a 
+x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u 
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9 
+H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0 
+H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) 
+d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort 
+m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort 
+(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: 
+T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort 
+m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: 
+T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m))) 
+(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g 
+a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: 
+T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
+C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e 
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: 
+nat).(getl n0 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) 
+(getl_conf_le d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 
+(le_S_n n d0 (le_S_n (S n) (S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 
+H6))))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda 
+(e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) u) e2)) (\lambda (e2: 
+C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 
+(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 
+u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 
+(CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let H10 \def 
+(eq_ind nat (minus d0 n) (\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind 
+Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 \def 
+(csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T 
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind Abbr) u2)))) 
+(\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda 
+(_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift 
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind Abbr) 
+x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14: 
+(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c1: 
+C).(getl n a0 c1)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind 
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 (S (plus n (minus d0 (S 
+n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18: 
+(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S 
+n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t0: T).(subst1 (minus d0 (S 
+n)) u0 u t0)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 
+e u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u 
+(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift 
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1 
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S 
+n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S 
+n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0 
+(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4 
+x5)).(let H25 \def (eq_ind T x4 (\lambda (t0: T).(ty3 g x3 t0 x5)) H24 x2 
+(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r 
+nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S 
+n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O) 
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
+u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5) 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0)) 
+(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) 
+d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda 
+(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0)) 
+(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n) 
+H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n))) 
+(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus 
+d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0 
+H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt 
+Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11)))))) 
+(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda 
+(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n0: nat).(drop (S 
+O) n0 a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n0: 
+nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 
+(\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind 
+nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d 
+(Bind Abbr) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Abbr) u0) 
+(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in 
+(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 with [(CSort _) 
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) 
+(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e 
+(Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T (\lambda (e0: C).(match 
+e0 with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d 
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) 
+n H0 (CHead e (Bind Abbr) u0) H9)) in (\lambda (H13: (eq C d e)).(let H14 
+\def (eq_ind_r T u0 (\lambda (t0: T).(getl n c0 (CHead e (Bind Abbr) t0))) 
+H10 u H12) in (let H15 \def (eq_ind_r T u0 (\lambda (t0: T).(csubst1 n t0 c0 
+a0)) H8 u H12) in (eq_ind T u (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 n t0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n t0 (lift (S n) O t) (lift (S O) n y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (eq_ind_r 
+C e (\lambda (c1: C).(getl n c0 (CHead c1 (Bind Abbr) u))) H14 d H13) in 
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 n u (TLRef n) (lift 
+(S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n u (lift (S n) O t) 
+(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(lift n O u) (lift n O t) (subst1_single n u (TLRef n) (lift (S O) n (lift n 
+O u)) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t0: T).(subst0 n u 
+(TLRef n) t0)) (subst0_lref u n) (lift (S O) n (lift n O u)) (lift_free u n 
+(S O) O n (le_plus_r O n) (le_O_n n)))) (eq_ind_r T (lift (plus (S O) n) O t) 
+(\lambda (t0: T).(subst1 n u (lift (S n) O t) t0)) (subst1_refl n u (lift (S 
+n) O t)) (lift (S O) n (lift n O t)) (lift_free t n (S O) O n (le_plus_r O n) 
+(le_O_n n))) (ty3_lift g d u t H1 a O n (getl_conf_ge_drop Abbr a0 d u n 
+(csubst1_getl_ge n n (le_n n) c0 a0 u H15 (CHead d (Bind Abbr) u) H16) a 
+H7)))) u0 H12))))) H11))) d0 H6))))) (\lambda (H6: (lt d0 n)).(eq_ind_r nat 
+(S (plus O (minus n (S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) 
+(minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S O)) 
+(S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 
+d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) (lift 
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O 
+t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) 
+(S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n (S O)) (S O))) 
+t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 
+(TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus 
+d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 
+u0 (lift (S n) O t) t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 
+(lift n O t)) (lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) 
+(plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: 
+nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S 
+O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 
+n (le_S_n d0 n (le_S_n (S d0) (S n) (le_S (S (S d0)) (S n) (le_n_S (S d0) n 
+H6)))) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat 
+(plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S O)) (plus_sym d0 
+(S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H6)))) (plus 
+(S O) (minus n (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n 
+(S O)))) (refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
+(le_lt_trans O d0 n (le_O_n d0) H6))))))))))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
+d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d0 u0 d a0) \to 
+(\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
+(u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Abbr) 
+u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 a0)).(\lambda (a: 
+C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) 
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H6: 
+(lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
+(CHead d (Bind Abst) u) (CHead e (Bind Abbr) u0))) (getl_conf_le d0 (CHead e 
+(Bind Abbr) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S_n (S n) 
+(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H6))))) (S (minus d0 (S n))) 
+(minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 n) u0 
+(CHead d (Bind Abst) u) e2)) (\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift 
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x: C).(\lambda (H8: (csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) 
+x)).(\lambda (H9: (getl n a0 x)).(let H10 \def (eq_ind nat (minus d0 n) 
+(\lambda (n0: nat).(csubst1 n0 u0 (CHead d (Bind Abst) u) x)) H8 (S (minus d0 
+(S n))) (minus_x_Sy d0 n H6)) in (let H11 \def (csubst1_gen_head (Bind Abst) 
+d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T C (\lambda (u2: T).(\lambda 
+(c2: C).(eq C x (CHead c2 (Bind Abst) u2)))) (\lambda (u2: T).(\lambda (_: 
+C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda (_: T).(\lambda (c2: 
+C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
+C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst) x0))).(\lambda (H13: (subst1 
+(minus d0 (S n)) u0 u x0)).(\lambda (H14: (csubst1 (minus d0 (S n)) u0 d 
+x1)).(let H15 \def (eq_ind C x (\lambda (c1: C).(getl n a0 c1)) H9 (CHead x1 
+(Bind Abst) x0) H12) in (let H16 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
+(S O) n0 a0 a)) H5 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H6)) in 
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T x0 (lift (S O) (minus d0 
+(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) x1 e0))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S 
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H17: (eq T x0 (lift (S O) (minus 
+d0 (S n)) x2))).(\lambda (H18: (getl n a (CHead x3 (Bind Abst) x2))).(\lambda 
+(H19: (drop (S O) (minus d0 (S n)) x1 x3)).(let H20 \def (eq_ind T x0 
+(\lambda (t0: T).(subst1 (minus d0 (S n)) u0 u t0)) H13 (lift (S O) (minus d0 
+(S n)) x2) H17) in (let H21 \def (H2 e u0 (minus d0 (S n)) (getl_gen_S (Bind 
+Abst) d (CHead e (Bind Abbr) u0) u (minus d0 (S n)) H7) x1 H14 x3 H19) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u 
+(lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
+(minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g x3 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H22: (subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n)) 
+x4))).(\lambda (_: (subst1 (minus d0 (S n)) u0 t (lift (S O) (minus d0 (S n)) 
+x5))).(\lambda (H24: (ty3 g x3 x4 x5)).(let H25 \def (eq_ind T x4 (\lambda 
+(t0: T).(ty3 g x3 t0 x5)) H24 x2 (subst1_confluence_lift u x4 u0 (minus d0 (S 
+n)) H22 x2 H20)) in (eq_ind_r nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: 
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0 u0 (lift (S 
+n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (eq_ind_r nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n)) 
+u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O) 
+(plus (S n) (minus d0 (S n))) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))) (TLRef n) (lift (S n) O x2) (eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(subst1 d0 u0 (TLRef n) t0)) (subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 
+(TLRef n)) (lift_lref_lt n (S O) d0 H6)) (eq_ind_r T (lift (S n) O (lift (S 
+O) (minus d0 (S n)) x2)) (\lambda (t0: T).(subst1 (plus (minus d0 (S n)) (S 
+n)) u0 (lift (S n) O u) t0)) (subst1_lift_ge u (lift (S O) (minus d0 (S n)) 
+x2) u0 (minus d0 (S n)) (S n) H20 O (le_O_n (minus d0 (S n)))) (lift (S O) 
+(plus (S n) (minus d0 (S n))) (lift (S n) O x2)) (lift_d x2 (S O) (S n) 
+(minus d0 (S n)) O (le_O_n (minus d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 
+H25)) d0 (le_plus_minus (S n) d0 H6)) d0 (le_plus_minus_sym (S n) d0 
+H6)))))))) H21)))))))) (getl_drop_conf_lt Abst a0 x1 x0 n H15 a (S O) (minus 
+d0 (S n)) H16))))))))) H11)))))) (csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead 
+d (Bind Abst) u) H0)))) (\lambda (H6: (eq nat n d0)).(let H7 \def (eq_ind_r 
+nat d0 (\lambda (n0: nat).(drop (S O) n0 a0 a)) H5 n H6) in (let H8 \def 
+(eq_ind_r nat d0 (\lambda (n0: nat).(csubst1 n0 u0 c0 a0)) H4 n H6) in (let 
+H9 \def (eq_ind_r nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Abbr) 
+u0))) H3 n H6) in (eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 n0 u0 (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) n0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C 
+(CHead d (Bind Abst) u) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind 
+Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) 
+H9)) in (let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow 
+(match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow False | 
+Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow 
+False])])) I (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n 
+H0 (CHead e (Bind Abbr) u0) H9)) in (False_ind (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 n u0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 n u0 (lift (S n) O u) (lift (S O) n y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H11))) d0 H6))))) 
+(\lambda (H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda 
+(n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef 
+n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 
+(lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: 
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S 
+n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (eq_ind_r nat (plus (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift 
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) 
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus 
+(minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O u) 
+(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(subst1 d0 
+u0 (TLRef (plus (minus n (S O)) (S O))) t0)) (subst1_refl d0 u0 (TLRef (plus 
+(minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H6))) (eq_ind_r T 
+(lift (plus (S O) n) O u) (\lambda (t0: T).(subst1 d0 u0 (lift (S n) O u) 
+t0)) (subst1_refl d0 u0 (lift (S n) O u)) (lift (S O) d0 (lift n O u)) 
+(lift_free u n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H6)) 
+(le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a 
+(TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u 
+(getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 n (le_S_n 
+d0 n (le_S_n (S d0) (S n) (le_S (S (S d0)) (S n) (le_n_S (S d0) n H6)))) c0 
+a0 u0 H4 (CHead d (Bind Abst) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) 
+d0) (\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S O)) (plus_sym d0 (S O)))) t 
+H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H6)))) (plus (S O) (minus 
+n (S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
+(le_lt_trans O d0 n (le_O_n d0) H6))))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: 
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: ((\forall 
+(e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) 
+(CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 (CHead c0 (Bind 
+b) u) a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
+(u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) 
+u0))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u0 c0 a0)).(\lambda (a: 
+C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H1 e u0 d H4 a0 H5 a H6) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S 
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 u (lift (S O) d 
+x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d x1))).(\lambda (H10: (ty3 g a 
+x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
+Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u 
+(lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
+a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S 
+d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S 
+d) u0 t4 (lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+(CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S O) (S d) 
+x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) x3))).(\lambda 
+(H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S 
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Bind 
+b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) (lift (S O) d 
+x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u 
+t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 (lift (S O) (S 
+d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) 
+d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) 
+(\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head u0 u 
+(lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S O) d 
+(THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 H10 
+b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: 
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: 
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let 
+H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d 
+x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d 
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w 
+(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d 
+x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind 
+Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S 
+O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u 
+x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S 
+O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) 
+x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 
+(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 
+(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 
+(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 
+x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead 
+(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in 
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat 
+Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead 
+(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) 
+(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead 
+(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v 
+(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat 
+Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift 
+(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d 
+x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind 
+Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S 
+d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) 
+t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t 
+(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) 
+(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead 
+(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) 
+(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S 
+O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda 
+(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 
+t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl 
+d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S 
+O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda 
+(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let 
+H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead 
+(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d 
+u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def 
+(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift 
+t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) 
+x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d 
+x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) 
+t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) 
+H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) 
+(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda 
+(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S 
+O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat 
+Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 
+H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
+
+lemma ty3_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c 
+(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T 
+T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead 
+e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3: 
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: 
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to 
+(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u 
+t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
+d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4 
+t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d 
+c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0 
+a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9: 
+(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def 
+(eq_ind T t4 (\lambda (t0: T).(pc3 c0 t0 t3)) H4 (lift (S O) d x1) H9) in 
+(let H12 \def (eq_ind T t4 (\lambda (t0: T).(ty3 g c0 u t0)) H2 (lift (S O) d 
+x1) H9) in (let H13 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S 
+O) d x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0 
+d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: 
+(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d 
+x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t0: 
+T).(ty3 g c0 t3 t0)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3 
+(\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x3))) H18 (lift (S O) d x2) H15) 
+in (let H20 \def (eq_ind T t3 (\lambda (t0: T).(pc3 c0 (lift (S O) d x1) t0)) 
+H11 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift 
+(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1 
+H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u 
+H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda 
+(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e 
+(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0 
+a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m)) 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T 
+(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m 
+(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g 
+m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m))) 
+(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
+d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: 
+T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) 
+u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt 
+n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
+(CHead d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e 
+(Bind Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S_n (S n) 
+(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H5))))) (S (minus d0 (S n))) 
+(minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
+(S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H5)) in 
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift (S O) (minus d0 
+(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abbr) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) d e0))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: C).(\lambda (H8: (eq T u (lift (S O) (minus d0 (S n)) 
+x0))).(\lambda (H9: (getl n a (CHead x1 (Bind Abbr) x0))).(\lambda (H10: 
+(drop (S O) (minus d0 (S n)) d x1)).(let H11 \def (eq_ind T u (\lambda (t0: 
+T).(\forall (e0: C).(\forall (u1: T).(\forall (d1: nat).((getl d1 d (CHead e0 
+(Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d1 d a0) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S O) d1 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t (lift (S O) d1 y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift (S O) (minus d0 (S n)) x0) H8) in 
+(let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g d t0 t)) H1 (lift (S O) 
+(minus d0 (S n)) x0) H8) in (let H13 \def (H11 e u0 (minus d0 (S n)) 
+(getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) u (minus d0 (S n)) H6) x1 
+H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) 
+(minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: 
+(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) 
+x2))).(\lambda (H15: (eq T t (lift (S O) (minus d0 (S n)) x3))).(\lambda 
+(H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t (\lambda (t0: T).(ty3 g d 
+(lift (S O) (minus d0 (S n)) x0) t0)) H12 (lift (S O) (minus d0 (S n)) x3) 
+H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) (\lambda (t0: T).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H18 \def 
+(eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S 
+O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S 
+n))) (lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O x3)) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x3)) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T 
+(TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) 
+(lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O 
+x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) 
+(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3)) 
+(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t 
+H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0 
+(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r 
+nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in 
+(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u) 
+(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 
+(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def 
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow True | Abst \Rightarrow False | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
+Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) 
+H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
+n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+t) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S 
+O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift 
+(S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus 
+(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S 
+O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda 
+(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef 
+(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T 
+(lift (plus (S O) n) O t) (\lambda (t0: T).(eq T (lift (S n) O t) t0)) 
+(refl_equal T (lift (S n) O t)) (lift (S O) d0 (lift n O t)) (lift_free t n 
+(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) 
+(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
+(S O))) (lift n0 O t))) (ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge 
+n (CHead d (Bind Abbr) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) 
+n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
+(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
+(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n 
+c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl 
+d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: 
+T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void) 
+u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt 
+n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n0: nat).(getl n0 
+(CHead d (Bind Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e 
+(Bind Void) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S_n (S n) 
+(S d0) (le_S (S (S n)) (S d0) (le_n_S (S n) d0 H5))))) (S (minus d0 (S n))) 
+(minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat d0 (\lambda (n0: nat).(drop 
+(S O) n0 c0 a)) H4 (S (plus n (minus d0 (S n)))) (lt_plus_minus n d0 H5)) in 
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift (S O) (minus d0 
+(S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) d e0))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: C).(\lambda (H8: (eq T u (lift (S O) (minus d0 (S n)) 
+x0))).(\lambda (H9: (getl n a (CHead x1 (Bind Abst) x0))).(\lambda (H10: 
+(drop (S O) (minus d0 (S n)) d x1)).(let H11 \def (eq_ind T u (\lambda (t0: 
+T).(\forall (e0: C).(\forall (u1: T).(\forall (d1: nat).((getl d1 d (CHead e0 
+(Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d1 d a0) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S O) d1 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t (lift (S O) d1 y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a0 y1 y2))))))))))) H2 (lift (S O) (minus d0 (S n)) x0) H8) in 
+(let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g d t0 t)) H1 (lift (S O) 
+(minus d0 (S n)) x0) H8) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x0) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) 
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+t0) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (let H13 \def (H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abst) d 
+(CHead e (Bind Void) u0) u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift 
+(S O) (minus d0 (S n)) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift 
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) 
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) 
+(minus d0 (S n)) x0)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T 
+(lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda 
+(H15: (eq T t (lift (S O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 
+x3)).(let H17 \def (eq_ind T t (\lambda (t0: T).(ty3 g d (lift (S O) (minus 
+d0 (S n)) x0) t0)) H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def 
+(eq_ind_r T x2 (\lambda (t0: T).(ty3 g x1 t0 x3)) H16 x0 (lift_inj x0 x2 (S 
+O) (minus d0 (S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S 
+n))) (lift (S n) O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(TLRef n) (lift (S n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T 
+(TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) 
+(lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O 
+x0))) (ty3_abst g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n))) 
+(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x0)) 
+(lift_d x0 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))))))))) 
+H13)) u H8)))))))) (getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S 
+n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 
+(\lambda (n0: nat).(drop (S O) n0 c0 a)) H4 n H5) in (let H7 \def (eq_ind_r 
+nat d0 (\lambda (n0: nat).(getl n0 c0 (CHead e (Bind Void) u0))) H3 n H5) in 
+(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
+T (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abst) u) 
+(\lambda (c1: C).(getl n c0 c1)) H0 (CHead e (Bind Void) u0) (getl_mono c0 
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def 
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee with [(CSort _) 
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind b) 
+\Rightarrow (match b with [Abbr \Rightarrow False | Abst \Rightarrow True | 
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind 
+Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind Void) u0) 
+H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef 
+n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O 
+u) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S 
+O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift 
+(S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus 
+(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O))) (lift (S O) d0 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S 
+O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda 
+(t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef 
+(plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) 
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T 
+(lift (plus (S O) n) O u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) 
+(refl_equal T (lift (S n) O u)) (lift (S O) d0 (lift n O u)) (lift_free u n 
+(S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) 
+(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
+(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge 
+n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) 
+n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
+(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
+(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda 
+(H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda 
+(t4: T).(\lambda (H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: 
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind 
+b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 
+(Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: 
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind 
+Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def 
+(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d 
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S 
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda 
+(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
+(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in 
+(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) 
+H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: 
+T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 
+(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 
+(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
+y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda 
+(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in 
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 
+(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) 
+(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T 
+T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S 
+O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S 
+O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: 
+(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
+(Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) 
+(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind 
+b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead 
+(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) 
+(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) 
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 
+(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u 
+H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda 
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall 
+(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall 
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v 
+(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: 
+C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind 
+Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
+(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: 
+(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) 
+d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d 
+x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: 
+T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in 
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead 
+(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u 
+(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14 
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H9 (THead (Bind Abst) x2 
+x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d 
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
+Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda 
+(t0: T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 c0 
+(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d0 y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d0 y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H1 (lift (S O) d x2) H12) in 
+(eq_ind_r T (lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
+(Bind Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18: 
+(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4 
+x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead 
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r 
+T x5 (\lambda (t0: T).(ty3 g a x4 t0)) H19 x2 (lift_inj x2 x5 (S O) d H18)) 
+in (eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d 
+x4) (lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d 
+(THead (Flat Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind 
+Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) 
+x2 x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4 
+(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4 
+(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 
+x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2 
+x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d 
+x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind 
+Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d 
+x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2) 
+(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u 
+H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7))))))) 
+H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall 
+(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall 
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3: 
+((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind 
+Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda 
+(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a: 
+C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in 
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (lift (S O) d x0))).(\lambda (H8: 
+(eq T t0 (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
+(eq_ind T t0 (\lambda (t: T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in 
+(eq_ind_r T (lift (S O) d x1) (\lambda (t: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) t t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H11 \def 
+(eq_ind T t4 (\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) 
+d x0) H7) in (let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e0: 
+C).(\forall (u0: T).(\forall (d0: nat).((getl d0 c0 (CHead e0 (Bind Void) 
+u0)) \to (\forall (a0: C).((drop (S O) d0 c0 a0) \to (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a0 y1 y2))))))))))) H1 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T t4 
+(\lambda (t: T).(ty3 g c0 t3 t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T 
+(lift (S O) d x0) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T (THead (Flat Cast) t t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) t) (lift (S O) 
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def 
+(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d 
+x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S 
+O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T 
+(THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S O) d x2))).(\lambda 
+(H16: (eq T (lift (S O) d x0) (lift (S O) d x3))).(\lambda (H17: (ty3 g a x2 
+x3)).(let H18 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t (lift (S O) d 
+x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda 
+(t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) 
+(lift (S O) d x0) t) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(eq T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (lift (S O) 
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def 
+(eq_ind_r T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d 
+H16)) in (eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: 
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Cast) (lift (S O) d x1) 
+(lift (S O) d x0)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
+T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat Cast) x1 x0)) 
+(\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) 
+d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g 
+a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S 
+O) d (THead (Flat Cast) x0 x2)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
+(y2: T).(eq T (lift (S O) d (THead (Flat Cast) x1 x0)) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2) 
+(THead (Flat Cast) x1 x0) (refl_equal T (lift (S O) d (THead (Flat Cast) x0 
+x2))) (refl_equal T (lift (S O) d (THead (Flat Cast) x1 x0))) (ty3_cast g a 
+x2 x0 H19 x1 H9)) (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) 
+(lift_flat Cast x1 x0 (S O) d)) (THead (Flat Cast) (lift (S O) d x0) (lift (S 
+O) d x2)) (lift_flat Cast x0 x2 (S O) d))) t3 H15))))))) H14)) t4 H7)))) t0 
+H8))))))) H6)))))))))))))))) c t1 t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/defs.ma
new file mode 100644 (file)
index 0000000..c7dafeb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/defs.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/pr0/defs.ma".
+
+include "basic_1A/C/defs.ma".
+
+inductive wcpr0: C \to (C \to Prop) \def
+| wcpr0_refl: \forall (c: C).(wcpr0 c c)
+| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall 
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k 
+u1) (CHead c2 k u2)))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/fwd.ma
new file mode 100644 (file)
index 0000000..bcdd0ee
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/fwd.ma
@@ -0,0 +1,105 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wcpr0/defs.ma".
+
+implied rec lemma wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P 
+c c))) (f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) 
+\to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P 
+(CHead c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on 
+w: P c c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp 
+c1 c2 w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1 
+u2 p k)].
+
+lemma wcpr0_gen_sort:
+ \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort 
+n))))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) 
+x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C 
+x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: 
+C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c: 
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e: 
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0)) 
+(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda 
+(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2 
+c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C 
+(CHead c1 k u1) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False 
+| (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C 
+(CHead c2 k u2) (CHead c1 k u1)) H5))))))))))) y x H0))) H))).
+
+lemma wcpr0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 
+(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda 
+(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c: 
+C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda 
+(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: 
+C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2: 
+C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))) 
+(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal 
+C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1) 
+(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 
+c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C 
+(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: 
+T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: 
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) 
+(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to 
+(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead 
+c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0 
+k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 
+u0) (CHead c1 k u1) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 
+k0 u0) (CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) 
+(CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K k0 k)).(\lambda 
+(H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C (CHead c2 k1 u2) 
+(CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead 
+c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10 \def (eq_ind T u0 
+(\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1 (\lambda (t: T).(or 
+(eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda (c3: C).(\lambda 
+(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
+(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let 
+H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k u1)) \to (or (eq C 
+c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c2 (CHead c3 k 
+u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def (eq_ind C c0 
+(\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1 (\lambda (c: C).(or 
+(eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda 
+(u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda 
+(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) 
+(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: 
+C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: 
+C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
+u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k 
+u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2 
+k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x H0))) 
+H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/getl.ma
new file mode 100644 (file)
index 0000000..c9f09e7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wcpr0/getl.ma
@@ -0,0 +1,448 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wcpr0/fwd.ma".
+
+include "basic_1A/getl/props.ma".
+
+lemma wcpr0_drop:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead 
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
+c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1) 
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u1) 
+(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in 
+((let H5 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 
+u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u1 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl 
+(CHead c3 k u1) (CHead e1 k0 u0) H3)) in (\lambda (H7: (eq K k k0)).(\lambda 
+(H8: (eq C c3 e1)).(eq_ind K k (\lambda (k1: K).(ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k1 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u0 u3))))) (eq_ind T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t 
+u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 
+u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead 
+c4 k u2) (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 (drop_refl (CHead c4 k 
+u2)) H0 H2) e1 H8) u0 H6) k0 H7)))) H5)) H4)))))) (K_ind (\lambda (k0: 
+K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: 
+K).((drop n O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(drop n O (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 
+u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop 
+(S n) O (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(drop (S n) O (CHead c4 k0 u2) (CHead e2 k1 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 
+(Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
+T).(drop n O (CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
+u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
+(drop (S n) O (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
+u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(drop n O c4 (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c4 
+(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H6: (drop n O c4 (CHead x0 k0 x1))).(\lambda 
+(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Bind b) n c4 (CHead x0 k0 x1) H6 u2) 
+H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 
+(Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
+T).(drop n O (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 
+u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
+(drop (S n) O (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S 
+n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c4 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop (S n) O c4 
+(CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop 
+(Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) 
+c1 c2 H))).
+
+lemma wcpr0_drop_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead 
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 
+u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead 
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead 
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead 
+c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2) 
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with 
+[(CSort _) \Rightarrow c4 | (CHead c _ _) \Rightarrow c])) (CHead c4 k u2) 
+(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in 
+((let H5 \def (f_equal C K (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 
+u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def 
+(f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead 
+_ _ t) \Rightarrow t])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl 
+(CHead c4 k u2) (CHead e1 k0 u0) H3)) in (\lambda (H7: (eq K k k0)).(\lambda 
+(H8: (eq C c4 e1)).(eq_ind K k (\lambda (k1: K).(ex3_2 C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k1 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u0))))) (eq_ind T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: 
+T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) 
+(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) 
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 
+H2) e1 H8) u0 H6) k0 H7)))) H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall 
+(n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop n O 
+(CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u4: T).(drop n O (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
+u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S 
+n) O (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(drop (S n) O (CHead c3 k0 u1) (CHead e2 k1 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 
+(Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
+T).(drop n O (CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
+(drop (S n) O (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 
+u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T 
+(\lambda (e2: C).(\lambda (u3: T).(drop n O c3 (CHead e2 k0 u3)))) (\lambda 
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 
+(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H6: (drop n O c3 (CHead x0 k0 x1))).(\lambda 
+(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: 
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Bind b) n c3 (CHead x0 k0 x1) H6 u1) 
+H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: 
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 
+(Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: 
+T).(drop n O (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: 
+(drop (S n) O (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S 
+n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in 
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop (S n) O c3 
+(CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead 
+c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop 
+(Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) 
+c2 c1 H))).
+
+lemma wcpr0_getl:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k 
+u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1) 
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1 
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1 
+u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda 
+(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
+C (\lambda (e: C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) 
+\Rightarrow c])) (CHead e1 k0 u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 
+(CHead e1 k0 u0) u1 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 
+k0 u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) 
+in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c3 
+(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in (\lambda (H8: 
+(eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K (Bind b) (\lambda 
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind 
+b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: 
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind b) 
+u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c3 (\lambda 
+(c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind 
+b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Bind b) u2) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2 (getl_refl b c4 u2) H0 H2) e1 H9) u0 
+H7) k0 H8)))) H6)) H5)))) (\lambda (f: F).(\lambda (H4: (clear (CHead c3 
+(Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 
+(CHead e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u1 
+H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c4 (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c4 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f) u2) (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f 
+u2) H7 H8)))))) H5)))) k (getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) 
+H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((getl n (CHead c3 k0 u1) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k0 
+u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u1) (CHead e1 
+k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead 
+c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: 
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((getl n (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 (Bind b) u2) (CHead e2 k0 
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u1) (CHead 
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 
+u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c4 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead 
+x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let 
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) 
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead 
+x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4 
+(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat 
+f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 
+H))).
+
+lemma wcpr0_getl_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: 
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall 
+(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda 
+(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda 
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k 
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl 
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3 
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: 
+T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda 
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall 
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1 
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k 
+u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: 
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2) 
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1 
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 
+u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda 
+(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C 
+C (\lambda (e: C).(match e with [(CSort _) \Rightarrow e1 | (CHead c _ _) 
+\Rightarrow c])) (CHead e1 k0 u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 
+(CHead e1 k0 u0) u2 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match 
+e with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 
+k0 u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) 
+in ((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c4 
+(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in (\lambda (H8: 
+(eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K (Bind b) (\lambda 
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind 
+b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: 
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) 
+u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c4 (\lambda 
+(c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind 
+b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T 
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u1) (CHead e2 
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1 (getl_refl b c3 u1) H0 H2) e1 H9) u0 
+H7) k0 H8)))) H6)) H5)))) (\lambda (f: F).(\lambda (H4: (clear (CHead c4 
+(Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c4 
+(CHead e1 k0 u0) c4 (drop_refl c4) (clear_gen_flat f c4 (CHead e1 k0 u0) u2 
+H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u1) (CHead 
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f 
+u1) H7 H8)))))) H5)))) k (getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) 
+H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((getl n (CHead c4 k0 u2) (CHead e1 k1 
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 
+u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) 
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1: 
+C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c4 k0 u2) (CHead e1 
+k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead 
+c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: 
+B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall 
+(k0: K).((getl n (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T 
+(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u1) (CHead e2 k0 
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c4 (Bind b) u2) (CHead 
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 
+u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda 
+(u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro 
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) 
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda 
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead 
+x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: 
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl 
+n (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: 
+C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) 
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda 
+(u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: 
+K).(\lambda (H4: (getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let 
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) 
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: 
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: 
+T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: 
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead 
+x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 
+(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat 
+f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 
+H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/clear.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/clear.ma
new file mode 100644 (file)
index 0000000..09abc1f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/clear.ma
@@ -0,0 +1,87 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wf3/fwd.ma".
+
+include "basic_1A/clear/fwd.ma".
+
+lemma wf3_clear_conf:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (g: G).(\forall 
+(c2: C).((wf3 g c1 c2) \to (wf3 g c c2))))))
+\def
+ \lambda (c1: C).(\lambda (c: C).(\lambda (H: (clear c1 c)).(clear_ind 
+(\lambda (c0: C).(\lambda (c2: C).(\forall (g: G).(\forall (c3: C).((wf3 g c0 
+c3) \to (wf3 g c2 c3)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) 
+c2)).H0)))))) (\lambda (e: C).(\lambda (c0: C).(\lambda (_: (clear e 
+c0)).(\lambda (H1: ((\forall (g: G).(\forall (c2: C).((wf3 g e c2) \to (wf3 g 
+c0 c2)))))).(\lambda (f: F).(\lambda (u: T).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (H2: (wf3 g (CHead e (Flat f) u) c2)).(let H_y \def 
+(wf3_gen_flat1 g e c2 u f H2) in (H1 g c2 H_y))))))))))) c1 c H))).
+
+lemma clear_wf3_trans:
+ \forall (c1: C).(\forall (d1: C).((clear c1 d1) \to (\forall (g: G).(\forall 
+(d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda 
+(c2: C).(clear c2 d2))))))))
+\def
+ \lambda (c1: C).(\lambda (d1: C).(\lambda (H: (clear c1 d1)).(clear_ind 
+(\lambda (c: C).(\lambda (c0: C).(\forall (g: G).(\forall (d2: C).((wf3 g c0 
+d2) \to (ex2 C (\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(clear c2 
+d2)))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (g: 
+G).(\lambda (d2: C).(\lambda (H0: (wf3 g (CHead e (Bind b) u) d2)).(let H_x 
+\def (wf3_gen_bind1 g e d2 u b H0) in (let H1 \def H_x in (or_ind (ex3_2 C T 
+(\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda 
+(c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g 
+e u w)))) (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) 
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) 
+\to False)))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) 
+(\lambda (c2: C).(clear c2 d2))) (\lambda (H2: (ex3_2 C T (\lambda (c2: 
+C).(\lambda (_: T).(eq C d2 (CHead c2 (Bind b) u)))) (\lambda (c2: 
+C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g e u 
+w))))).(ex3_2_ind C T (\lambda (c2: C).(\lambda (_: T).(eq C d2 (CHead c2 
+(Bind b) u)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g e c2))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g e u w))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e 
+(Bind b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H3: (eq C d2 (CHead x0 (Bind b) u))).(\lambda (H4: (wf3 g e 
+x0)).(\lambda (H5: (ty3 g e u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda 
+(c: C).(ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: 
+C).(clear c2 c)))) (ex_intro2 C (\lambda (c2: C).(wf3 g (CHead e (Bind b) u) 
+c2)) (\lambda (c2: C).(clear c2 (CHead x0 (Bind b) u))) (CHead x0 (Bind b) u) 
+(wf3_bind g e x0 H4 u x1 H5 b) (clear_bind b x0 u)) d2 H3)))))) H2)) (\lambda 
+(H2: (ex3 C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) (TSort O)))) 
+(\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: T).((ty3 g e u w) 
+\to False))))).(ex3_ind C (\lambda (c2: C).(eq C d2 (CHead c2 (Bind Void) 
+(TSort O)))) (\lambda (c2: C).(wf3 g e c2)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g e u w) \to False))) (ex2 C (\lambda (c2: C).(wf3 g (CHead e (Bind 
+b) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda (x0: C).(\lambda (H3: 
+(eq C d2 (CHead x0 (Bind Void) (TSort O)))).(\lambda (H4: (wf3 g e 
+x0)).(\lambda (H5: ((\forall (w: T).((ty3 g e u w) \to False)))).(eq_ind_r C 
+(CHead x0 (Bind Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (c2: C).(wf3 
+g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 c)))) (ex_intro2 C 
+(\lambda (c2: C).(wf3 g (CHead e (Bind b) u) c2)) (\lambda (c2: C).(clear c2 
+(CHead x0 (Bind Void) (TSort O)))) (CHead x0 (Bind Void) (TSort O)) (wf3_void 
+g e x0 H4 u H5 b) (clear_bind Void x0 (TSort O))) d2 H3))))) H2)) H1))))))))) 
+(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1: 
+((\forall (g: G).(\forall (d2: C).((wf3 g c d2) \to (ex2 C (\lambda (c2: 
+C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)))))))).(\lambda (f: 
+F).(\lambda (u: T).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g c 
+d2)).(let H_x \def (H1 g d2 H2) in (let H3 \def H_x in (ex2_ind C (\lambda 
+(c2: C).(wf3 g e c2)) (\lambda (c2: C).(clear c2 d2)) (ex2 C (\lambda (c2: 
+C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 d2))) (\lambda 
+(x: C).(\lambda (H4: (wf3 g e x)).(\lambda (H5: (clear x d2)).(ex_intro2 C 
+(\lambda (c2: C).(wf3 g (CHead e (Flat f) u) c2)) (\lambda (c2: C).(clear c2 
+d2)) x (wf3_flat g e x H4 u f) H5)))) H3)))))))))))) c1 d1 H))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/defs.ma
new file mode 100644 (file)
index 0000000..263f236
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/defs.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/ty3/defs.ma".
+
+inductive wf3 (g: G): C \to (C \to Prop) \def
+| wf3_sort: \forall (m: nat).(wf3 g (CSort m) (CSort m))
+| wf3_bind: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
+T).(\forall (t: T).((ty3 g c1 u t) \to (\forall (b: B).(wf3 g (CHead c1 (Bind 
+b) u) (CHead c2 (Bind b) u))))))))
+| wf3_void: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
+T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(wf3 g 
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))
+| wf3_flat: \forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: 
+T).(\forall (f: F).(wf3 g (CHead c1 (Flat f) u) c2))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/fwd.ma
new file mode 100644 (file)
index 0000000..972e33f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/fwd.ma
@@ -0,0 +1,377 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wf3/defs.ma".
+
+implied rec lemma wf3_ind (g: G) (P: (C \to (C \to Prop))) (f: (\forall (m: 
+nat).(P (CSort m) (CSort m)))) (f0: (\forall (c1: C).(\forall (c2: C).((wf3 g 
+c1 c2) \to ((P c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c1 u t) \to 
+(\forall (b: B).(P (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))))))))))) (f1: 
+(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall 
+(u: T).(((\forall (t: T).((ty3 g c1 u t) \to False))) \to (\forall (b: B).(P 
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O))))))))))) (f2: (\forall 
+(c1: C).(\forall (c2: C).((wf3 g c1 c2) \to ((P c1 c2) \to (\forall (u: 
+T).(\forall (f2: F).(P (CHead c1 (Flat f2) u) c2)))))))) (c: C) (c0: C) (w: 
+wf3 g c c0) on w: P c c0 \def match w with [(wf3_sort m) \Rightarrow (f m) | 
+(wf3_bind c1 c2 w0 u t t0 b) \Rightarrow (f0 c1 c2 w0 ((wf3_ind g P f f0 f1 
+f2) c1 c2 w0) u t t0 b) | (wf3_void c1 c2 w0 u f3 b) \Rightarrow (f1 c1 c2 w0 
+((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3 b) | (wf3_flat c1 c2 w0 u f3) 
+\Rightarrow (f2 c1 c2 w0 ((wf3_ind g P f f0 f1 f2) c1 c2 w0) u f3)].
+
+lemma wf3_gen_sort1:
+ \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to 
+(eq C x (CSort m)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort 
+m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c: 
+C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda 
+(c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0: 
+nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat 
+(\lambda (e: C).(match e with [(CSort n) \Rightarrow n | (CHead _ _ _) 
+\Rightarrow m0])) (CSort m0) (CSort m) H1) in (eq_ind_r nat m (\lambda (n: 
+nat).(eq C (CSort n) (CSort n))) (refl_equal C (CSort m)) m0 H2)))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 
+(CSort m)) \to (eq C c2 c1)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) 
+(CSort m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: 
+C).(match ee with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow 
+True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 
+(Bind b) u)) H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: 
+(wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 
+c1)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to 
+False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort 
+m))).(let H5 \def (eq_ind C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee 
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I 
+(CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O)) (CHead c1 
+(Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 
+g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u: 
+T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat f) u) (CSort m))).(let 
+H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m) 
+H3) in (False_ind (eq C c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
+
+lemma wf3_gen_bind1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b: 
+B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2: 
+C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda 
+(_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 
+C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: 
+C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to 
+False))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b: 
+B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind 
+b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda 
+(c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: 
+C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) 
+(\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g 
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or 
+(ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v)))) 
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void) 
+(TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C 
+(CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda 
+(ee: C).(match ee with [(CSort _) \Rightarrow True | (CHead _ _ _) 
+\Rightarrow False])) I (CHead c1 (Bind b) v) H1) in (False_ind (or (ex3_2 C T 
+(\lambda (c2: C).(\lambda (_: T).(eq C (CSort m) (CHead c2 (Bind b) v)))) 
+(\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C (CSort m) (CHead c2 (Bind 
+Void) (TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c1 v w) \to False))))) H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) 
+v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (H3: (ty3 g c0 u t)).(\lambda (b0: B).(\lambda (H4: (eq C 
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
+H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
+H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r B b 
+(\lambda (b1: B).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead 
+c2 (Bind b1) u) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: 
+T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C 
+(\lambda (c3: C).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind Void) (TSort 
+O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g 
+c1 v w) \to False)))))) (let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 
+t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0: T).(or (ex3_2 C T (\lambda 
+(c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) t0) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) t0) 
+(CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda 
+(_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H11 \def (eq_ind 
+C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1 H9) in (let H12 \def (eq_ind C c0 
+(\lambda (c: C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda 
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
+C).(wf3 g c c2)) H1 c1 H9) in (or_introl (ex3_2 C T (\lambda (c3: C).(\lambda 
+(_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind 
+Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c1 v w) \to False)))) (ex3_2_intro C T (\lambda (c3: 
+C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w))) c2 t (refl_equal C (CHead c2 (Bind b) v)) H13 H11))))) u 
+H7)) b0 H8)))) H6)) H5))))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda 
+(H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Bind b) v)) \to (or 
+(ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) 
+(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall 
+(t: T).((ty3 g c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C 
+(CHead c0 (Bind b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
+\Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 
+\def (f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow b0 | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
+\Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let 
+H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u | 
+(CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) 
+H4) in (\lambda (_: (eq B b0 b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def 
+(eq_ind T u (\lambda (t: T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 
+v H7) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c 
+v t) \to False))) H10 c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3: 
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 
+v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) 
+(\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v 
+w) \to False))))))) H2 c1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c: 
+C).(wf3 g c c2)) H1 c1 H9) in (or_intror (ex3_2 C T (\lambda (c3: C).(\lambda 
+(_: T).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind b) v)))) 
+(\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: 
+T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind Void) 
+(TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))) (ex3_intro C 
+(\lambda (c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) 
+(TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c1 v w) \to False))) c2 (refl_equal C (CHead c2 (Bind Void) (TSort 
+O))) H13 H11))))))))) H6)) H5)))))))))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) 
+\to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind 
+b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead 
+c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c1 v w) \to False)))))))).(\lambda (u: T).(\lambda 
+(f: F).(\lambda (H3: (eq C (CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let 
+H4 \def (eq_ind C (CHead c0 (Flat f) u) (\lambda (ee: C).(match ee with 
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind 
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead c1 (Bind b) v) 
+H3) in (False_ind (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 
+(CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) 
+(\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq 
+C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H4))))))))) y 
+x H0))) H)))))).
+
+lemma wf3_gen_flat1:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f: 
+F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f: 
+F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat 
+f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y: 
+C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0: 
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m: 
+nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def 
+(eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow 
+True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Flat f) v) H1) in 
+(False_ind (wf3 g c1 (CSort m)) H2)))) (\lambda (c0: C).(\lambda (c2: 
+C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v)) 
+\to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
+t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat 
+f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match 
+ee with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k 
+with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 
+(Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2 (Bind b) u)) H5))))))))))) 
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: 
+(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
+T).(\lambda (_: ((\forall (t: T).((ty3 g c0 u t) \to False)))).(\lambda (b: 
+B).(\lambda (H4: (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 
+\def (eq_ind C (CHead c0 (Bind b) u) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead c1 (Flat f) v) 
+H4) in (False_ind (wf3 g c1 (CHead c2 (Bind Void) (TSort O))) H5)))))))))) 
+(\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: 
+(((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u: 
+T).(\lambda (f0: F).(\lambda (H3: (eq C (CHead c0 (Flat f0) u) (CHead c1 
+(Flat f) v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e with [(CSort 
+_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) 
+(CHead c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow (match 
+k with [(Bind _) \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 
+(Flat f0) u) (CHead c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow 
+t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda (_: (eq F 
+f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0 (\lambda (c: 
+C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8) in (let H10 
+\def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in H10))))) H5)) 
+H4))))))))) y x H0))) H)))))).
+
+lemma wf3_gen_head2:
+ \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k: 
+K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k: 
+K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda 
+(c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind 
+b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_: 
+C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
+k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k 
+v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee with [(CSort 
+_) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c k v) H1) 
+in (False_ind (ex B (\lambda (b: B).(eq K k (Bind b)))) H2)))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (u: 
+T).(\lambda (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda 
+(H4: (eq C (CHead c2 (Bind b) u) (CHead c k v))).(let H5 \def (f_equal C C 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def 
+(f_equal C K (\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind b) | 
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead 
+c k v) H4) in (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 
+c)).(let H10 \def (eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in 
+(let H11 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex 
+B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C 
+c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k 
+(\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0 
+(Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B 
+(\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K 
+(Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6)) 
+H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 
+c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K 
+k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u 
+t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void) 
+(TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e 
+with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 
+(Bind Void) (TSort O)) (CHead c k v) H4) in ((let H6 \def (f_equal C K 
+(\lambda (e: C).(match e with [(CSort _) \Rightarrow (Bind Void) | (CHead _ 
+k0 _) \Rightarrow k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in 
+((let H7 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind Void) 
+(TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void) k)).(\lambda 
+(H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0: C).((eq C c0 
+(CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0)))))) H2 c H9) in 
+(let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let 
+H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B 
+(\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in (eq_ind K 
+(Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind b0))))) 
+(let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind Void) t)) 
+\to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12 (TSort O) H7) 
+in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0))) Void 
+(refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 
+(CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_: 
+T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def 
+(f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind 
+C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq 
+K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda 
+(c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k 
+v))))))))))))) x y H0))) H)))))).
+
+theorem wf3_mono:
+ \forall (g: G).(\forall (c: C).(\forall (c1: C).((wf3 g c c1) \to (\forall 
+(c2: C).((wf3 g c c2) \to (eq C c1 c2))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (c1: C).(\lambda (H: (wf3 g c 
+c1)).(wf3_ind g (\lambda (c0: C).(\lambda (c2: C).(\forall (c3: C).((wf3 g c0 
+c3) \to (eq C c2 c3))))) (\lambda (m: nat).(\lambda (c2: C).(\lambda (H0: 
+(wf3 g (CSort m) c2)).(let H_y \def (wf3_gen_sort1 g c2 m H0) in (eq_ind_r C 
+(CSort m) (\lambda (c0: C).(eq C (CSort m) c0)) (refl_equal C (CSort m)) c2 
+H_y))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 
+c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 
+c4))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c2 u 
+t)).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g (CHead c2 (Bind b) 
+u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in (let H4 \def H_x in 
+(or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind 
+b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq C c0 (CHead 
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 (Bind b) u) 
+c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead 
+c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda 
+(_: C).(\lambda (w: T).(ty3 g c2 u w))))).(ex3_2_ind C T (\lambda (c4: 
+C).(\lambda (_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: 
+C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 
+u w))) (eq C (CHead c3 (Bind b) u) c0) (\lambda (x0: C).(\lambda (x1: 
+T).(\lambda (H6: (eq C c0 (CHead x0 (Bind b) u))).(\lambda (H7: (wf3 g c2 
+x0)).(\lambda (_: (ty3 g c2 u x1)).(eq_ind_r C (CHead x0 (Bind b) u) (\lambda 
+(c4: C).(eq C (CHead c3 (Bind b) u) c4)) (f_equal3 C K T C CHead c3 x0 (Bind 
+b) (Bind b) u u (H1 x0 H7) (refl_equal K (Bind b)) (refl_equal T u)) c0 
+H6)))))) H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind 
+Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall 
+(w: T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 
+(CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda 
+(_: C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind b) 
+u) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) (TSort 
+O)))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: ((\forall (w: T).((ty3 g c2 u 
+w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c4: 
+C).(eq C (CHead c3 (Bind b) u) c4)) (let H_x0 \def (H8 t H2) in (let H9 \def 
+H_x0 in (False_ind (eq C (CHead c3 (Bind b) u) (CHead x0 (Bind Void) (TSort 
+O))) H9))) c0 H6))))) H5)) H4))))))))))))) (\lambda (c2: C).(\lambda (c3: 
+C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: ((\forall (c4: C).((wf3 g c2 c4) 
+\to (eq C c3 c4))))).(\lambda (u: T).(\lambda (H2: ((\forall (t: T).((ty3 g 
+c2 u t) \to False)))).(\lambda (b: B).(\lambda (c0: C).(\lambda (H3: (wf3 g 
+(CHead c2 (Bind b) u) c0)).(let H_x \def (wf3_gen_bind1 g c2 c0 u b H3) in 
+(let H4 \def H_x in (or_ind (ex3_2 C T (\lambda (c4: C).(\lambda (_: T).(eq C 
+c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) 
+(\lambda (_: C).(\lambda (w: T).(ty3 g c2 u w)))) (ex3 C (\lambda (c4: C).(eq 
+C c0 (CHead c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) 
+(\lambda (_: C).(\forall (w: T).((ty3 g c2 u w) \to False)))) (eq C (CHead c3 
+(Bind Void) (TSort O)) c0) (\lambda (H5: (ex3_2 C T (\lambda (c4: C).(\lambda 
+(_: T).(eq C c0 (CHead c4 (Bind b) u)))) (\lambda (c4: C).(\lambda (_: 
+T).(wf3 g c2 c4))) (\lambda (_: C).(\lambda (w: T).(ty3 g c2 u 
+w))))).(ex3_2_ind C T (\lambda (c4: C).(\lambda (_: T).(eq C c0 (CHead c4 
+(Bind b) u)))) (\lambda (c4: C).(\lambda (_: T).(wf3 g c2 c4))) (\lambda (_: 
+C).(\lambda (w: T).(ty3 g c2 u w))) (eq C (CHead c3 (Bind Void) (TSort O)) 
+c0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c0 (CHead x0 (Bind 
+b) u))).(\lambda (_: (wf3 g c2 x0)).(\lambda (H8: (ty3 g c2 u x1)).(eq_ind_r 
+C (CHead x0 (Bind b) u) (\lambda (c4: C).(eq C (CHead c3 (Bind Void) (TSort 
+O)) c4)) (let H_x0 \def (H2 x1 H8) in (let H9 \def H_x0 in (False_ind (eq C 
+(CHead c3 (Bind Void) (TSort O)) (CHead x0 (Bind b) u)) H9))) c0 H6)))))) 
+H5)) (\lambda (H5: (ex3 C (\lambda (c4: C).(eq C c0 (CHead c4 (Bind Void) 
+(TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: C).(\forall (w: 
+T).((ty3 g c2 u w) \to False))))).(ex3_ind C (\lambda (c4: C).(eq C c0 (CHead 
+c4 (Bind Void) (TSort O)))) (\lambda (c4: C).(wf3 g c2 c4)) (\lambda (_: 
+C).(\forall (w: T).((ty3 g c2 u w) \to False))) (eq C (CHead c3 (Bind Void) 
+(TSort O)) c0) (\lambda (x0: C).(\lambda (H6: (eq C c0 (CHead x0 (Bind Void) 
+(TSort O)))).(\lambda (H7: (wf3 g c2 x0)).(\lambda (_: ((\forall (w: T).((ty3 
+g c2 u w) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda 
+(c4: C).(eq C (CHead c3 (Bind Void) (TSort O)) c4)) (f_equal3 C K T C CHead 
+c3 x0 (Bind Void) (Bind Void) (TSort O) (TSort O) (H1 x0 H7) (refl_equal K 
+(Bind Void)) (refl_equal T (TSort O))) c0 H6))))) H5)) H4)))))))))))) 
+(\lambda (c2: C).(\lambda (c3: C).(\lambda (_: (wf3 g c2 c3)).(\lambda (H1: 
+((\forall (c4: C).((wf3 g c2 c4) \to (eq C c3 c4))))).(\lambda (u: 
+T).(\lambda (f: F).(\lambda (c0: C).(\lambda (H2: (wf3 g (CHead c2 (Flat f) 
+u) c0)).(let H_y \def (wf3_gen_flat1 g c2 c0 u f H2) in (H1 c0 H_y)))))))))) 
+c c1 H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/getl.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/getl.ma
new file mode 100644 (file)
index 0000000..3b49f4c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/getl.ma
@@ -0,0 +1,199 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wf3/clear.ma".
+
+include "basic_1A/ty3/dec.ma".
+
+lemma wf3_getl_conf:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall 
+(v: T).((getl i c1 (CHead d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: 
+C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda 
+(d2: C).(getl i c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
+d2)))))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: 
+C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 (Bind b) v)) \to 
+(\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall (w: T).((ty3 g 
+d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind b) v))) 
+(\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda (c1: C).(\lambda (d1: 
+C).(\lambda (v: T).(\lambda (H: (getl O c1 (CHead d1 (Bind b) v))).(\lambda 
+(g: G).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda (w: T).(\lambda 
+(H1: (ty3 g d1 v w)).(let H_y \def (wf3_clear_conf c1 (CHead d1 (Bind b) v) 
+(getl_gen_O c1 (CHead d1 (Bind b) v) H) g c2 H0) in (let H_x \def 
+(wf3_gen_bind1 g d1 c2 v b H_y) in (let H2 \def H_x in (or_ind (ex3_2 C T 
+(\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda 
+(c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 
+g d1 v w0)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort 
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 
+g d1 v w0) \to False)))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind 
+b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H3: (ex3_2 C T (\lambda 
+(c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g d1 
+v w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
+(Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g d1 c3))) (\lambda (_: 
+C).(\lambda (w0: T).(ty3 g d1 v w0))) (ex2 C (\lambda (d2: C).(getl O c2 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H4: (eq C c2 (CHead x0 (Bind b) v))).(\lambda 
+(H5: (wf3 g d1 x0)).(\lambda (_: (ty3 g d1 v x1)).(eq_ind_r C (CHead x0 (Bind 
+b) v) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) 
+v))) (\lambda (d2: C).(wf3 g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(getl O 
+(CHead x0 (Bind b) v) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) 
+x0 (getl_refl b x0 v) H5) c2 H4)))))) H3)) (\lambda (H3: (ex3 C (\lambda (c3: 
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g d1 
+c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g d1 v w0) \to 
+False))))).(ex3_ind C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort 
+O)))) (\lambda (c3: C).(wf3 g d1 c3)) (\lambda (_: C).(\forall (w0: T).((ty3 
+g d1 v w0) \to False))) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind b) 
+v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: C).(\lambda (H4: (eq C c2 
+(CHead x0 (Bind Void) (TSort O)))).(\lambda (_: (wf3 g d1 x0)).(\lambda (H6: 
+((\forall (w0: T).((ty3 g d1 v w0) \to False)))).(eq_ind_r C (CHead x0 (Bind 
+Void) (TSort O)) (\lambda (c: C).(ex2 C (\lambda (d2: C).(getl O c (CHead d2 
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H6 w H1) in 
+(let H7 \def H_x0 in (False_ind (ex2 C (\lambda (d2: C).(getl O (CHead x0 
+(Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
+d2))) H7))) c2 H4))))) H3)) H2))))))))))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (c1: C).(\forall (d1: C).(\forall (v: T).((getl n c1 (CHead d1 
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c1 c2) \to (\forall 
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind 
+b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))))).(\lambda (c1: C).(C_ind 
+(\lambda (c: C).(\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead d1 
+(Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to (\forall 
+(w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))))))))) (\lambda (n0: 
+nat).(\lambda (d1: C).(\lambda (v: T).(\lambda (H0: (getl (S n) (CSort n0) 
+(CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (wf3 g 
+(CSort n0) c2)).(\lambda (w: T).(\lambda (_: (ty3 g d1 v w)).(getl_gen_sort 
+n0 (S n) (CHead d1 (Bind b) v) H0 (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))) (\lambda 
+(c: C).(\lambda (H0: ((\forall (d1: C).(\forall (v: T).((getl (S n) c (CHead 
+d1 (Bind b) v)) \to (\forall (g: G).(\forall (c2: C).((wf3 g c c2) \to 
+(\forall (w: T).((ty3 g d1 v w) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))))))))))))).(\lambda 
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (v: T).(\lambda (H1: (getl 
+(S n) (CHead c k t) (CHead d1 (Bind b) v))).(\lambda (g: G).(\lambda (c2: 
+C).(\lambda (H2: (wf3 g (CHead c k t) c2)).(\lambda (w: T).(\lambda (H3: (ty3 
+g d1 v w)).(K_ind (\lambda (k0: K).((wf3 g (CHead c k0 t) c2) \to ((getl (r 
+k0 n) c (CHead d1 (Bind b) v)) \to (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))))) (\lambda (b0: 
+B).(\lambda (H4: (wf3 g (CHead c (Bind b0) t) c2)).(\lambda (H5: (getl (r 
+(Bind b0) n) c (CHead d1 (Bind b) v))).(let H_x \def (wf3_gen_bind1 g c c2 t 
+b0 H4) in (let H6 \def H_x in (or_ind (ex3_2 C T (\lambda (c3: C).(\lambda 
+(_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: C).(\lambda (_: 
+T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t w0)))) (ex3 C 
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: 
+C).(wf3 g c c3)) (\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to 
+False)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) 
+(\lambda (d2: C).(wf3 g d1 d2))) (\lambda (H7: (ex3_2 C T (\lambda (c3: 
+C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b0) t)))) (\lambda (c3: 
+C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: C).(\lambda (w0: T).(ty3 g c t 
+w0))))).(ex3_2_ind C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 
+(Bind b0) t)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c c3))) (\lambda (_: 
+C).(\lambda (w0: T).(ty3 g c t w0))) (ex2 C (\lambda (d2: C).(getl (S n) c2 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2))) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H8: (eq C c2 (CHead x0 (Bind b0) t))).(\lambda 
+(H9: (wf3 g c x0)).(\lambda (_: (ty3 g c t x1)).(eq_ind_r C (CHead x0 (Bind 
+b0) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 
+(Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g 
+x0 H9 w H3) in (let H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 
+(CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: 
+C).(getl (S n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: 
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind 
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S 
+n) (CHead x0 (Bind b0) t) (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
+d2)) x (getl_head (Bind b0) n x0 (CHead x (Bind b) v) H12 t) H13)))) H11))) 
+c2 H8)))))) H7)) (\lambda (H7: (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 
+(Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) (\lambda (_: 
+C).(\forall (w0: T).((ty3 g c t w0) \to False))))).(ex3_ind C (\lambda (c3: 
+C).(eq C c2 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c c3)) 
+(\lambda (_: C).(\forall (w0: T).((ty3 g c t w0) \to False))) (ex2 C (\lambda 
+(d2: C).(getl (S n) c2 (CHead d2 (Bind b) v))) (\lambda (d2: C).(wf3 g d1 
+d2))) (\lambda (x0: C).(\lambda (H8: (eq C c2 (CHead x0 (Bind Void) (TSort 
+O)))).(\lambda (H9: (wf3 g c x0)).(\lambda (_: ((\forall (w0: T).((ty3 g c t 
+w0) \to False)))).(eq_ind_r C (CHead x0 (Bind Void) (TSort O)) (\lambda (c0: 
+C).(ex2 C (\lambda (d2: C).(getl (S n) c0 (CHead d2 (Bind b) v))) (\lambda 
+(d2: C).(wf3 g d1 d2)))) (let H_x0 \def (H c d1 v H5 g x0 H9 w H3) in (let 
+H11 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl n x0 (CHead d2 (Bind b) 
+v))) (\lambda (d2: C).(wf3 g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) 
+(CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: 
+C).(wf3 g d1 d2))) (\lambda (x: C).(\lambda (H12: (getl n x0 (CHead x (Bind 
+b) v))).(\lambda (H13: (wf3 g d1 x)).(ex_intro2 C (\lambda (d2: C).(getl (S 
+n) (CHead x0 (Bind Void) (TSort O)) (CHead d2 (Bind b) v))) (\lambda (d2: 
+C).(wf3 g d1 d2)) x (getl_head (Bind Void) n x0 (CHead x (Bind b) v) H12 
+(TSort O)) H13)))) H11))) c2 H8))))) H7)) H6)))))) (\lambda (f: F).(\lambda 
+(H4: (wf3 g (CHead c (Flat f) t) c2)).(\lambda (H5: (getl (r (Flat f) n) c 
+(CHead d1 (Bind b) v))).(let H_y \def (wf3_gen_flat1 g c c2 t f H4) in (H0 d1 
+v H5 g c2 H_y w H3))))) k H2 (getl_gen_S k c (CHead d1 (Bind b) v) t n 
+H1)))))))))))))) c1)))) i)).
+
+lemma getl_wf3_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).((getl i c1 d1) \to 
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: 
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl i c2 d2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: 
+C).((getl n c1 d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to 
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 
+d2)))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (H: (getl O c1 
+d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H0: (wf3 g d1 d2)).(let H_x 
+\def (clear_wf3_trans c1 d1 (getl_gen_O c1 d1 H) g d2 H0) in (let H1 \def H_x 
+in (ex2_ind C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(clear c2 d2)) 
+(ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 d2))) 
+(\lambda (x: C).(\lambda (H2: (wf3 g c1 x)).(\lambda (H3: (clear x 
+d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(getl O c2 
+d2)) x H2 (getl_intro O x d2 x (drop_refl x) H3))))) H1))))))))) (\lambda (n: 
+nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).((getl n c1 d1) \to 
+(\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: 
+C).(wf3 g c1 c2)) (\lambda (c2: C).(getl n c2 d2))))))))))).(\lambda (c1: 
+C).(C_ind (\lambda (c: C).(\forall (d1: C).((getl (S n) c d1) \to (\forall 
+(g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda (c2: C).(wf3 g c 
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))))))))) (\lambda (n0: nat).(\lambda 
+(d1: C).(\lambda (H0: (getl (S n) (CSort n0) d1)).(\lambda (g: G).(\lambda 
+(d2: C).(\lambda (_: (wf3 g d1 d2)).(getl_gen_sort n0 (S n) d1 H0 (ex2 C 
+(\lambda (c2: C).(wf3 g (CSort n0) c2)) (\lambda (c2: C).(getl (S n) c2 
+d2)))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).((getl (S n) c 
+d1) \to (\forall (g: G).(\forall (d2: C).((wf3 g d1 d2) \to (ex2 C (\lambda 
+(c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)))))))))).(\lambda 
+(k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (H1: (getl (S n) (CHead c k 
+t) d1)).(\lambda (g: G).(\lambda (d2: C).(\lambda (H2: (wf3 g d1 d2)).(K_ind 
+(\lambda (k0: K).((getl (r k0 n) c d1) \to (ex2 C (\lambda (c2: C).(wf3 g 
+(CHead c k0 t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))))) (\lambda (b: 
+B).(\lambda (H3: (getl (r (Bind b) n) c d1)).(let H_x \def (H c d1 H3 g d2 
+H2) in (let H4 \def H_x in (ex2_ind C (\lambda (c2: C).(wf3 g c c2)) (\lambda 
+(c2: C).(getl n c2 d2)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) 
+c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 
+g c x)).(\lambda (H6: (getl n x d2)).(let H_x0 \def (ty3_inference g c t) in 
+(let H7 \def H_x0 in (or_ind (ex T (\lambda (t2: T).(ty3 g c t t2))) (\forall 
+(t2: T).((ty3 g c t t2) \to False)) (ex2 C (\lambda (c2: C).(wf3 g (CHead c 
+(Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2))) (\lambda (H8: (ex T 
+(\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 g c t t2)) 
+(ex2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda (c2: 
+C).(getl (S n) c2 d2))) (\lambda (x0: T).(\lambda (H9: (ty3 g c t 
+x0)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) (\lambda 
+(c2: C).(getl (S n) c2 d2)) (CHead x (Bind b) t) (wf3_bind g c x H5 t x0 H9 
+b) (getl_head (Bind b) n x d2 H6 t)))) H8)) (\lambda (H8: ((\forall (t2: 
+T).((ty3 g c t t2) \to False)))).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead 
+c (Bind b) t) c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (CHead x (Bind Void) 
+(TSort O)) (wf3_void g c x H5 t H8 b) (getl_head (Bind Void) n x d2 H6 (TSort 
+O)))) H7)))))) H4))))) (\lambda (f: F).(\lambda (H3: (getl (r (Flat f) n) c 
+d1)).(let H_x \def (H0 d1 H3 g d2 H2) in (let H4 \def H_x in (ex2_ind C 
+(\lambda (c2: C).(wf3 g c c2)) (\lambda (c2: C).(getl (S n) c2 d2)) (ex2 C 
+(\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) (\lambda (c2: C).(getl (S 
+n) c2 d2))) (\lambda (x: C).(\lambda (H5: (wf3 g c x)).(\lambda (H6: (getl (S 
+n) x d2)).(ex_intro2 C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) 
+(\lambda (c2: C).(getl (S n) c2 d2)) x (wf3_flat g c x H5 t f) H6)))) H4))))) 
+k (getl_gen_S k c d1 t n H1))))))))))) c1)))) i).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/props.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/props.ma
new file mode 100644 (file)
index 0000000..a79898a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/props.ma
@@ -0,0 +1,153 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wf3/ty3.ma".
+
+include "basic_1A/app/defs.ma".
+
+lemma wf3_total:
+ \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2: 
+C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g 
+(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C 
+(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def 
+H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g 
+(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind 
+(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda 
+(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex 
+T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to 
+False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda 
+(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3 
+g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda 
+(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g 
+(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4 
+b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to 
+False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2)) 
+(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda 
+(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x 
+(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
+
+lemma ty3_shift1:
+ \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall 
+(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c 
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c 
+(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall 
+(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 
+t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0: 
+C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3 
+g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2)))))))) 
+(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda 
+(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C 
+c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g 
+(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C 
+(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C 
+C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) 
+\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind b) u) H4) in (let H7 
+\def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: 
+T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 
+t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 \def (eq_ind_r C c2 (\lambda (c0: 
+C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T (\lambda (t0: T).(ty3 g (CHead c1 
+(Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind b) u t1)) 
+(app1 c1 (THead (Bind b) u t2))) (\lambda (x: T).(\lambda (_: (ty3 g (CHead 
+c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) (THead (Bind b) u t1) (THead 
+(Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 H5)))) (ty3_correct g (CHead c1 
+(Bind b) u) t1 t2 H5))))))))))))))))) (\lambda (c1: C).(\lambda (c2: 
+C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall 
+(t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 
+c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: 
+T).((ty3 g c1 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead 
+c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead 
+c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort 
+O)) H4) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort 
+_) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0) 
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 
+(Bind Void) (TSort O)) H4) in ((let H8 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) 
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9: 
+(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b 
+(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in 
+(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u))) 
+(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12 
+\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11 
+(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0: 
+T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O) 
+(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1 
+(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def 
+(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall 
+(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1 
+t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g 
+c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) 
+(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort 
+O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda 
+(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1) 
+(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g 
+c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12)))) 
+(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b 
+H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: 
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall 
+(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1 
+t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 
+(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead 
+c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1 
+(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 
+c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort 
+(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in 
+(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1 
+(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in 
+(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3 
+g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
+t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def 
+(eq_ind K (Flat f) (\lambda (ee: K).(match ee with [(Bind _) \Rightarrow 
+False | (Flat _) \Rightarrow True])) I (Bind x) H9) in (False_ind (ty3 g 
+(CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u 
+t2))) H10)))) H8)))))))))))))))) y c H0))) H))).
+
+lemma wf3_idem:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g 
+c2 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1 
+c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m: 
+nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g 
+c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g 
+c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_: 
+(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_: 
+((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4 
+c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda 
+(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 
+c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
+
+lemma wf3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t 
+u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t 
+u)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind 
+C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) 
+(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1 
+x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t 
+u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_1A/wf3/ty3.ma b/matita/matita/contribs/lambdadelta/basic_1A/wf3/ty3.ma
new file mode 100644 (file)
index 0000000..0023934
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_1A/wf3/ty3.ma
@@ -0,0 +1,126 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "basic_1A/wf3/getl.ma".
+
+lemma wf3_pr2_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
+u) \to (pr2 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2 
+c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda 
+(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u: 
+T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c: 
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c 
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: 
+C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 
+u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def 
+(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in 
+(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x: 
+T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0 
+g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2 
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t) 
+(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda 
+(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7))))) 
+H5)))))))))))))))))) c1 t1 t2 H))))).
+
+lemma wf3_pr3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1 
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1 
+u) \to (pr3 c2 t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall 
+(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t 
+t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1 
+c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t)))))) 
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1 
+c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda 
+(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1 
+t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2 
+H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
+
+lemma wf3_pc3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1 
+u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda 
+(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g 
+c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t)) 
+(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4: 
+(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf 
+g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2))))) 
+H3)))))))))))).
+
+lemma wf3_ty3_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 
+t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda 
+(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c: 
+C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda 
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u: 
+T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall 
+(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4 
+t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0: 
+T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4 
+x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4 
+c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c: 
+C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g 
+c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
+(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g 
+c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def 
+(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind 
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: 
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: 
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d 
+x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n: 
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c 
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u 
+t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u 
+t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def 
+(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind 
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: 
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x: 
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d 
+x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c: 
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: 
+((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: 
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) 
+t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to 
+(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g 
+c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t 
+H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda 
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g 
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead 
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g 
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c 
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda 
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0: 
+T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c 
+c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c 
+c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre.ma
new file mode 100644 (file)
index 0000000..99be99c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predeval_4.ma".
+include "basic_2A/computation/cprs.ma".
+include "basic_2A/computation/csx.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
+
+definition cpre: relation4 genv lenv term term ≝
+                 λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive parallel reduction (term)"
+   'PRedEval G L T1 T2 = (cpre G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: nf2_sn3 *)
+lemma csx_cpre: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
+* #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
+#T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre_cpre.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre_cpre.ma
new file mode 100644 (file)
index 0000000..672c9b0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpre_cpre.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cprs_cprs.ma".
+include "basic_2A/computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
+
+(* Main properties *********************************************************)
+
+(* Basic_1: was: nf2_pr3_confluence *)
+theorem cpre_mono: ∀G,L,T,T1. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
+#G #L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2
+elim (cprs_conf … H1T1 … H1T2) -T #T #HT1
+>(cprs_inv_cnr1 … HT1 H2T1) -T1 #HT2
+>(cprs_inv_cnr1 … HT2 H2T2) -T2 //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma
new file mode 100644 (file)
index 0000000..8d71a75
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs.ma
@@ -0,0 +1,144 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predstar_4.ma".
+include "basic_2A/reduction/cnr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Basic_1: includes: pr1_pr0 *)
+definition cprs: relation4 genv lenv term term ≝
+                 λG. LTC … (cpr G).
+
+interpretation "context-sensitive parallel computation (term)"
+   'PRedStar G L T1 T2 = (cprs G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cprs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2.
+#G #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cprs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1.
+#G #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: pr3_pr2 *)
+lemma cpr_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+/2 width=1 by inj/ qed.
+
+(* Basic_1: was: pr3_refl *)
+lemma cprs_refl: ∀G,L,T. ⦃G, L⦄ ⊢ T ➡* T.
+/2 width=1 by cpr_cprs/ qed.
+
+lemma cprs_strap1: ∀G,L,T1,T,T2.
+                   ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+normalize /2 width=3 by step/ qed-.
+
+(* Basic_1: was: pr3_step *)
+lemma cprs_strap2: ∀G,L,T1,T,T2.
+                   ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr.
+/3 width=5 by lsubr_cpr_trans, LTC_lsub_trans/
+qed-.
+
+(* Basic_1: was: pr3_pr1 *)
+lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+/2 width=3 by lsubr_cprs_trans/ qed.
+
+lemma cprs_bind_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
+/3 width=3 by cprs_strap2, cpr_cprs, cpr_pair_sn, cpr_bind/ qed.
+
+(* Basic_1: was only: pr3_thin_dx *)
+lemma cprs_flat_dx: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2
+/3 width=5 by cprs_strap1, cpr_flat, cpr_cprs, cpr_pair_sn/
+qed.
+
+lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_strap1, cpr_cprs, cpr_pair_sn, cpr_flat/
+qed.
+
+lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* T2.
+#G #L #V #T1 #T #T2 #HT2 #H @(cprs_ind_dx … H) -T1
+/3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/
+qed.
+
+lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
+#G #L #T1 #T2 #H @(cprs_ind … H) -T2
+/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
+qed.
+
+lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+                    ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2
+/4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/
+qed.
+
+lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+                     ⦃G, L⦄ ⊢ V1 ➡ V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
+/4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: pr3_gen_sort *)
+lemma cprs_inv_sort1: ∀G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡* U2 → U2 = ⋆k.
+#G #L #U2 #k #H @(cprs_ind … H) -U2 //
+#U2 #U #_ #HU2 #IHU2 destruct
+>(cpr_inv_sort1 … HU2) -HU2 //
+qed-.
+
+(* Basic_1: was: pr3_gen_cast *)
+lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
+#G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_intror/
+#U2 #U #_ #HU2 * /3 width=3 by cprs_strap1, or_introl/ *
+#W #T #HW1 #HT1 #H destruct
+elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3 by cprs_strap1, or_introl/ *
+#W2 #T2 #HW2 #HT2 #H destruct /4 width=5 by cprs_strap1, ex3_2_intro, or_intror/
+qed-.
+
+(* Basic_1: was: nf2_pr3_unfold *)
+lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U.
+#G #L #T #U #H @(cprs_ind_dx … H) -T //
+#T0 #T #H1T0 #_ #IHT #H2T0
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: removed theorems 13:
+   pr1_head_1 pr1_head_2 pr1_comp
+   clear_pr3_trans pr3_cflat pr3_gen_bind
+   pr3_head_1 pr3_head_2 pr3_head_21 pr3_head_12
+   pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma
new file mode 100644 (file)
index 0000000..7950057
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_cprs.ma
@@ -0,0 +1,155 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpr_lpr.ma".
+include "basic_2A/computation/cprs_lift.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr3_t *)
+(* Basic_1: includes: pr1_t *)
+theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
+normalize /2 width=3 by trans_TC/ qed-.
+
+(* Basic_1: was: pr3_confluence *)
+(* Basic_1: includes: pr1_confluence *)
+theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
+normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
+
+theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=5 by cprs_trans, cprs_bind_dx/
+qed.
+
+(* Basic_1: was: pr3_flat *)
+theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
+#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
+/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
+qed.
+
+theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+                      ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
+                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
+#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
+@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
+qed.
+
+theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+                   ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
+#V #V2 #_ #HV2 #IHV1
+@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
+qed.
+
+theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+                       ⦃G, L⦄ ⊢ V1 ➡ V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+                       ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cprs_ind … H) -W2
+/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
+qed.
+
+theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1
+/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr3_gen_appl *)
+lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
+                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
+                                        U2 = ⓐV2. T2
+                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
+                                        ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
+                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⬆[0,1] V0 ≡ V2 &
+                                        ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
+                                        ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
+#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+  elim (cpr_inv_appl1 … HU2) -HU2 *
+  [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
+  | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
+    lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
+    lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+    /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_beta, ex2_3_intro, or3_intro1/
+  | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
+    /5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/
+  ]
+| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
+]
+qed-.
+
+(* Properties concerning sn parallel reduction on local environments ********)
+
+(* Basic_1: was just: pr3_pr2_pr2_t *)
+(* Basic_1: includes: pr3_pr0_pr2_t *)
+lemma lpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lpr G).
+#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+  elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+  elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
+  /4 width=6 by cprs_strap2, cprs_delta/
+|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
+|4,6: /3 width=1 by cprs_flat, cprs_eps/
+|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
+]
+qed-.
+
+lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
+lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G).
+#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+qed-.
+
+(* Basic_1: was: pr3_strip *)
+(* Basic_1: includes: pr1_strip *)
+lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
+normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
+
+lemma cprs_lpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                        ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/
+#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01)
+#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0
+#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T
+/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
+qed-.
+
+lemma cprs_lpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+                        ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                        ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lpr_cprs_trans, ex2_intro/
+qed-.
+
+lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_lift.ma
new file mode 100644 (file)
index 0000000..635f1e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cprs_lift.ma
@@ -0,0 +1,60 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpr_lift.ma".
+include "basic_2A/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma cprs_delta: ∀G,L,K,V,V2,i.
+                  ⬇[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡* V2 →
+                  ∀W2. ⬆[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
+#G #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6 by cpr_cprs, cpr_delta/ ]
+#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
+lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr3_gen_lref *)
+lemma cprs_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
+                      T2 = #i ∨
+                      ∃∃K,V1,T1. ⬇[i] L ≡ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡* T1 &
+                                 ⬆[0, i + 1] T1 ≡ T2.
+#G #L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+  elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+  * /4 width=6 by cpr_cprs, ex3_3_intro, or_intror/
+| * #K #V1 #T1 #HLK #HVT1 #HT1
+  lapply (drop_fwd_drop2 … HLK) #H0LK
+  elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+  /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr3_lift *)
+lemma cprs_lift: ∀G. d_liftable (cprs G).
+/3 width=10 by d_liftable_LTC, cpr_lift/ qed.
+
+(* Basic_1: was: pr3_gen_lift *)
+lemma cprs_inv_lift1: ∀G. d_deliftable_sn (cprs G).
+/3 width=6 by d_deliftable_sn_LTC, cpr_inv_lift1/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxe.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxe.ma
new file mode 100644 (file)
index 0000000..1bec07a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxe.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predeval_6.ma".
+include "basic_2A/computation/cpxs.ma".
+include "basic_2A/computation/csx.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION ON TERMS ****)
+
+definition cpxe: ∀h. sd h → relation4 genv lenv term term ≝
+                 λh,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 ∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive extended parallel reduction (term)"
+   'PRedEval h g G L T1 T2 = (cpxe h g G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma csx_cpxe: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] 𝐍⦃T2⦄.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnx_dec h g G L T1) /3 width=3 by ex_intro, conj/
+* #T #H1T1 #H2T1 elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1
+#T2 * /4 width=3 by cpxs_strap2, ex_intro, conj/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma
new file mode 100644 (file)
index 0000000..173808b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs.ma
@@ -0,0 +1,181 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predstar_6.ma".
+include "basic_2A/reduction/cnx.ma".
+include "basic_2A/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
+                 λh,g,G. LTC … (cpx h g G).
+
+interpretation "extended context-sensitive parallel computation (term)"
+   'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2.
+#h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1.
+#h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
+/2 width=1 by inj/ qed.
+
+lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+/2 width=1 by inj/ qed.
+
+lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+normalize /2 width=3 by step/ qed.
+
+lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+normalize /2 width=3 by TC_strap/ qed.
+
+lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
+/3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
+qed-.
+
+lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
+qed.
+
+lemma cpxs_sort: ∀h,g,G,L,k,d1. deg h g k d1 →
+                 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] ⋆((next h)^d2 k).
+#h #g #G #L #k #d1 #Hkd1 #d2 @(nat_ind_plus … d2) -d2 /2 width=1 by cpx_cpxs/
+#d2 #IHd2 #Hd21 >iter_SO
+@(cpxs_strap1 … (⋆(iter d2 ℕ (next h) k)))
+[ /3 width=3 by lt_to_le/
+| @(cpx_st … (d1-d2-1)) <plus_minus_m_m
+  /2 width=1 by deg_iter, monotonic_le_minus_r/
+]
+qed.
+
+lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
+qed.
+
+lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
+qed.
+
+lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
+qed.
+
+lemma cpxs_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, g] ②{I}V2.T.
+#h #g #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_pair_sn/
+qed.
+
+lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
+#h #g #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
+qed.
+
+lemma cpxs_eps: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
+qed.
+
+lemma cpxs_ct: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
+#h #g #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ct/
+qed.
+
+lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
+                    ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
+/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
+qed.
+
+lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                     ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 
+/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
+                      ∃∃n,d. deg h g k (n+d) & U2 = ⋆((next h)^n k).
+#h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2
+[ elim (deg_total h g k) #d #Hkd
+  @(ex2_2_intro … 0 … Hkd) -Hkd //
+| #U #U2 #_ #HU2 * #n #d #Hknd #H destruct
+  elim (cpx_inv_sort1 … HU2) -HU2
+  [ #H destruct /2 width=4 by ex2_2_intro/
+  | * #d0 #Hkd0 #H destruct -d
+    @(ex2_2_intro … (n+1) d0) /2 width=1 by deg_inv_prec/ >iter_SO //
+  ]
+]
+qed-.
+
+lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
+                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
+                       | ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
+                       | ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
+#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
+#W #T #HW1 #HT1 #H destruct
+elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
+#W2 #T2 #HW2 #HT2 #H destruct
+lapply (cpxs_strap1 … HW1 … HW2) -W
+lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
+qed-.
+
+lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → T = U.
+#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
+#T0 #T #H1T0 #_ #IHT #H2T0
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
+qed-.
+
+lemma cpxs_neq_inv_step_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
+                            ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+[ #H elim H -H //
+| #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
+  [ -H1 -H2 /3 width=1 by/
+  | -IH2 /3 width=4 by ex3_intro/ (**) (* auto fails without clear *)
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_aaa.ma
new file mode 100644 (file)
index 0000000..f753b28
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_aaa.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpx_aaa.ma".
+include "basic_2A/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma cpxs_aaa_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+#h #g #G #L #T1 #A #HT1 #T2 #HT12
+@(TC_Conf3 … HT1 ? HT12) -A -T1 -T2 /2 width=5 by cpx_aaa_conf/
+qed-.
+
+lemma cprs_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/3 width=5 by cpxs_aaa_conf, cprs_cpxs/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma
new file mode 100644 (file)
index 0000000..6100252
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_cpxs.ma
@@ -0,0 +1,187 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpx_drop.ma".
+include "basic_2A/computation/cpxs_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Main properties **********************************************************)
+
+theorem cpxs_trans: ∀h,g,G,L. Transitive … (cpxs h g G L).
+normalize /2 width=3 by trans_TC/ qed-.
+
+theorem cpxs_bind: ∀h,g,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
+                   ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+#h #g #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_trans, cpxs_bind_dx/
+qed.
+
+theorem cpxs_flat: ∀h,g,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                   ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_trans, cpxs_flat_dx/
+qed.
+
+theorem cpxs_beta_rc: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
+                      ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
+                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
+/4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
+qed.
+
+theorem cpxs_beta: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
+                   ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
+/4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
+qed.
+
+theorem cpxs_theta_rc: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                       ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⬆[0, 1] V ≡ V2 →
+                       ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
+                       ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
+/3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
+qed.
+
+theorem cpxs_theta: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
+                    ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpxs_inv_appl1: ∀h,g,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h, g] U2 →
+                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 &
+                                        U2 = ⓐV2. T2
+                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, g] U2
+                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V0 & ⬆[0,1] V0 ≡ V2 &
+                                        ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U2.
+#h #g #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+  elim (cpx_inv_appl1 … HU2) -HU2 *
+  [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cpxs_strap1, or3_intro0, ex3_2_intro/
+  | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
+    lapply (cpxs_strap1 … HV10 … HV02) -V0 #HV12
+    lapply (lsubr_cpx_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+    /5 width=5 by cpxs_bind, cpxs_flat_dx, cpx_cpxs, lsubr_beta, ex2_3_intro, or3_intro1/
+  | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
+    /5 width=10 by cpxs_flat_sn, cpxs_bind_dx, cpxs_strap1, ex4_5_intro, or3_intro2/
+  ]
+| /4 width=9 by cpxs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cpxs_strap1, or3_intro2, ex4_5_intro/
+]
+qed-.
+
+(* Properties on sn extended parallel reduction for local environments ******)
+
+lemma lpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpx h g G).
+#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| /3 width=2 by cpx_cpxs, cpx_st/
+| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+  elim (lpx_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+  elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
+  /4 width=7 by cpxs_delta, cpxs_strap2/
+|4,9: /4 width=1 by cpxs_beta, cpxs_bind, lpx_pair/
+|5,7,8: /3 width=1 by cpxs_flat, cpxs_ct, cpxs_eps/
+| /4 width=3 by cpxs_zeta, lpx_pair/
+| /4 width=3 by cpxs_theta, cpxs_strap1, lpx_pair/
+]
+qed-.
+
+lemma cpx_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, g] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lpx_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpx h g G).
+#h #g #G @s_r_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
+qed-.
+
+lemma cpxs_bind2_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+/4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+  #U2 #HVU2 @(ex3_intro … U2)
+  [1,3: /3 width=7 by fqu_drop, cpxs_delta, drop_pair, drop_drop/
+  | #H destruct /2 width=7 by lift_inv_lref2_be/
+  ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+  [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+  [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+  [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
+  #U2 #HTU2 @(ex3_intro … U2)
+  [1,3: /2 width=10 by cpxs_lift, fqu_drop/
+  | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+  #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_neq … H1 … HTU1 H) -T1
+  /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lift.ma
new file mode 100644 (file)
index 0000000..cf538d2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lift.ma
@@ -0,0 +1,124 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqus_fqus.ma".
+include "basic_2A/reduction/cpx_lift.ma".
+include "basic_2A/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpxs_delta: ∀h,g,I,G,L,K,V,V2,i.
+                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, g] V2 →
+                  ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, g] W2.
+#h #g #I #G #L #K #V #V2 #i #HLK #H elim H -V2
+[ /3 width=9 by cpx_cpxs, cpx_delta/
+| #V1 lapply (drop_fwd_drop2 … HLK) -HLK
+  elim (lift_total V1 0 (i+1)) /4 width=12 by cpx_lift, cpxs_strap1/
+]
+qed.
+
+lemma lstas_cpxs: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+                  ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
+[ /3 width=3 by cpxs_sort, da_inv_sort/
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+  elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpxs_delta/
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+  elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+  #HV1 #H destruct lapply (le_plus_to_le_r … Hd21) -Hd21
+  /3 width=7 by cpxs_delta/
+| /4 width=3 by cpxs_bind_dx, da_inv_bind/
+| /4 width=3 by cpxs_flat_dx, da_inv_flat/
+| /4 width=3 by cpxs_eps, da_inv_flat/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpxs_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, g] T2 →
+                      T2 = #i ∨
+                      ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, g] T1 &
+                                   ⬆[0, i+1] T1 ≡ T2.
+#h #g #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+  elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+  * /4 width=7 by cpx_cpxs, ex3_4_intro, or_intror/
+| * #I #K #V1 #T1 #HLK #HVT1 #HT1
+  lapply (drop_fwd_drop2 … HLK) #H0LK
+  elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+  /4 width=7 by cpxs_strap1, ex3_4_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cpxs_lift: ∀h,g,G. d_liftable (cpxs h g G).
+/3 width=10 by cpx_lift, cpxs_strap1, d_liftable_LTC/ qed.
+
+lemma cpxs_inv_lift1: ∀h,g,G. d_deliftable_sn (cpxs h g G).
+/3 width=6 by d_deliftable_sn_LTC, cpx_inv_lift1/
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+                      ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
+#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+lemma fquq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+                       ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fquq_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                        ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
+                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d2 → d1 ≤ d2 →
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
+
+lemma fqup_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+                       ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
+#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+lemma fqus_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+                       ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fqus_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                        ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
+                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d2 → d1 ≤ d2 →
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+/3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lleq.ma
new file mode 100644 (file)
index 0000000..610f884
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lleq.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_lleq.ma".
+include "basic_2A/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_cpxs_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+                       ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+/4 width=6 by cpx_lleq_conf_dx, lleq_cpx_trans, cpxs_strap2/
+qed-.
+
+lemma cpxs_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+                      ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
+/3 width=3 by lleq_cpxs_trans, lleq_sym/ qed-.
+
+lemma cpxs_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+                         ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
+#h #g #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lleq_conf_dx/
+qed-.
+
+lemma cpxs_lleq_conf_sn: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
+                         ∀L2. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
+/4 width=6 by cpxs_lleq_conf_dx, lleq_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma
new file mode 100644 (file)
index 0000000..025cfdc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_lreq.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_lreq.ma".
+include "basic_2A/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) (lreq 0 (∞)).
+/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts.ma
new file mode 100644 (file)
index 0000000..afe33af
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts.ma
@@ -0,0 +1,107 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/tsts.ma".
+include "basic_2A/computation/lpxs_cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Forward lemmas involving same top term structure *************************)
+
+lemma cpxs_fwd_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≂ U.
+#h #g #G #L #T #HT #U #H
+>(cpxs_inv_cnx1 … H HT) -G -L -T //
+qed-.
+
+lemma cpxs_fwd_sort: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
+                     ⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
+#h #g #G #L #U #k #H
+elim (cpxs_inv_sort1 … H) -H #n #d generalize in match k; -k @(nat_ind_plus … n) -n
+[ #k #_ #H -d destruct /2 width=1 by or_introl/
+| #n #IHn #k >plus_plus_comm_23 #Hnd #H destruct
+  lapply (deg_next_SO … Hnd) -Hnd #Hnd
+  elim (IHn … Hnd) -IHn
+  [ #H lapply (tsts_inv_atom1 … H) -H #H >H -H /2 width=1 by or_intror/
+  | generalize in match Hnd; -Hnd @(nat_ind_plus … n) -n
+    /4 width=3 by cpxs_strap2, cpx_st, or_intror/
+  | >iter_SO >iter_n_Sm //
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_beta *)
+lemma cpxs_fwd_beta: ∀h,g,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, g] U →
+                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, g] U.
+#h #g #a #G #L #V #W #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #b #W0 #T0 #HT0 #HU
+  elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
+  lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
+  /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_beta, or_intror/
+| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
+  elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
+]
+qed-.
+
+(* Note: probably this is an inversion lemma *)
+lemma cpxs_fwd_delta: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+                      ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
+                      ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, g] U →
+                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
+#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
+elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
+* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
+lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+/4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
+qed-.
+
+lemma cpxs_fwd_theta: ∀h,g,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
+                      ∀V2. ⬆[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
+                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U.
+#h #g #a #G #L #V1 #V #T #U #H #V2 #HV12
+elim (cpxs_inv_appl1 … H) -H *
+[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #b #W #T0 #HT0 #HU
+  elim (cpxs_inv_abbr1 … HT0) -HT0 *
+  [ #V3 #T3 #_ #_ #H destruct
+  | #X #HT2 #H #H0 destruct
+    elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
+    @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+    @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+    @(cpxs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
+    /4 width=7 by cpx_zeta, lift_bind, lift_flat/
+  ]
+| #b #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
+  @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
+  elim (cpxs_inv_abbr1 … HT0) -HT0 *
+  [ #V5 #T5 #HV5 #HT5 #H destruct
+    lapply (cpxs_lift … HV13 (L.ⓓV) … HV12 … HV34) -V1 -V3
+    /3 width=2 by cpxs_flat, cpxs_bind, drop_drop/
+  | #X #HT1 #H #H0 destruct
+    elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
+    lapply (cpxs_lift … HV13 (L.ⓓV0) … HV12 … HV34) -V3 /2 width=2 by drop_drop/ #HV24
+    @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+    @(cpxs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V5 -T5
+    @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
+  ]
+]
+qed-.
+
+lemma cpxs_fwd_cast: ∀h,g,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, g] U →
+                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ➡*[h, g] U | ⦃G, L⦄ ⊢ W ➡*[h, g] U.
+#h #g #G #L #W #T #U #H
+elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
+#W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts_vector.ma
new file mode 100644 (file)
index 0000000..70a66b9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/cpxs_tsts_vector.ma
@@ -0,0 +1,190 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/tsts_vector.ma".
+include "basic_2A/substitution/lift_vector.ma".
+include "basic_2A/computation/cpxs_tsts.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Vector form of forward lemmas involving same top term structure **********)
+
+(* Basic_1: was just: nf2_iso_appls_lref *)
+lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
+                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≂ U.
+#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair/
+| #a #W0 #T0 #HT0 #HU
+  lapply (IHVs … HT0) -IHVs -HT0 #HT0
+  elim (tsts_inv_bind_applv_simple … HT0) //
+| #a #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
+  lapply (IHVs … HT0) -IHVs -HT0 #HT0
+  elim (tsts_inv_bind_applv_simple … HT0) //
+]
+qed-.
+
+lemma cpxs_fwd_sort_vector: ∀h,g,G,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
+                            ⒶVs.⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
+#h #g #G #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #a #W1 #T1 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tsts_inv_bind_applv_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
+  ]
+| #a #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tsts_inv_bind_applv_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
+  ]
+]
+qed-.
+
+
+(* Basic_1: was just: pr3_iso_appls_beta *)
+lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
+                            ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
+#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #b #W1 #T1 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tsts_inv_bind_applv_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV0.ⓛ{b}W1.T1)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
+  ]
+| #b #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (tsts_inv_bind_applv_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV1.ⓓ{b}V3.T1)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
+  ]
+]
+qed-.
+
+lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+                             ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
+                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
+                             ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
+#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
+#V #Vs #IHVs #U #H -K -V1
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #b #W0 #T0 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (tsts_inv_bind_applv_simple … HT0) //
+  | @or_intror -i (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /3 width=1 by cpxs_flat_dx, cpr_cpx, cpr_beta/
+  ]
+| #b #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (tsts_inv_bind_applv_simple … HT0) //
+  | @or_intror -i (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V3.T0)) /3 width=3 by cpxs_flat, cpr_cpx, cpr_theta/
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_abbr *)
+lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
+                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
+                             ⒶV1s. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
+#h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1 by or_intror/
+#V1s #V2s #V1a #V2a #HV12a #HV12s #a
+generalize in match HV12a; -HV12a
+generalize in match V2a; -V2a
+generalize in match V1a; -V1a
+elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
+#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+| #b #W0 #T0 #HT0 #HU
+  elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
+  [ -HV12a -HV12b -HU
+    elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
+  | @or_intror -V1s (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    elim (cpxs_inv_abbr1 … HT0) -HT0 *
+    [ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
+    | -V1b #X #HT1 #H #H0 destruct
+      elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+      @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0))
+      /4 width=7 by cpxs_beta_dx, cpx_zeta, lift_bind, lift_flat/
+    ]
+  ]
+| #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
+  elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
+  [ -HV12a -HV10a -HV0a -HU
+    elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
+  | @or_intror -V1s -V1b (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    elim (cpxs_inv_abbr1 … HT0) -HT0 *
+    [ #V1 #T1 #HV1 #HT1 #H destruct
+      lapply (cpxs_lift … HV10a (L.ⓓV) (Ⓕ) … HV12a … HV0a) -V1a -V0a [ /2 width=1 by drop_drop/ ] #HV2a
+      @(cpxs_trans … (ⓓ{a}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
+    | #X #HT1 #H #H0 destruct
+      elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
+      lapply (cpxs_lift … HV10a (L.ⓓV0) (Ⓕ) … HV12a … HV0a) -V0a [ /2 width=1 by drop_drop/ ] #HV2a
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+      @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1
+      @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
+    ]
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_cast *)
+lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
+                            ∨∨ ⒶVs. ⓝW. T ≂ U
+                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
+                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
+#h #g #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+#V #Vs #IHVs #W #T #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
+| #b #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (tsts_inv_bind_applv_simple … HT0) //
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+  | @or3_intro2 -T (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV.ⓛ{b}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+  ]
+| #b #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (tsts_inv_bind_applv_simple … HT0) //
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+  | @or3_intro2 -T (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -U
+    @(cpxs_strap1 … (ⓐV0.ⓓ{b}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx.ma
new file mode 100644 (file)
index 0000000..b77127d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx.ma
@@ -0,0 +1,133 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/sn_5.ma".
+include "basic_2A/reduction/cnx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+definition csx: ∀h. sd h → relation3 genv lenv term ≝
+                λh,g,G,L. SN … (cpx h g G L) (eq …).
+
+interpretation
+   "context-sensitive extended strong normalization (term)"
+   'SN h g G L T = (csx h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
+               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
+                     R T1
+               ) →
+               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H elim H -T1
+/5 width=1 by SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: sn3_pr2_intro *)
+lemma csx_intro: ∀h,g,G,L,T1.
+                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
+                 ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+/4 width=1 by SN_intro/ qed.
+
+lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
+qed-.
+
+(* Basic_1: was just: sn3_nf2 *)
+lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/2 width=1 by NF_to_SN/ qed.
+
+lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
+#h #g #G #L #k elim (deg_total h g k)
+#d generalize in match k; -k @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #k #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
+[ #H destruct elim HX //
+| -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
+]
+qed.
+
+(* Basic_1: was just: sn3_cast *)
+lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
+#h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_cast1 … H1) -H1
+[ * #W0 #T0 #HLW0 #HLT0 #H destruct
+  elim (eq_false_inv_tpair_sn … H2) -H2
+  [ /3 width=3 by csx_cpx_trans/
+  | -HLW0 * #H destruct /3 width=1 by/
+  ]
+|2,3: /3 width=3 by csx_cpx_trans/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #V2 #HLV2 #HV2
+@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_head *)
+lemma csx_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
+
+fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_bind *)
+lemma csx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
+
+fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_flat *)
+lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
+
+lemma csx_fwd_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
+
+lemma csx_fwd_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
+
+(* Basic_1: removed theorems 14:
+            sn3_cdelta
+            sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
+            sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
+            sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_aaa.ma
new file mode 100644 (file)
index 0000000..7330de6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_aaa.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/gcp_aaa.ma".
+include "basic_2A/computation/cpxs_aaa.ma".
+include "basic_2A/computation/csx_tsts_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Main properties on atomic arity assignment *******************************)
+
+theorem aaa_csx: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #A #H
+@(gcr_aaa … (csx_gcp h g) (csx_gcr h g) … H)
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+fact aaa_ind_csx_aux: ∀h,g,G,L,A. ∀R:predicate term.
+                      (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                      ) →
+                      ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #g #G #L #A #R #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
+qed-.
+
+lemma aaa_ind_csx: ∀h,g,G,L,A. ∀R:predicate term.
+                   (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                   ) →
+                   ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
+/5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-.
+
+fact aaa_ind_csx_alt_aux: ∀h,g,G,L,A. ∀R:predicate term.
+                          (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                                (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                          ) →
+                          ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #g #G #L #A #R #IH #T #H @(csx_ind_alt … H) -T /4 width=5 by cpxs_aaa_conf/
+qed-.
+
+lemma aaa_ind_csx_alt: ∀h,g,G,L,A. ∀R:predicate term.
+                       (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                             (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                       ) →
+                       ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
+/5 width=9 by aaa_ind_csx_alt_aux, aaa_csx/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_alt.ma
new file mode 100644 (file)
index 0000000..396dc0e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_alt.ma
@@ -0,0 +1,107 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/snalt_5.ma".
+include "basic_2A/computation/cpxs.ma".
+include "basic_2A/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* alternative definition of csx *)
+definition csxa: ∀h. sd h → relation3 genv lenv term ≝
+                 λh,g,G,L. SN … (cpxs h g G L) (eq …).
+
+interpretation
+   "context-sensitive extended strong normalization (term) alternative"
+   'SNAlt h g G L T = (csxa h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csxa_ind: ∀h,g,G,L. ∀R:predicate term.
+                (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+                      (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                ) →
+                ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma csx_intro_cpxs: ∀h,g,G,L,T1.
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
+                      ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+/4 width=1 by cpx_cpxs, csx_intro/ qed.
+
+(* Basic_1: was just: sn3_intro *)
+lemma csxa_intro: ∀h,g,G,L,T1.
+                  (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
+                  ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+/4 width=1 by SN_intro/ qed.
+
+fact csxa_intro_aux: ∀h,g,G,L,T1. (
+                        ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+                     ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+/4 width=3 by csxa_intro/ qed-.
+
+(* Basic_1: was just: sn3_pr3_trans (old version) *)
+lemma csxa_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+                       ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2.
+#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+@csxa_intro #T #HLT2 #HT2
+elim (eq_term_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1 by/
+| -HT1 -HT2 /3 width=4 by/
+qed.
+
+(* Basic_1: was just: sn3_pr2_intro (old version) *)
+lemma csxa_intro_cpx: ∀h,g,G,L,T1. (
+                         ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+                      ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+#h #g #G #L #T1 #H
+@csxa_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
+[ -H #H destruct #H
+  elim H //
+| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
+  elim (eq_term_dec T0 T) #HT0
+  [ -HLT1 -HLT2 -H /3 width=1 by/
+  | -IHT -HT12 /4 width=3 by csxa_cpxs_trans/
+  ]
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem csx_csxa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
+#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
+qed.
+
+theorem csxa_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
+qed.
+
+(* Basic_1: was just: sn3_pr3_trans *)
+lemma csx_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                      ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
+qed-.
+
+(* Main eliminators *********************************************************)
+
+lemma csx_ind_alt: ∀h,g,G,L. ∀R:predicate term.
+                   (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                   ) →
+                   ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
+#h #g #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_fpbs.ma
new file mode 100644 (file)
index 0000000..0d2f18a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_fpbs.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbs.ma".
+include "basic_2A/computation/csx_lleq.ma".
+include "basic_2A/computation/csx_lpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Advanced properties ******************************************************)
+
+lemma csx_fpb_conf: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
+/2 width=5 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_lleq_conf/
+qed-.
+
+lemma csx_fpbs_conf: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                     ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+/2 width=5 by csx_fpb_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma
new file mode 100644 (file)
index 0000000..6bda543
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lift.ma
@@ -0,0 +1,119 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cnx_lift.ma".
+include "basic_2A/computation/gcp.ma".
+include "basic_2A/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was just: sn3_lift *)
+lemma csx_lift: ∀h,g,G,L2,L1,T1,s,l,m. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+                ∀T2. ⬇[s, l, m] L2 ≡ L1 → ⬆[l, m] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+@csx_intro #T #HLT2 #HT2
+elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1 by/
+qed.
+
+(* Basic_1: was just: sn3_gen_lift *)
+lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,s,l,m. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
+                    ∀T2. ⬇[s, l, m] L1 ≡ L2 → ⬆[l, m] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+@csx_intro #T #HLT2 #HT2
+elim (lift_total T l m) #T0 #HT0
+lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: sn3_gen_def *)
+lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V →
+                         ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
+#h #g #I #G #L #K #V #i #HLK #Hi
+elim (lift_total V 0 (i+1))
+/4 width=9 by csx_inv_lift, csx_cpx_trans, cpx_delta, drop_fwd_drop2/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_abbr *)
+lemma csx_lref_bind: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
+#h #g #I #G #L #K #V #i #HLK #HV
+@csx_intro #X #H #Hi
+elim (cpx_inv_lref1 … H) -H
+[ #H destruct elim Hi //
+| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
+  lapply (drop_mono … HLK0 … HLK) -HLK #H destruct
+  /3 width=8 by csx_lift, csx_cpx_trans, drop_fwd_drop2/
+]
+qed.
+
+lemma csx_appl_simple: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1.
+                       (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
+                       𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
+#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
+@csx_intro #X #H1 #H2
+elim (cpx_inv_appl1_simple … H1) // -H1
+#V0 #T0 #HLV0 #HLT10 #H destruct
+elim (eq_false_inv_tpair_dx … H2) -H2
+[ -IHV -HT1 /4 width=3 by csx_cpx_trans, cpx_pair_sn/
+| -HLT10 * #H #HV0 destruct
+  @IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto 17s *)
+]
+qed.
+
+lemma csx_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/2 width=8 by csx_inv_lref_bind, csx_inv_lift, csx_fwd_flat_dx, csx_fwd_bind_dx, csx_fwd_pair_sn/
+qed-.
+
+lemma csx_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
+[ /2 width=5 by csx_fqu_conf/
+| * #HG #HL #HT destruct //
+]
+qed-.
+
+lemma csx_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by csx_fqu_conf/
+qed-.
+
+lemma csx_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
+[ /2 width=5 by csx_fqup_conf/
+| * #HG #HL #HT destruct //
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem csx_gcp: ∀h,g. gcp (cpx h g) (eq …) (csx h g).
+#h #g @mk_gcp
+[ normalize /3 width=13 by cnx_lift/
+| #G #L elim (deg_total h g 0) /3 width=8 by cnx_sort_iter, ex_intro/
+| /2 width=8 by csx_lift/
+| /2 width=3 by csx_fwd_flat_dx/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lleq.ma
new file mode 100644 (file)
index 0000000..6e23a71
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lleq.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_lleq.ma".
+include "basic_2A/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma csx_lleq_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
+                     ∀L2. L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #T #H @(csx_ind … H) -T
+/4 width=6 by csx_intro, cpx_lleq_conf_dx, lleq_cpx_trans/
+qed-.
+
+lemma csx_lleq_trans: ∀h,g,G,L1,L2,T.
+                      L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T → ⦃G, L1⦄ ⊢ ⬊*[h, g] T.
+/3 width=3 by csx_lleq_conf, lleq_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpx.ma
new file mode 100644 (file)
index 0000000..72e0bc1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpx.ma
@@ -0,0 +1,138 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/tsts_tsts.ma".
+include "basic_2A/computation/cpxs_cpxs.ma".
+include "basic_2A/computation/csx_alt.ma".
+include "basic_2A/computation/csx_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Advanced properties ******************************************************)
+
+lemma csx_lpx_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+                    ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T
+/4 width=3 by csx_intro, lpx_cpx_trans/
+qed-.
+
+lemma csx_abst: ∀h,g,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+                ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
+#h #g #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_abst1 … H1) -H1
+#W0 #T0 #HLW0 #HLT0 #H destruct
+elim (eq_false_inv_tpair_sn … H2) -H2
+[ -IHT #H lapply (csx_cpx_trans … HLT0) // -HT
+  #HT0 lapply (csx_lpx_conf … (L.ⓛW0) … HT0) -HT0 /3 width=1 by lpx_pair/
+| -IHW -HLW0 -HT * #H destruct /3 width=1 by/
+]
+qed.
+
+lemma csx_abbr: ∀h,g,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
+                ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
+#h #g #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_abbr1 … H1) -H1 *
+[ #V1 #T1 #HLV1 #HLT1 #H destruct
+  elim (eq_false_inv_tpair_sn … H2) -H2
+  [ /4 width=5 by csx_cpx_trans, csx_lpx_conf, lpx_pair/
+  | -IHV -HLV1 * #H destruct /3 width=1 by cpx_cpxs/
+  ]
+| -IHV -IHT -H2
+  /3 width=8 by csx_cpx_trans, csx_inv_lift, drop_drop/
+]
+qed.
+
+fact csx_appl_beta_aux: ∀h,g,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
+                        ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
+#h #g #a #G #L #X #H @(csx_ind … H) -X
+#X #HT1 #IHT1 #V #W #T1 #H1 destruct
+@csx_intro #X #H1 #H2
+elim (cpx_inv_appl1 … H1) -H1 *
+[ -HT1 #V0 #Y #HLV0 #H #H0 destruct
+  elim (cpx_inv_abst1 … H) -H #W0 #T0 #HLW0 #HLT0 #H destruct
+  @IHT1 -IHT1 [4: // | skip |3: #H destruct /2 width=1 by/ ] -H2
+  lapply (lsubr_cpx_trans … HLT0 (L.ⓓⓝW.V) ?) -HLT0 /3 width=1 by cpx_bind, cpx_flat, lsubr_beta/
+| -IHT1 -H2 #b #V0 #W0 #W2 #T0 #T2 #HLV0 #HLW02 #HLT02 #H1 #H3 destruct
+  lapply (lsubr_cpx_trans … HLT02 (L.ⓓⓝW0.V) ?) -HLT02
+  /4 width=5 by csx_cpx_trans, cpx_bind, cpx_flat, lsubr_beta/
+| -HT1 -IHT1 -H2 #b #V0 #V1 #W0 #W1 #T0 #T3 #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: sn3_beta *)
+lemma csx_appl_beta: ∀h,g,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
+/2 width=3 by csx_appl_beta_aux/ qed.
+
+fact csx_appl_theta_aux: ∀h,g,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⬆[0, 1] V1 ≡ V2 →
+                         ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
+#h #g #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+lapply (csx_fwd_pair_sn … HVT) #HV
+lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1 … HL) -HL *
+[ -HV #V0 #Y #HLV10 #HL #H0 destruct
+  elim (cpx_inv_abbr1 … HL) -HL *
+  [ #V3 #T3 #HV3 #HLT3 #H0 destruct
+    elim (lift_total V0 0 1) #V4 #HV04
+    elim (eq_term_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V3.ⓐV4.T3))
+    [ -IHVT #H0 destruct
+      elim (eq_false_inv_tpair_sn … H) -H
+      [ -HLV10 -HV3 -HLT3 -HVT
+        >(lift_inj … HV12 … HV04) -V4
+        #H elim H //
+      | * #_ #H elim H //
+      ]
+    | -H -HVT #H
+      lapply (cpx_lift … HLV10 (L.ⓓV) (Ⓕ) … HV12 … HV04) -HLV10 -HV12 /2 width=1 by drop_drop/ #HV24
+      @(IHVT … H … HV04) -IHVT /4 width=1 by cpx_cpxs, cpx_bind, cpx_flat/
+    ]
+  | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
+    lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1 by cpx_flat/ -T #HVT0
+    lapply (csx_inv_lift … L … (Ⓕ) … 1 HVT0 ? ? ?) -HVT0
+    /3 width=5 by csx_cpx_trans, cpx_pair_sn, drop_drop, lift_flat/
+  ]
+| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
+| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
+  lapply (cpx_lift … HLV10 (L. ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /2 width=2 by drop_drop/ #HLV23
+  @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
+  @(csx_lpx_conf … (L.ⓓW0)) /2 width=1 by lpx_pair/ -W1
+  /4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
+]
+qed-.
+
+lemma csx_appl_theta: ∀h,g,a,V1,V2. ⬆[0, 1] V1 ≡ V2 →
+                      ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
+/2 width=5 by csx_appl_theta_aux/ qed.
+
+(* Basic_1: was just: sn3_appl_appl *)
+lemma csx_appl_simple_tsts: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≂ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
+                            𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
+#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1_simple … HL) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (eq_false_inv_tpair_sn … H) -H
+[ -IHT1 #HV0
+  @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
+  @IHV -IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/
+| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
+  elim (tsts_dec T1 T0) #H2T10
+  [ @IHT1 -IHT1 /4 width=3 by cpxs_strap2, cpxs_strap1, tsts_canc_sn, simple_tsts_repl_dx/
+  | -IHT1 -H3T1 -H1T10 /3 width=1 by cpx_cpxs/
+  ]
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpxs.ma
new file mode 100644 (file)
index 0000000..4128a4e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_lpxs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/csx_lpx.ma".
+include "basic_2A/computation/lpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+(* Properties on sn extended parallel computation for local environments ****)
+
+lemma csx_lpxs_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                     ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L1 #L2 #H @(lpxs_ind … H) -L2 /3 by lpxs_strap1, csx_lpx_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma
new file mode 100644 (file)
index 0000000..51345fb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_tsts_vector.ma
@@ -0,0 +1,128 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/gcp_cr.ma".
+include "basic_2A/computation/cpxs_tsts_vector.ma".
+include "basic_2A/computation/csx_lpx.ma".
+include "basic_2A/computation/csx_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_appls_lref *)
+lemma csx_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
+                     ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
+#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
+#V #Vs #IHV #H
+elim (csxv_inv_cons … H) -H #HV #HVs
+@csx_appl_simple_tsts /2 width=1 by applv_simple/ -IHV -HV -HVs
+#X #H #H0
+lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
+elim (H0) -H0 //
+qed.
+
+lemma csx_applv_sort: ∀h,g,G,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
+#h #g #G #L #k elim (deg_total h g k)
+#d generalize in match k; -k @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
+#d #IHd #k #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd #Vs elim Vs -Vs /2 width=1 by/
+#V #Vs #IHVs #HVVs
+elim (csxv_inv_cons … HVVs) #HV #HVs
+@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_atom/ -IHVs -HV -HVs
+#X #H #H0
+elim (cpxs_fwd_sort_vector … H) -H #H
+[ elim H0 -H0 //
+| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1 by cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_beta *)
+lemma csx_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
+#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+#V0 #Vs #IHV #V #W #T #H1T
+lapply (csx_fwd_pair_sn … H1T) #HV0
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
+#X #H #H0
+elim (cpxs_fwd_beta_vector … H) -H #H
+[ -H1T elim H0 -H0 //
+| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+lemma csx_applv_delta: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+                       ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
+                       ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
+#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_fwd_drop2/
+| #V #Vs #IHV #H1T
+  lapply (csx_fwd_pair_sn … H1T) #HV
+  lapply (csx_fwd_flat_dx … H1T) #H2T
+  @csx_appl_simple_tsts /2 width=1 by applv_simple, simple_atom/ -IHV -HV  -H2T
+  #X #H #H0
+  elim (cpxs_fwd_delta_vector … HLK … HV12 … H) -HLK -HV12 -H #H
+  [ -H1T elim H0 -H0 //
+  | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+  ]
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_abbr *)
+lemma csx_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
+                       ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
+                       ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
+#h #g #a #G #L #V1s #V2s * -V1s -V2s /2 width=1 by/
+#V1s #V2s #V1 #V2 #HV12 #H
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1s -V2s /2 width=3 by csx_appl_theta/
+#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H
+lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
+lapply (csx_fwd_pair_sn … H) #HW1
+lapply (csx_fwd_flat_dx … H) #H1
+@csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12s -HW1 -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1 by liftv_cons/ -HV12s -HV12
+[ -H #H elim H2 -H2 //
+| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_cast *)
+lemma csx_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
+#h #g #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+#V #Vs #IHV #W #T #H1W #H1T
+lapply (csx_fwd_pair_sn … H1W) #HV
+lapply (csx_fwd_flat_dx … H1W) #H2W
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2W -H2T
+#X #H #H0
+elim (cpxs_fwd_cast_vector … H) -H #H
+[ -H1W -H1T elim H0 -H0 //
+| -H1W -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+theorem csx_gcr: ∀h,g. gcr (cpx h g) (eq …) (csx h g) (csx h g).
+#h #g @mk_gcr //
+[ /3 width=1 by csx_applv_cnx/
+|2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
+| /2 width=7 by csx_applv_delta/
+| #G #L #V1s #V2s #HV12s #a #V #T #H #HV
+  @(csx_applv_theta … HV12s) -HV12s
+  @csx_abbr //
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma
new file mode 100644 (file)
index 0000000..b45bc8b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/csx_vector.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/term_vector.ma".
+include "basic_2A/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
+                 λh,g,G,L. all … (csx h g G L).
+
+interpretation
+   "context-sensitive strong normalization (term vector)"
+   'SN h g G L Ts = (csxv h g G L Ts).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma csxv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
+                     ⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
+normalize // qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma csx_fwd_applv: ∀h,g,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Ⓐ Vs.T →
+                     ⦃G, L⦄ ⊢ ⬊*[h, g] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
+#V #Vs #IHVs #HVs
+lapply (csx_fwd_pair_sn … HVs) #HV
+lapply (csx_fwd_flat_dx … HVs) -HVs #HVs
+elim (IHVs HVs) -IHVs -HVs /3 width=1 by conj/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg.ma
new file mode 100644 (file)
index 0000000..0fc2771
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lazybtpredstarproper_8.ma".
+include "basic_2A/reduction/fpb.ma".
+include "basic_2A/computation/fpbs.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
+                 λh,g,G1,L1,T1,G2,L2,T2.
+                 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+
+interpretation "'qrst' proper parallel computation (closure)"
+   'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpb_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by ex2_3_intro/ qed.
+
+lemma fpbg_fpbq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+/3 width=9 by fpbs_strap1, ex2_3_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fleq.ma
new file mode 100644 (file)
index 0000000..a4f2708
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fleq.ma
@@ -0,0 +1,73 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fleq_fleq.ma".
+include "basic_2A/reduction/fpbq_alt.ma".
+include "basic_2A/computation/fpbg.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbq_trans, fleq_fpbq/ qed-.
+
+lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
+elim (fleq_fpb_trans …  H1 … H0) -G -L -T
+/4 width=9 by fpbs_strap2, fleq_fpbq, ex2_3_intro/
+qed-.
+
+(* alternative definition of fpbs *******************************************)
+
+lemma fleq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
+                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by lleq_fpbs/
+qed.
+
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
+                     ⦃G1, L1, T1⦄ >≡[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+/3 width=5 by fpbs_strap2, fpb_fpbq/
+qed-.
+
+lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+                 ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 @(fpbq_ind_alt … H2) -H2 #H2
+  [ /3 width=5 by fleq_trans, or_introl/
+  | elim (fleq_fpb_trans … H1 … H2) -G -L -T
+    /4 width=5 by ex2_3_intro, or_intror, fleq_fpbs/
+  | /3 width=5 by fpbg_fleq_trans, or_intror/
+  | /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
+  ]
+]
+qed-.
+
+(* Advanced properties of "qrst" parallel computation on closures ***********)
+
+lemma fpbs_fpb_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
+                      ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+                      ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+#h #g #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H2 elim (fleq_fpb_trans … H12 … H2) -F2 -K2 -T2
+  /3 width=5 by fleq_fpbs, ex2_3_intro/
+| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
+  @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbg.ma
new file mode 100644 (file)
index 0000000..9387b5f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbg.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbg_fpbs.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Main properties **********************************************************)
+
+theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
+/3 width=5 by fpbg_fpbs_trans, fpbg_fwd_fpbs/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbs.ma
new file mode 100644 (file)
index 0000000..366819f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbs.ma
@@ -0,0 +1,68 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/lpxs_lleq.ma".
+include "basic_2A/computation/fpbs_lift.ma".
+include "basic_2A/computation/fpbg_fleq.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on "qrst" parallel reduction on closures **********************)
+
+lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+                      ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+                      ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
+
+lemma fpbq_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
+                       ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
+/2 width=5 by fleq_fpbg_trans, fpb_fpbg_trans/
+qed-.
+
+(* Properties on "qrst" parallel compuutation on closures *******************)
+
+lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
+qed.
+
+lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                 (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
+qed.
+
+lemma lstas_fpbg: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
+                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
+
+lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                 (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_lift.ma
new file mode 100644 (file)
index 0000000..aa27d1f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_lift.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/fpb_lift.ma".
+include "basic_2A/computation/fpbg.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpbg: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 →
+                ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/4 width=2 by fpb_fpbg, sta_fpb/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs.ma
new file mode 100644 (file)
index 0000000..e72a56c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs.ma
@@ -0,0 +1,161 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btpredstar_8.ma".
+include "basic_2A/multiple/fqus.ma".
+include "basic_2A/reduction/fpbq.ma".
+include "basic_2A/computation/cpxs.ma".
+include "basic_2A/computation/lpxs.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
+                 λh,g. tri_TC … (fpbq h g).
+
+interpretation "'qrst' parallel computation (closure)"
+   'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpbs_ind: ∀h,g,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
+                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+/3 width=8 by tri_TC_star_ind/ qed-.
+
+lemma fpbs_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+/3 width=8 by tri_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
+/2 width=1 by tri_inj/ qed.
+
+lemma fpbq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fpbs_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
+
+lemma fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 
+/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
+qed.
+
+lemma fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 
+/3 width=5 by fpbq_fquq, tri_step/
+qed.
+
+lemma cpxs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpbq_cpx, fpbs_strap1/
+qed.
+
+lemma lpxs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
+/3 width=5 by fpbq_lpx, fpbs_strap1/
+qed.
+
+lemma lleq_fpbs: ∀h,g,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
+
+lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
+
+lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
+
+lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
+/3 width=5 by fpbs_strap1, fpbq_fquq/
+qed-.
+
+lemma fpbs_fqup_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
+
+lemma fpbs_cpxs_trans: ∀h,g,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpbs_strap1, fpbq_cpx/
+qed-.
+
+lemma fpbs_lpxs_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
+#h #g #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
+/3 width=5 by fpbs_strap1, fpbq_lpx/
+qed-.
+
+lemma fpbs_lleq_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
+
+lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
+/3 width=5 by fpbs_strap2, fpbq_fquq/
+qed-.
+
+lemma cpxs_fpbs_trans: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
+/3 width=5 by fpbs_strap2, fpbq_cpx/
+qed-.
+
+lemma lpxs_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1⦄ ⊢ ➡*[h, g] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
+/3 width=5 by fpbs_strap2, fpbq_lpx/
+qed-.
+
+lemma lleq_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
+
+lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                      ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+
+lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                      ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
+
+lemma fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
+                      ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
+
+lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                           ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
+
+lemma lpxs_lleq_fpbs: ∀h,g,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L →
+                      L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=3 by lpxs_fpbs_trans, lleq_fpbs/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+                    ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_aaa.ma
new file mode 100644 (file)
index 0000000..58d3424
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_aaa.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/fpbq_aaa.ma".
+include "basic_2A/computation/fpbs.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma fpbs_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                     ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A
+/2 width=8 by fpbq_aaa_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_alt.ma
new file mode 100644 (file)
index 0000000..872928d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_alt.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btpredstaralt_8.ma".
+include "basic_2A/multiple/lleq_fqus.ma".
+include "basic_2A/computation/cpxs_lleq.ma".
+include "basic_2A/computation/lpxs_lleq.ma".
+include "basic_2A/computation/fpbs.ma".
+
+(* "QREST" PARALLEL COMPUTATION FOR CLOSURES ********************************)
+
+(* Note: alternative definition of fpbs *)
+definition fpbsa: ∀h. sd h → tri_relation genv lenv term ≝
+                  λh,g,G1,L1,T1,G2,L2,T2.
+                  ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+                         ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
+                         ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+
+interpretation "'big tree' parallel computation (closure) alternative"
+   'BTPRedStarAlt h g G1 L1 T1 G2 L2 T2 = (fpbsa h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpb_fpbsa_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 | #L #HL1 ]
+#G2 #L2 #T2 * #L00 #L0 #T0 #HT0 #HG2 #HL00 #HL02
+[ elim (fquq_cpxs_trans … HT0 … HG1) -T
+  /3 width=7 by fqus_strap2, ex4_3_intro/
+| /3 width=7 by cpxs_strap2, ex4_3_intro/
+| lapply (lpx_cpxs_trans … HT0 … HL1) -HT0 #HT10
+  elim (lpx_fqus_trans … HG2 … HL1) -L
+  /3 width=7 by lpxs_strap2, cpxs_trans, ex4_3_intro/
+| lapply (lleq_cpxs_trans … HT0 … HL1) -HT0 #HT0
+  lapply (cpxs_lleq_conf_sn … HT0 … HL1) -HL1 #HL1
+  elim (lleq_fqus_trans … HG2 … HL1) -L #K00 #HG12 #HKL00
+  elim (lleq_lpxs_trans … HL00 … HKL00) -L00
+  /3 width=9 by lleq_trans, ex4_3_intro/
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fpbs_fpbsa: ∀h,g,G1,G2,L1,L2,T1,T2.
+                    ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+/2 width=7 by fpb_fpbsa_trans, ex4_3_intro/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem fpbsa_inv_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
+                        ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_lleq/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpbs_intro_alt: ∀h,g,G1,G2,L1,L0,L,L2,T1,T,T2.
+                      ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
+                      ⦃G2, L0⦄ ⊢ ➡*[h, g] L → L ≡[T2, 0] L2 →  ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ .
+/3 width=7 by fpbsa_inv_fpbs, ex4_3_intro/ qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+lemma fpbs_inv_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                    ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+                              ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
+                              ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+/2 width=1 by  fpbs_fpbsa/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpb.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpb.ma
new file mode 100644 (file)
index 0000000..634c21a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpb.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/fpbq_alt.ma".
+include "basic_2A/computation/fpbs_alt.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Properties on extended context-sensitive parallel computation for terms **)
+
+lemma fpbs_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_alt … H) -H
+#L00 #L0 #T0 #HT10 #H10 #HL00 #HL02 lapply (lleq_cpx_trans … HTU2 … HL02) -HTU2
+#HTU2 lapply (cpx_lleq_conf_sn … HTU2 … HL02) -HL02
+#HL02 lapply (lpxs_cpx_trans … HTU2 … HL00) -HTU2
+#HTU2 elim (fqus_cpxs_trans_neq … H10 … HTU2 HnTU2) -H10 -HTU2 -HnTU2
+#U0 #HTU0 #HnTU0 #HU02 elim (eq_term_dec T1 T0) #HnT10 destruct
+[ -HT10 elim (cpxs_neq_inv_step_sn … HTU0 HnTU0) -HTU0 -HnTU0
+| -HnTU0 elim (cpxs_neq_inv_step_sn … HT10 HnT10) -HT10 -HnT10
+]
+/4 width=10 by fpbs_intro_alt, cpxs_trans, ex3_intro/
+qed-.
+
+(* Properties on "rst" proper parallel reduction on closures ****************)
+
+lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbq_fpbs, fpb_fpbq/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpbs.ma
new file mode 100644 (file)
index 0000000..1f9a654
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_fpbs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbs.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Main properties **********************************************************)
+
+theorem fpbs_trans: ∀h,g. tri_transitive … (fpbs h g).
+/2 width=5 by tri_TC_transitive/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_lift.ma
new file mode 100644 (file)
index 0000000..4b538ed
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_lift.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cpxs_lift.ma".
+include "basic_2A/computation/fpbs.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_fpbs: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+/3 width=5 by cpxs_fpbs, lstas_cpxs/ qed.
+
+lemma sta_fpbs: ∀h,g,G,L,T,U,d.
+                ⦃G, L⦄ ⊢ T ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
+                ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
+/2 width=5 by lstas_fpbs/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma cpr_lpr_sta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,d2.
+                        ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+                        ⦃G, L2⦄ ⊢ T2 ▪[h, g] d2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
+                        ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
+/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb.ma
new file mode 100644 (file)
index 0000000..3425fd9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb.ma
@@ -0,0 +1,47 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btsn_5.ma".
+include "basic_2A/reduction/fpb.ma".
+include "basic_2A/computation/csx.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+inductive fsb (h) (g): relation3 genv lenv term ≝
+| fsb_intro: ∀G1,L1,T1. (
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
+             ) → fsb h g G1 L1 T1
+.
+
+interpretation
+   "'qrst' strong normalization (closure)"
+   'BTSN h g G L T = (fsb h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma fsb_ind_alt: ∀h,g. ∀R: relation3 …. (
+                      ∀G1,L1,T1. ⦥[h,g] ⦃G1, L1, T1⦄ → (
+                         ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+                      ) → R G1 L1 T1
+                   ) →
+                   ∀G,L,T. ⦥[h, g] ⦃G, L, T⦄ → R G L T.
+#h #g #R #IH #G #L #T #H elim H -G -L -T
+/4 width=1 by fsb_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fsb_inv_csx: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpb_cpx/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_aaa.ma
new file mode 100644 (file)
index 0000000..ff566e3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_aaa.ma
@@ -0,0 +1,71 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbs_aaa.ma".
+include "basic_2A/computation/csx_aaa.ma".
+include "basic_2A/computation/fsb_csx.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+(* Main properties **********************************************************)
+
+(* Note: this is the "big tree" theorem ("RST" version) *)
+theorem aaa_fsb: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥[h, g] ⦃G, L, T⦄.
+/3 width=2 by aaa_csx, csx_fsb/ qed.
+
+(* Note: this is the "big tree" theorem ("QRST" version) *)
+theorem aaa_fsba: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥⦥[h, g] ⦃G, L, T⦄.
+/3 width=2 by fsb_fsba, aaa_fsb/ qed.
+
+(* Advanced eliminators on atomica arity assignment for terms ***************)
+
+fact aaa_ind_fpb_aux: ∀h,g. ∀R:relation3 genv lenv term.
+                      (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+                                    (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                    R G1 L1 T1
+                      ) →
+                      ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+#h #g #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
+#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
+/2 width=2 by fpb_fpbs/
+qed-.
+
+lemma aaa_ind_fpb: ∀h,g. ∀R:relation3 genv lenv term.
+                   (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+                                 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                 R G1 L1 T1
+                   ) →
+                   ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+/4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
+
+fact aaa_ind_fpbg_aux: ∀h,g. ∀R:relation3 genv lenv term.
+                       (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+                                     (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                     R G1 L1 T1
+                       ) →
+                       ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+#h #g #R #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
+#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
+/2 width=2 by fpbg_fwd_fpbs/
+qed-.
+
+lemma aaa_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
+                                  (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                  R G1 L1 T1
+                    ) →
+                    ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+/4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_alt.ma
new file mode 100644 (file)
index 0000000..e483e09
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_alt.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btsnalt_5.ma".
+include "basic_2A/computation/fpbg_fpbs.ma".
+include "basic_2A/computation/fsb.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+(* Note: alternative definition of fsb *)
+inductive fsba (h) (g): relation3 genv lenv term ≝
+| fsba_intro: ∀G1,L1,T1. (
+                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → fsba h g G2 L2 T2
+              ) → fsba h g G1 L1 T1.
+
+interpretation
+   "'big tree' strong normalization (closure) alternative"
+   'BTSNAlt h g G L T = (fsba h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
+                       ∀G1,L1,T1. ⦥⦥[h,g] ⦃G1, L1, T1⦄ → (
+                          ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+                       ) → R G1 L1 T1
+                    ) →
+                    ∀G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → R G L T.
+#h #g #R #IH #G #L #T #H elim H -G -L -T
+/4 width=1 by fsba_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦥⦥[h, g] ⦃G1, L1, T1⦄ →
+                       ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥⦥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
+/4 width=5 by fsba_intro, fpbs_fpbg_trans/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fsb_fsba: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄  → ⦥⦥[h, g] ⦃G, L, T⦄.
+#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
+#G1 #L1 #T1 #_ #IH @fsba_intro
+#G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → ⦥[h, g] ⦃G, L, T⦄.
+#h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
+/4 width=1 by fsb_intro, fpb_fpbg/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
+                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
+/4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
+                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                R G1 L1 T1
+                    ) →
+                    ∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ → R G1 L1 T1.
+#h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
+/3 width=1 by fsba_inv_fsb, fsb_fsba/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_csx.ma
new file mode 100644 (file)
index 0000000..074b53e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_csx.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbs_fpb.ma".
+include "basic_2A/computation/fpbs_fpbs.ma".
+include "basic_2A/computation/csx_fpbs.ma".
+include "basic_2A/computation/lsx_csx.ma".
+include "basic_2A/computation/fsb_alt.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+(* Advanced propreties on context-sensitive extended normalizing terms ******)
+
+lemma csx_fsb_fpbs: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #L1 #T1 #H @(csx_ind … H) -T1
+#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind … G2 L2 T2) -G2 -L2 -T2
+#G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf … H10) // -HT1
+#HT0 generalize in match IHu; -IHu generalize in match H10; -H10
+@(lsx_ind … (csx_lsx … HT0 0)) -L0
+#L0 #_ #IHd #H10 #IHu @fsb_intro
+#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd |  ]
+[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
+| #T2 #HT02 #HnT02 elim (fpbs_cpx_trans_neq … H10 … HT02 HnT02) -T0
+  /3 width=4 by/
+| #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
+  [ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
+  | #G3 #L3 #T3 #H03 #_ elim (lpx_fqup_trans … H03 … HL02) -L2
+    #L4 #T4 elim (eq_term_dec T0 T4) [ -IHc | -IHu ]
+    [ #H destruct /5 width=5 by fsb_fpbs_trans, lpxs_fpbs, fpbs_fqup_trans, lpx_lpxs/
+    | #HnT04 #HT04 #H04 #HL43 elim (cpxs_neq_inv_step_sn … HT04 HnT04) -HT04 -HnT04
+      #T2 #HT02 #HnT02 #HT24 elim (fpbs_cpx_trans_neq … H10 … HT02 HnT02) -T0
+      lapply (fpbs_intro_alt … G3 … L4 … L3 L3 … T3 … HT24 ? ? ?) -HT24
+      /3 width=8 by fpbs_trans, lpx_lpxs, fqup_fqus/ (**) (* full auto too slow *)
+    ]
+  ]
+]
+qed.
+
+lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦥[h, g] ⦃G, L, T⦄.
+/2 width=5 by csx_fsb_fpbs/ qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma csx_ind_fpb: ∀h,g. ∀R:relation3 genv lenv term.
+                   (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                               (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                               R G1 L1 T1
+                   ) →
+                   ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
+
+lemma csx_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                R G1 L1 T1
+                    ) →
+                    ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp.ma
new file mode 100644 (file)
index 0000000..2a217b5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/multiple/drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+                λG,L,T. NF … (RR G L) RS T.
+
+definition candidate: Type[0] ≝ relation3 genv lenv term.
+
+definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+                 ∀G. d_liftable1 (nf RR RS G) (Ⓕ).
+
+definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+                 ∀G,L. ∃k. NF … (RR G L) RS (⋆k).
+
+definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G) (Ⓕ).
+
+definition CP3 ≝ λRP:candidate.
+                 ∀G,L,T,k. RP G L (ⓐ⋆k.T) → RP G L T.
+
+(* requirements for generic computation properties *)
+record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
+{ cp0: CP0 RR RS;
+  cp1: CP1 RR RS;
+  cp2: CP2 RP;
+  cp3: CP3 RP
+}.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_lift1 *)
+lemma gcp0_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftables1 (nf RR RS G) (Ⓕ).
+#RR #RS #RP #H #G @d1_liftable_liftables @(cp0 … H)
+qed.
+
+lemma gcp2_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftables1 (RP G) (Ⓕ).
+#RR #RS #RP #H #G @d1_liftable_liftables @(cp2 … H)
+qed.
+
+(* Basic_1: was only: sns3_lifts1 *)
+lemma gcp2_lifts_all: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftables1_all (RP G) (Ⓕ).
+#RR #RS #RP #H #G @d1_liftables_liftables_all /2 width=7 by gcp2_lifts/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma
new file mode 100644 (file)
index 0000000..1bbe0d5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_aaa.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lifts_lifts.ma".
+include "basic_2A/multiple/drops_drops.ma".
+include "basic_2A/static/aaa_lifts.ma".
+include "basic_2A/static/aaa_aaa.ma".
+include "basic_2A/computation/lsubc_drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
+                             gcp RR RS RP → gcr RR RS RP RP →
+                             ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L0,cs. ⬇*[Ⓕ, cs] L0 ≡ L1 →
+                             ∀T0. ⬆*[cs] T ≡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
+                             ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
+#RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
+[ #G #L #k #L0 #cs #HL0 #X #H #L2 #HL20
+  >(lifts_inv_sort1 … H) -H
+  lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
+  lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #cs #HL01 #X #H #L2 #HL20
+  lapply (acr_gcr … H1RP H2RP B) #HB
+  elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
+  lapply (drop_fwd_drop2 … HLK1) #HK1b
+  elim (drops_drop_trans … HL01 … HLK1) #X #cs1 #i0 #HL0 #H #Hi0 #Hcs1
+  >(at_mono … Hi1 … Hi0) -i1
+  elim (drops_inv_skip2 … Hcs1 … H) -cs1 #K0 #V0 #cs0 #Hcs0 #HK01 #HV10 #H destruct
+  elim (lsubc_drop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H
+  elim (lsubc_inv_pair2 … H) -H *
+  [ #K2 #HK20 #H destruct
+    elim (lift_total V0 0 (i0 +1)) #V #HV0
+    elim (lifts_lift_trans  … Hi0 … Hcs0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
+    lapply (s5 … HB ? G ? ? (◊) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
+  | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hcs0
+    #K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
+    lapply (drop_fwd_drop2 … HLK2) #HLK2b
+    lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
+    lapply (aaa_mono … H2KV0A … HKV0B) #H destruct -H2KV0A -HKV0B
+    elim (lift_total V0 0 (i0 +1)) #V3 #HV03
+    elim (lift_total V2 0 (i0 +1)) #V #HV2
+    lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
+    lapply (s7 … HB G L2 (◊)) /3 width=7 by gcr_lift/
+  ]
+| #a #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
+  elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+  lapply (acr_gcr … H1RP H2RP A) #HA
+  lapply (acr_gcr … H1RP H2RP B) #HB
+  lapply (s1 … HB) -HB #HB
+  lapply (s6 … HA G L2 (◊) (◊)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
+| #a #G #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL02
+  elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+  @(acr_abst  … H1RP H2RP) /2 width=5 by/
+  #L3 #V3 #W3 #T3 #cs3 #HL32 #HW03 #HT03 #H1B #H2B
+  elim (drops_lsubc_trans … H1RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
+  lapply (aaa_lifts … L2 W3 … (cs @@ cs3) … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
+  @(IHA (L2. ⓛW3) … (cs + 1 @@ cs3 + 1)) -IHA
+  /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
+| #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
+  elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+  /3 width=10 by drops_nil, lifts_nil/
+| #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #cs #HL0 #X #H #L2 #HL20
+  elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+  lapply (acr_gcr … H1RP H2RP A) #HA
+  lapply (s7 … HA G L2 (◊)) /3 width=5 by/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+               ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+/2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
+
+lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+               ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
+lapply (acr_gcr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4 by acr_aaa/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma
new file mode 100644 (file)
index 0000000..a35bc63
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/gcp_cr.ma
@@ -0,0 +1,169 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/ineint_5.ma".
+include "basic_2A/grammar/aarity.ma".
+include "basic_2A/multiple/mr2_mr2.ma".
+include "basic_2A/multiple/lifts_lift_vector.ma".
+include "basic_2A/multiple/drops_drop.ma".
+include "basic_2A/computation/gcp.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:candidate.
+                ∀G,L,T. C G L T → RP G L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
+                ∀G,L,Vs. all … (RP G L) Vs →
+                ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+
+(* Note: this generalizes Tait's ii *)
+definition S3 ≝ λC:candidate.
+                ∀a,G,L,Vs,V,T,W.
+                C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
+
+definition S4 ≝ λRP,C:candidate.
+                ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
+
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
+                C G L (ⒶVs.V2) → ⬆[0, i+1] V1 ≡ V2 →
+                ⬇[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
+
+definition S6 ≝ λRP,C:candidate.
+                ∀G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
+                ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
+
+definition S7 ≝ λC:candidate.
+                ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
+
+(* requirements for the generic reducibility candidate *)
+record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
+{ s1: S1 RP C;
+  s2: S2 RR RS RP C;
+  s3: S3 C;
+  s4: S4 RP C;
+  s5: S5 C;
+  s6: S6 RP C;
+  s7: S7 C
+}.
+
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+                 λC1,C2,G,K,T. ∀L,W,U,cs.
+                 ⬇*[Ⓕ, cs] L ≡ K → ⬆*[cs] T ≡ U → C1 G L W → C2 G L (ⓐW.U).
+
+(* the reducibility candidate associated to an atomic arity *)
+let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom     ⇒ RP
+| APair B A ⇒ cfun (acr RP B) (acr RP A)
+].
+
+interpretation
+   "candidate of reducibility of an atomic arity (abstract)"
+   'InEInt RP G L T A = (acr RP A G L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic 1: was: sc3_lift *)
+lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G) (Ⓕ).
+#RR #RS #RP #H #A elim A -A
+/3 width=8 by cp2, drops_cons, lifts_cons/
+qed.
+
+(* Basic_1: was: sc3_lift1 *)
+lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftables1 (acr RP A G) (Ⓕ).
+#RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/
+qed.
+
+(* Basic_1: was:
+   sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
+*)
+lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+               ∀A. gcr RR RS RP (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A //
+#B #A #IHB #IHA @mk_gcr
+[ #G #L #T #H
+  elim (cp1 … H1RP G L) #k #HK
+  lapply (H L (⋆k) T (◊) ? ? ?) -H //
+  [ lapply (s2 … IHB G L (◊) … HK) //
+  | /3 width=6 by s1, cp3/
+  ]
+| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+  lapply (s1 … IHB … HB) #HV0
+  @(s2 … IHA … (V0 @ V0s))
+  /3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
+| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
+  elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
+  @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #k #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  >(lifts_inv_sort1 … HY) -Y
+  lapply (s1 … IHB … HB) #HV0
+  @(s4 … IHA … (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/
+| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
+  elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
+  >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
+  elim (drops_inv_skip2 … Hcs0 … H) -H -cs0 #L2 #W1 #cs0 #Hcs0 #HLK #HVW1 #H destruct
+  elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
+  elim (lifts_lift_trans  … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
+  >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
+  @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
+| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+  elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
+  elim (lift_total V10 0 1) #V20 #HV120
+  elim (liftv_total 0 1 V10s) #V20s #HV120s
+  @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
+  @(HA … (cs + 1)) /2 width=2 by drops_skip/
+  [ @lifts_applv //
+    elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
+    >(liftv_mono … HV12s … HV10s) -V1s //
+  | @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
+  ]
+| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
+  @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
+]
+qed.
+
+lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+                ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+                   ∀L0,V0,W0,T0,cs. ⬇*[Ⓕ, cs] L0 ≡ L → ⬆*[cs] W ≡ W0 → ⬆*[cs + 1] T ≡ T0 →
+                                   ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
+                ) →
+                ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #cs #HL0 #H #HB
+lapply (acr_gcr … H1RP H2RP A) #HCA
+lapply (acr_gcr … H1RP H2RP B) #HCB
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
+lapply (s3 … HCA … a G L0 (◊)) #H @H -H
+lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+  lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
+]
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma
new file mode 100644 (file)
index 0000000..c779101
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx.ma
@@ -0,0 +1,77 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/cosn_5.ma".
+include "basic_2A/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
+
+inductive lcosx (h) (g) (G): relation2 ynat lenv ≝
+| lcosx_sort: ∀l. lcosx h g G l (⋆)
+| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
+| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, g, T, l] L →
+              lcosx h g G l L → lcosx h g G (⫯l) (L.ⓑ{I}T)
+.
+
+interpretation
+   "sn extended strong conormalization (local environment)"
+   'CoSN h g l G L = (lcosx h g G l L).
+
+(* Basic properties *********************************************************)
+
+lemma lcosx_O: ∀h,g,G,L. G ⊢ ~⬊*[h, g, 0] L.
+#h #g #G #L elim L /2 width=1 by lcosx_skip/
+qed.
+
+lemma lcosx_drop_trans_lt: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, l] L →
+                            ∀I,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → i < l →
+                            G ⊢ ~⬊*[h, g, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, g, V, ⫰(l-i)] K.
+#h #g #G #L #l #H elim H -L -l
+[ #l #J #K #V #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false … H) -H //
+| #I #L #T #l #HT #HL #IHL #J #K #V #i #H #Hil
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK destruct
+  [ >ypred_succ /2 width=1 by conj/
+  | lapply (ylt_pred … Hil ?) -Hil /2 width=1 by ylt_inj/ >ypred_succ #Hil
+    elim (IHL … HLK ?) -IHL -HLK <yminus_inj >yminus_SO2 //
+    <(ypred_succ l) in ⊢ (%→%→?); >yminus_pred /2 width=1 by ylt_inj, conj/
+  ]
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ~⬊*[h, g, x] L → ∀l. x = ⫯l →
+                         L = ⋆ ∨
+                         ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
+                                  G ⊢ ⬊*[h, g, V, l] K.
+#h #g #G #L #l * -L -l /2 width=1 by or_introl/
+[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
+| #I #L #T #l #HT #HL #x #H <(ysucc_inj … H) -x
+  /3 width=6 by ex3_3_intro, or_intror/
+]
+qed-.
+
+lemma lcosx_inv_succ: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, ⫯l] L → L = ⋆ ∨
+                      ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
+                               G ⊢ ⬊*[h, g, V, l] K.
+/2 width=3 by lcosx_inv_succ_aux/ qed-.
+
+lemma lcosx_inv_pair: ∀h,g,I,G,L,T,l. G ⊢ ~⬊*[h, g, ⫯l] L.ⓑ{I}T →
+                      G ⊢ ~⬊*[h, g, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
+#h #g #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
+[ #H destruct
+| * #Z #Y #X #H destruct /2 width=1 by conj/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma
new file mode 100644 (file)
index 0000000..47de7c8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lcosx_cpx.ma
@@ -0,0 +1,67 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_max.ma".
+include "basic_2A/computation/lsx_drop.ma".
+include "basic_2A/computation/lsx_lpx.ma".
+include "basic_2A/computation/lsx_lpxs.ma".
+include "basic_2A/computation/lcosx.ma".
+
+(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
+
+(* Properties on extended context-sensitive parallel reduction for term *****)
+
+lemma lsx_cpx_trans_lcosx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+                           ∀l. G ⊢ ~⬊*[h, g, l] L →
+                           G ⊢ ⬊*[h, g, T1, l] L → G ⊢ ⬊*[h, g, T2, l] L.
+#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
+[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #l #HL #H
+  elim (ylt_split i l) #Hli [ -H | -HL ]
+  [ <(ymax_pre_sn l (⫯i)) /2 width=1 by ylt_fwd_le_succ/
+    elim (lcosx_drop_trans_lt … HL … HLK) // -HL -Hli
+    lapply (drop_fwd_drop2 … HLK) -HLK /3 width=7 by lsx_lift_ge/
+  | lapply (lsx_fwd_lref_be … H … HLK) // -H -Hli
+    lapply (drop_fwd_drop2 … HLK) -HLK
+    /4 width=10 by lsx_ge, lsx_lift_le/
+  ]
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #HL #H
+  elim (lsx_inv_bind … H) -H #HV1 #HT1
+  @lsx_bind /2 width=2 by/ (**) (* explicit constructor *)
+  @(lsx_lreq_conf … (L.ⓑ{I}V1)) /3 width=1 by lcosx_pair, lreq_succ/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #HL #H
+  elim (lsx_inv_flat … H) -H /3 width=1 by lsx_flat/
+| #G #L #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #l #HL #H
+  elim (lsx_inv_bind … H) -H
+  /4 width=9 by lcosx_pair, lsx_inv_lift_ge, drop_drop/
+| #G #L #V #T1 #T2 #_ #IHT12 #l #HL #H
+  elim (lsx_inv_flat … H) -H /2 width=1 by/
+| #G #L #V1 #V2 #T #_ #IHV12 #l #HL #H
+  elim (lsx_inv_flat … H) -H /2 width=1 by/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #l #HL #H
+  elim (lsx_inv_flat … H) -H #HV1 #H
+  elim (lsx_inv_bind … H) -H #HW1 #HT1
+  @lsx_bind /3 width=1 by lsx_flat/ (**) (* explicit constructor *)
+  @(lsx_lreq_conf … (L.ⓛW1)) /3 width=1 by lcosx_pair, lreq_succ/
+| #a #G #L #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #HVU2 #_ #_ #IHV12 #IHW12 #IHT12 #l #HL #H
+  elim (lsx_inv_flat … H) -H #HV1 #H
+  elim (lsx_inv_bind … H) -H #HW1 #HT1
+  @lsx_bind /2 width=1 by/ (**) (* explicit constructor *)
+  @lsx_flat [ /3 width=7 by lsx_lift_ge, drop_drop/ ]
+  @(lsx_lreq_conf … (L.ⓓW1)) /3 width=1 by lcosx_pair, lreq_succ/
+]
+qed-.
+
+lemma lsx_cpx_trans_O: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+                       G ⊢ ⬊*[h, g, T1, 0] L → G ⊢ ⬊*[h, g, T2, 0] L.
+/2 width=3 by lsx_cpx_trans_lcosx/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs.ma
new file mode 100644 (file)
index 0000000..c529d58
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs.ma
@@ -0,0 +1,71 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predsnstar_3.ma".
+include "basic_2A/substitution/lpx_sn_tc.ma".
+include "basic_2A/reduction/lpr.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+definition lprs: relation3 genv lenv lenv ≝
+                 λG. TC … (lpr G).
+
+interpretation "parallel computation (local environment, sn variant)"
+   'PRedSnStar G L1 L2 = (lprs G L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
+                (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
+                ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
+#G #L1 #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
+                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
+                   ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
+#G #L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=1 by inj/ qed.
+
+lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
+/2 width=1 by lpr_lprs/ qed.
+
+lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=3 by step/ qed-.
+
+lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
+/2 width=3 by TC_strap/ qed-.
+
+lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
+/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+
+lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
+/2 width=2 by TC_lpx_sn_fwd_length/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma
new file mode 100644 (file)
index 0000000..a96eaa2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_cprs.ma
@@ -0,0 +1,142 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cprs_cprs.ma".
+include "basic_2A/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lprs_pair: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
+/2 width=1 by TC_lpx_sn_pair/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lprs_inv_pair1: ∀I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+                               L2 = K2.ⓑ{I}V2.
+/3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
+
+lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+                               L1 = K1.ⓑ{I}V1.
+/3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
+                    R (⋆) (⋆) → (
+                       ∀I,K1,K2,V1,V2.
+                       ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
+                       R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+                    ) →
+                    ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpr_cprs_trans/ qed-.
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma lprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lprs G).
+/3 width=5 by s_r_trans_LTC2, lpr_cprs_trans/ qed-.
+
+(* Basic_1: was just: pr3_pr3_pr3_t *)
+(* Note: alternative proof /3 width=5 by s_r_trans_LTC1, lprs_cpr_trans/ *)
+lemma lprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lprs G).
+#G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
+qed-.
+
+lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+                         ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                         ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #H @(lprs_ind … H) -L1 /2 width=3 by ex2_intro/
+#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
+elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2
+elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3
+elim (cprs_conf … HT2 … HT3) -T
+/3 width=5 by cprs_trans, ex2_intro/
+qed-.
+
+lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+                        ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                        ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
+
+(* Note: this can be proved on its own using lprs_ind_dx *)
+lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+                         ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                         ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
+/3 width=3 by lprs_cprs_trans, ex2_intro/
+qed-.
+
+lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+                        ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                        ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+/3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
+
+lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lprs_cprs_trans, lprs_pair, cprs_bind/ qed.
+
+(* Inversion lemmas on context-sensitive parallel computation for terms *****)
+
+(* Basic_1: was: pr3_gen_abst *)
+lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
+                               U2 = ⓛ{a}W2.T2.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro/
+qed-.
+
+lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
+#a #G #L #W1 #W2 #T1 #T2 #H elim (cprs_inv_abst1 … H) -H
+#W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic_1: was pr3_gen_abbr *)
+lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
+                               U2 = ⓓ{a}V2.T2
+                      ) ∨
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⬆[0, 1] U2 ≡ T2 & a = true.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+  elim (cpr_inv_abbr1 … HU02) -HU02 *
+  [ #V2 #T2 #HV02 #HT02 #H destruct
+    lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?)
+    /4 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
+  | #T2 #HT02 #HUT2
+    lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
+    /4 width=3 by lprs_pair, cprs_trans, ex3_intro, or_intror/
+  ]
+| #U1 #HTU1 #HU01 elim (lift_total U2 0 1)
+  #U #HU2 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
+  /4 width=3 by cprs_strap1, drop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* More advanced properties *************************************************)
+
+lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
+/3 width=3 by lprs_pair, lprs_cprs_trans/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_drop.ma
new file mode 100644 (file)
index 0000000..66465b6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_drop.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpr_drop.ma".
+include "basic_2A/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Properties on local environment slicing ***********************************)
+
+lemma lprs_drop_conf: ∀G. dropable_sn (lprs G).
+/3 width=3 by dropable_sn_TC, lpr_drop_conf/ qed-.
+
+lemma drop_lprs_trans: ∀G. dedropable_sn (lprs G).
+/3 width=3 by dedropable_sn_TC, drop_lpr_trans/ qed-.
+
+lemma lprs_drop_trans_O1: ∀G. dropable_dx (lprs G).
+/3 width=3 by dropable_dx_TC, lpr_drop_trans_O1/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_lprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_lprs.ma
new file mode 100644 (file)
index 0000000..cf5a613
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lprs_lprs.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpr_lpr.ma".
+include "basic_2A/computation/lprs.ma".
+
+(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lprs_strip: ∀G. confluent2 … (lprs G) (lpr G).
+/3 width=3 by TC_strip1, lpr_conf/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lprs_conf: ∀G. confluent2 … (lprs G) (lprs G).
+/3 width=3 by TC_confluent2, lpr_conf/ qed-.
+
+theorem lprs_trans: ∀G. Transitive … (lprs G).
+/2 width=3 by trans_TC/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs.ma
new file mode 100644 (file)
index 0000000..3aca39a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs.ma
@@ -0,0 +1,74 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predsnstar_5.ma".
+include "basic_2A/reduction/lpx.ma".
+include "basic_2A/computation/lprs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
+                 λh,g,G. TC … (lpx h g G).
+
+interpretation "extended parallel computation (local environment, sn variant)"
+   'PRedSnStar h g G L1 L2 = (lpxs h g G L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma lpxs_ind: ∀h,g,G,L1. ∀R:predicate lenv. R L1 →
+                (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → R L → R L2) →
+                ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L2.
+#h #g #G #L1 #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma lpxs_ind_dx: ∀h,g,G,L2. ∀R:predicate lenv. R L2 →
+                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → R L → R L1) →
+                   ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1.
+#h #g #G #L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lprs_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+/3 width=3 by lpr_lpx, monotonic_TC/ qed.
+
+lemma lpx_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+/2 width=1 by inj/ qed.
+
+lemma lpxs_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡*[h, g] L.
+/2 width=1 by lprs_lpxs/ qed.
+
+lemma lpxs_strap1: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+/2 width=3 by step/ qed.
+
+lemma lpxs_strap2: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+/2 width=3 by TC_strap/ qed.
+
+lemma lpxs_pair_refl: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V.
+/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpxs_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, g] L2 → L2 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+
+lemma lpxs_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, g] ⋆ → L1 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpxs_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → |L1| = |L2|.
+/2 width=2 by TC_lpx_sn_fwd_length/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_aaa.ma
new file mode 100644 (file)
index 0000000..682e6d7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_aaa.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpx_aaa.ma".
+include "basic_2A/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma lpxs_aaa_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+                     ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+#h #g #G #L1 #T #A #HT #L2 #HL12
+@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by lpx_aaa_conf/
+qed-.
+
+lemma lprs_aaa_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+                     ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by lprs_lpxs, lpxs_aaa_conf/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma
new file mode 100644 (file)
index 0000000..c4ba7cc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_cpxs.ma
@@ -0,0 +1,161 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cpxs_cpxs.ma".
+include "basic_2A/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Advanced properties ******************************************************)
+
+lemma lpxs_pair: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, g] V2 →
+                 ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+/2 width=1 by TC_lpx_sn_pair/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lpxs_inv_pair1: ∀h,g,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
+/3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
+
+lemma lpxs_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
+/3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lpxs_ind_alt: ∀h,g,G. ∀R:relation lenv.
+                    R (⋆) (⋆) → (
+                       ∀I,K1,K2,V1,V2.
+                       ⦃G, K1⦄ ⊢ ➡*[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 →
+                       R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+                    ) →
+                    ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.
+
+(* Properties on context-sensitive extended parallel computation for terms **)
+
+lemma lpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpxs h g G).
+/3 width=5 by s_r_trans_LTC2, lpx_cpxs_trans/ qed-.
+
+(* Note: alternative proof: /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ *)
+lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpxs h g G).
+#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
+qed-.
+
+lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+/4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
+
+(* Inversion lemmas on context-sensitive ext parallel computation for terms *)
+
+lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
+                               U2 = ⓛ{a}V2.T2.
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
+elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
+lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
+qed-.
+
+lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
+                               U2 = ⓓ{a}V2.T2
+                      ) ∨
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⬆[0, 1] U2 ≡ T2 & a = true.
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+  elim (cpx_inv_abbr1 … HU02) -HU02 *
+  [ #V2 #T2 #HV02 #HT02 #H destruct
+    lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?)
+    /4 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+  | #T2 #HT02 #HUT2
+    lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
+    /4 width=3 by lpxs_pair, cpxs_trans, ex3_intro, or_intror/
+  ]
+| #U1 #HTU1 #HU01
+  elim (lift_total U2 0 1) #U #HU2
+  /6 width=12 by cpxs_strap1, cpx_lift, drop_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* More advanced properties *************************************************)
+
+lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma lpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
+  /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+  #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
+  #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+  /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
+/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
+qed-.
+
+lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+  lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+  elim (lpx_fquq_trans … HT2 … HK1) -K
+  /3 width=7 by lpxs_strap2, cpxs_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+  lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+  elim (lpx_fqup_trans … HT2 … HK1) -K
+  /3 width=7 by lpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
+
+lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpxs_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
+/3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_drop.ma
new file mode 100644 (file)
index 0000000..0382fb6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_drop.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/lpx_drop.ma".
+include "basic_2A/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Properties on local environment slicing ***********************************)
+
+lemma lpxs_drop_conf: ∀h,g,G. dropable_sn (lpxs h g G).
+/3 width=3 by dropable_sn_TC, lpx_drop_conf/ qed-.
+
+lemma drop_lpxs_trans: ∀h,g,G. dedropable_sn (lpxs h g G).
+/3 width=3 by dedropable_sn_TC, drop_lpx_trans/ qed-.
+
+lemma lpxs_drop_trans_O1: ∀h,g,G. dropable_dx (lpxs h g G).
+/3 width=3 by dropable_dx_TC, lpx_drop_trans_O1/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lleq.ma
new file mode 100644 (file)
index 0000000..064ae90
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lleq.ma
@@ -0,0 +1,141 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_lleq.ma".
+include "basic_2A/reduction/lpx_lleq.ma".
+include "basic_2A/computation/cpxs_lreq.ma".
+include "basic_2A/computation/lpxs_drop.ma".
+include "basic_2A/computation/lpxs_cpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+                       ∀L1,T,l. L1 ≡[T, l] L2 →
+                       ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[T, l] K2.
+#h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
+#K #K2 #_ #HK2 #IH #L1 #T #l #HT elim (IH … HT) -L2
+#L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
+/3 width=3 by lpxs_strap1, ex2_intro/
+qed-.
+
+lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) →
+                              ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, g] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L0 & L0 ≡[T, l] L2.
+#h #g #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
+[ #H elim H -H //
+| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
+  [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
+    #L0 #L3 #H1 #H2 #H3 #H4 lapply (lleq_nlleq_trans … H … H2) -H2
+    #H2 elim (lleq_lpx_trans … H1 … H) -L
+    #L #H1 #H lapply (nlleq_lleq_div … H … H2) -H2
+    #H2 elim (lleq_lpxs_trans … H3 … H) -L0
+    /3 width=8 by lleq_trans, ex4_2_intro/
+  | -H12 -IH2 /3 width=6 by ex4_2_intro/
+  ]
+]
+qed-.
+
+lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
+  #K0 #V0 #H1KL1 #_ #H destruct
+  elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
+  #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
+  /2 width=4 by fqu_lref_O, ex3_intro/
+| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
+  [ elim (lleq_inv_bind … H)
+  | elim (lleq_inv_flat … H)
+  ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
+| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
+| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
+  /2 width=4 by fqu_flat_dx, ex3_intro/
+| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+  elim (drop_O1_le (Ⓕ) (m+1) K1)
+  [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
+    #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
+    #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
+    /3 width=4 by fqu_drop, ex3_intro/
+  | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
+    lapply (lleq_fwd_length … H2KL1) //
+  ]
+]
+qed-.
+
+lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fquq_inv_gen … H) -H
+[ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
+  #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
+  /3 width=5 by fqup_strap1, ex3_intro/
+]
+qed-.
+
+lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fqus_inv_gen … H) -H
+[ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+fact lreq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
+                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+                               ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+                                    (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
+[ #l #m #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+  /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
+]
+qed-.
+
+lemma lreq_lpxs_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+                            ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+                                 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+/2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lpxs.ma
new file mode 100644 (file)
index 0000000..09ddcfe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lpxs_lpxs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Main properties **********************************************************)
+
+theorem lpxs_trans: ∀h,g,G. Transitive … (lpxs h g G).
+/2 width=3 by trans_TC/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma
new file mode 100644 (file)
index 0000000..a0bb73d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc.ma
@@ -0,0 +1,114 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lrsubeqc_4.ma".
+include "basic_2A/static/lsubr.ma".
+include "basic_2A/static/aaa.ma".
+include "basic_2A/computation/gcp_cr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+inductive lsubc (RP) (G): relation lenv ≝
+| lsubc_atom: lsubc RP G (⋆) (⋆)
+| lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+              lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+  "local environment refinement (generic reducibility)"
+  'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
+/2 width=5 by lsubc_inv_atom1_aux/ qed-.
+
+fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+                          ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                      G ⊢ K1 ⫃[RP] K2 &
+                                      L2 = K2. ⓛW & X = ⓝW.V & I = Abbr.
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10 by ex7_4_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃[RP] L2 →
+                       (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+                       ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                   G ⊢ K1 ⫃[RP] K2 &
+                                   L2 = K2.ⓛW & X = ⓝW.V & I = Abbr.
+/2 width=3 by lsubc_inv_pair1_aux/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
+/2 width=5 by lsubc_inv_atom2_aux/ qed-.
+
+fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
+                          (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓑ{I} W) ∨
+                          ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                    G ⊢ K1 ⫃[RP] K2 &
+                                    L1 = K1.ⓓⓝW.V & I = Abst.
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8 by ex6_3_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_head_l *)
+lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⫃[RP] K2.ⓑ{I} W →
+                       (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓑ{I} W) ∨
+                       ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                 G ⊢ K1 ⫃[RP] K2 &
+                                 L1 = K1.ⓓⓝW.V & I = Abst.
+/2 width=3 by lsubc_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
+#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: csubc_refl *)
+lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
+#RP #G #L elim L -L /2 width=1 by lsubc_pair/
+qed.
+
+(* Basic_1: removed theorems 3:
+            csubc_clear_conf csubc_getl_conf csubc_csuba
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drop.ma
new file mode 100644 (file)
index 0000000..2eb945d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drop.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_lift.ma".
+include "basic_2A/computation/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Basic_1: was: csubc_drop_conf_O *)
+(* Note: the constant 0 can not be generalized *)
+lemma lsubc_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+                           ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
+#RP #G #L1 #L2 #H elim H -L1 -L2
+[ #X #s #m #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #X #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
+  [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
+    /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
+  [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
+    /3 width=8 by lsubc_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Basic_1: was: csubc_drop_conf_rev *)
+lemma drop_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
+                        ∀G,L1,K1,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+                        ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, m] L2 ≡ K2.
+#RR #RS #RP #Hgcp #G #L1 #K1 #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H elim (lsubc_inv_atom1 … H) -H
+  >Hm /2 width=3 by ex2_intro/
+| #L1 #I #V1 #X #H
+  elim (lsubc_inv_pair1 … H) -H *
+  [ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
+  | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
+    /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
+  ]
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12
+  elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
+| #I #L1 #K1 #V1 #V2 #l #m #HLK1 #HV21 #IHLK1 #X #H
+  elim (lsubc_inv_pair1 … H) -H *
+  [ #K2 #HK12 #H destruct
+    elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
+  | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
+    elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
+    elim (IHLK1 … HK12) #K #HL1K #HK2
+    lapply (gcr_lift … Hgcp … HV2 … HLK1 … HV3) -HV2
+    lapply (gcr_lift … Hgcp … H1W2 … HLK1 … HW23) -H1W2
+    /4 width=11 by lsubc_beta, aaa_lift, drop_skip, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drops.ma
new file mode 100644 (file)
index 0000000..41c3481
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drops.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/lsubc_drop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties concerning generic local environment slicing ******************)
+
+(* Basic_1: was: csubc_drop1_conf_rev *)
+lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
+                         ∀G,L1,K1,cs. ⬇*[Ⓕ, cs] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+                         ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[Ⓕ, cs] L2 ≡ K2.
+#RR #RS #RP #Hgcp #G #L1 #K1 #cs #H elim H -L1 -K1 -cs
+[ /2 width=3 by drops_nil, ex2_intro/
+| #L1 #L #K1 #cs #l #m #_ #HLK1 #IHL #K2 #HK12
+  elim (drop_lsubc_trans … Hgcp … HLK1 … HK12) -Hgcp -K1 #K #HLK #HK2
+  elim (IHL … HLK) -IHL -HLK /3 width=5 by drops_cons, ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_lsuba.ma
new file mode 100644 (file)
index 0000000..00a6d4d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_lsuba.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsuba.ma".
+include "basic_2A/computation/gcp_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* properties concerning lenv refinement for atomic arity assignment ********)
+
+lemma lsuba_lsubc: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+                   ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → G ⊢ L1 ⫃[RP] L2.
+#RR #RS #RP #H1RP #H2RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubc_pair/
+#L1 #L2 #W #V #A #H elim (aaa_inv_cast … H) -H /3 width=4 by acr_aaa, lsubc_beta/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx.ma
new file mode 100644 (file)
index 0000000..4f2afb7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx.ma
@@ -0,0 +1,109 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/sn_6.ma".
+include "basic_2A/multiple/lleq.ma".
+include "basic_2A/reduction/lpx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+definition lsx: ∀h. sd h → relation4 ynat term genv lenv ≝
+                λh,g,l,T,G. SN … (lpx h g G) (lleq l T).
+
+interpretation
+   "extended strong normalization (local environment)"
+   'SN h g l T G L = (lsx h g T l G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma lsx_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
+               (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                     R L1
+               ) →
+               ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsx_intro: ∀h,g,G,L1,T,l.
+                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, g, T, l] L2) →
+                 G ⊢ ⬊*[h, g, T, l] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+lemma lsx_atom: ∀h,g,G,T,l. G ⊢ ⬊*[h, g, T, l] ⋆.
+#h #g #G #T #l @lsx_intro
+#X #H #HT lapply (lpx_inv_atom1 … H) -H
+#H destruct elim HT -HT //
+qed.
+
+lemma lsx_sort: ∀h,g,G,L,l,k. G ⊢ ⬊*[h, g, ⋆k, l] L.
+#h #g #G #L1 #l #k @lsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=4 by lpx_fwd_length, lleq_sort/
+qed.
+
+lemma lsx_gref: ∀h,g,G,L,l,p. G ⊢ ⬊*[h, g, §p, l] L.
+#h #g #G #L1 #l #p @lsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=4 by lpx_fwd_length, lleq_gref/
+qed.
+
+lemma lsx_ge_up: ∀h,g,G,L,T,U,lt,l,m. lt ≤ yinj l + yinj m →
+                 ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L → G ⊢ ⬊*[h, g, U, l] L.
+#h #g #G #L #T #U #lt #l #m #Hltlm #HTU #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge_up/
+qed-.
+
+lemma lsx_ge: ∀h,g,G,L,T,l1,l2. l1 ≤ l2 →
+              G ⊢ ⬊*[h, g, T, l1] L → G ⊢ ⬊*[h, g, T, l2] L.
+#h #g #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsx_fwd_bind_sn: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
+qed-.
+
+lemma lsx_fwd_flat_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
+qed-.
+
+lemma lsx_fwd_flat_dx: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, T, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
+qed-.
+
+lemma lsx_fwd_pair_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ②{I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, V, l] L.
+#h #g * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lsx_inv_flat: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
+                    G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
+/3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_alt.ma
new file mode 100644 (file)
index 0000000..9468bed
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_alt.ma
@@ -0,0 +1,115 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/snalt_6.ma".
+include "basic_2A/computation/lpxs_lleq.ma".
+include "basic_2A/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* alternative definition of lsx *)
+definition lsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
+                 λh,g,l,T,G. SN … (lpxs h g G) (lleq l T).
+
+interpretation
+   "extended strong normalization (local environment) alternative"
+   'SNAlt h g l T G L = (lsxa h g T l G L).
+
+(* Basic eliminators ********************************************************)
+
+lemma lsxa_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
+                (∀L1. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                      R L1
+                ) →
+                ∀L. G ⊢ ⬊⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
+/5 width=1 by lleq_sym, SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsxa_intro: ∀h,g,G,L1,T,l.
+                  (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
+                  G ⊢ ⬊⬊*[h, g, T, l] L1.
+/5 width=1 by lleq_sym, SN_intro/ qed.
+
+fact lsxa_intro_aux: ∀h,g,G,L1,T,l.
+                     (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, g] L2 → L1 ≡[T, l] L → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
+                     G ⊢ ⬊⬊*[h, g, T, l] L1.
+/4 width=3 by lsxa_intro/ qed-.
+
+lemma lsxa_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                       ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊⬊*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1
+#L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
+#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
+/5 width=4 by lleq_canc_sn, lleq_trans/
+qed-.
+
+lemma lsxa_lpxs_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊⬊*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
+qed-.
+
+lemma lsxa_intro_lpx: ∀h,g,G,L1,T,l.
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
+                      G ⊢ ⬊⬊*[h, g, T, l] L1.
+#h #g #G #L1 #T #l #IH @lsxa_intro_aux
+#L #L2 #H @(lpxs_ind_dx … H) -L
+[ #H destruct #H elim H //
+| #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
+  #HnT #HL0 #HL2 #_ #HT #_ elim (lleq_lpx_trans … HL0 … HT) -L0
+  #L0 #HL10 #HL0 @(lsxa_lpxs_trans … HL2) -HL2
+  /5 width=3 by lsxa_lleq_trans, lleq_trans/
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsx_lsxa: ∀h,g,G,L,T,l. G ⊢ ⬊*[h, g, T, l] L → G ⊢ ⬊⬊*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsx_ind … H) -L
+/4 width=1 by lsxa_intro_lpx/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lsxa_inv_lsx: ∀h,g,G,L,T,l. G ⊢ ⬊⬊*[h, g, T, l] L → G ⊢ ⬊*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsxa_ind … H) -L
+/4 width=1 by lsx_intro, lpx_lpxs/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lsx_intro_alt: ∀h,g,G,L1,T,l.
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, g, T, l] L2) →
+                     G ⊢ ⬊*[h, g, T, l] L1.
+/6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
+
+lemma lsx_lpxs_trans: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
+                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
+/4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lsx_ind_alt: ∀h,g,G,T,l. ∀R:predicate lenv.
+                   (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
+                         (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+                         R L1
+                   ) →
+                   ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #IH #L #H @(lsxa_ind h g G T l … L)
+/4 width=1 by lsxa_inv_lsx, lsx_lsxa/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_csx.ma
new file mode 100644 (file)
index 0000000..1d19669
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_csx.ma
@@ -0,0 +1,59 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/csx_lpxs.ma".
+include "basic_2A/computation/lcosx_cpx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
+                        ∀K2. G ⊢ ⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
+                        ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L2.
+#h #g #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
+#V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
+#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
+#L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
+#Y #H #HLK2 elim (lpx_inv_pair1 … H) -H
+#K2 #V2 #HK02 #HV02 #H destruct
+elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
+[ /4 width=8 by lpxs_strap1, lleq_lref/
+| @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
+  /3 width=4 by lsx_cpx_trans_O, lsx_lpx_trans, lpxs_cpx_trans, lpxs_strap1/ (**) (* full auto too slow *)
+]
+qed.
+
+lemma lsx_lref_be: ∀h,g,I,G,K,V,i,l. l ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
+                   G ⊢ ⬊*[h, g, V, 0] K →
+                   ∀L. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L.
+/2 width=8 by lsx_lref_be_lpxs/ qed.
+
+(* Main properties **********************************************************)
+
+theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀l. G ⊢ ⬊*[h, g, T, l] L.
+#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
+#Z #Y #X #IH #G #L * * //
+[ #i #HG #HL #HT #H #l destruct
+  elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
+  elim (ylt_split i l) /2 width=1 by lsx_lref_skip/
+  #Hli #Hi elim (drop_O1_lt (Ⓕ) … Hi) -Hi
+  #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
+  /4 width=6 by lsx_lref_be, fqup_lref/
+| #a #I #V #T #HG #HL #HT #H #l destruct
+  elim (csx_fwd_bind … H) -H /3 width=1 by lsx_bind/
+| #I #V #T #HG #HL #HT #H #l destruct
+  elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_drop.ma
new file mode 100644 (file)
index 0000000..8c02ca9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_drop.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_drop.ma".
+include "basic_2A/reduction/lpx_drop.ma".
+include "basic_2A/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lsx_lref_free: ∀h,g,G,L,l,i. |L| ≤ i → G ⊢ ⬊*[h, g, #i, l] L.
+#h #g #G #L1 #l #i #HL1 @lsx_intro
+#L2 #HL12 #H elim H -H
+/4 width=6 by lpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
+qed.
+
+lemma lsx_lref_skip: ∀h,g,G,L,l,i. yinj i < l → G ⊢ ⬊*[h, g, #i, l] L.
+#h #g #G #L1 #l #i #HL1 @lsx_intro
+#L2 #HL12 #H elim H -H
+/3 width=4 by lpx_fwd_length, lleq_skip/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lsx_fwd_lref_be: ∀h,g,I,G,L,l,i. l ≤ yinj i → G ⊢ ⬊*[h, g, #i, l] L →
+                       ∀K,V. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, V, 0] K.
+#h #g #I #G #L #l #i #Hli #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
+#K2 #HK12 #HnK12 lapply (drop_fwd_drop2 … HLK1)
+#H2LK1 elim (drop_lpx_trans … H2LK1 … HK12) -H2LK1 -HK12
+#L2 #HL12 #H2LK2 #H elim (lreq_drop_conf_be … H … HLK1) -H /2 width=1 by ylt_inj/
+#Y #_ #HLK2 lapply (drop_fwd_drop2 … HLK2)
+#HY lapply (drop_mono … HY … H2LK2) -HY -H2LK2 #H destruct
+/4 width=10 by lleq_inv_lref_ge/
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma lsx_lift_le: ∀h,g,G,K,T,U,lt,l,m. lt ≤ yinj l →
+                   ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, T, lt] K →
+                   ∀L. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, U, lt] L.
+#h #g #G #K #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind … H) -K
+#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
+#L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
+/4 width=10 by lleq_lift_le/
+qed-.
+
+lemma lsx_lift_ge: ∀h,g,G,K,T,U,lt,l,m. yinj l ≤ lt →
+                   ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, T, lt] K →
+                   ∀L. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, U, lt + m] L.
+#h #g #G #K #T #U #lt #l #m #Hllt #HTU #H @(lsx_ind … H) -K
+#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
+#L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
+/4 width=9 by lleq_lift_ge/
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lsx_inv_lift_le: ∀h,g,G,L,T,U,lt,l,m. lt ≤ yinj l →
+                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, lt] K.
+#h #g #G #L #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
+#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
+/4 width=10 by lleq_inv_lift_le/
+qed-.
+
+lemma lsx_inv_lift_be: ∀h,g,G,L,T,U,lt,l,m. yinj l ≤ lt → lt ≤ l + m →
+                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, l] K.
+#h #g #G #L #T #U #lt #l #m #Hllt #Hltlm #HTU #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
+#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
+/4 width=11 by lleq_inv_lift_be/
+qed-.
+
+lemma lsx_inv_lift_ge: ∀h,g,G,L,T,U,lt,l,m. yinj l + yinj m ≤ lt →
+                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, lt-m] K.
+#h #g #G #L #T #U #lt #l #m #Hlmlt #HTU #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
+#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
+/4 width=9 by lleq_inv_lift_ge/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma
new file mode 100644 (file)
index 0000000..a50714b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma
@@ -0,0 +1,63 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_lleq.ma".
+include "basic_2A/reduction/lpx_lleq.ma".
+include "basic_2A/computation/lsx.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma lsx_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
+                      ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsx_ind … H) -L1
+#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
+#K2 #HLK2 #HnLK2 elim (lleq_lpx_trans … HLK2 … HL12) -HLK2
+/5 width=4 by lleq_canc_sn, lleq_trans/
+qed-.
+
+lemma lsx_lpx_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
+                     ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsx_lleq_trans/
+qed-.
+
+lemma lsx_lreq_conf: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
+                    ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, g, T, l] L2.
+#h #g #G #L1 #T #l #H @(lsx_ind … H) -L1
+#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
+#L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23
+#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
+#h #g #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#Y #H #HT elim (lpx_inv_pair1 … H) -H
+#L2 #V2 #HL12 #_ #H destruct
+@(lsx_lreq_conf … (L2.ⓑ{I}V1)) /2 width=1 by lreq_succ/
+@IHL1 // #H @HT -IHL1 -HL12 -HT
+@(lleq_lreq_trans … (L2.ⓑ{I}V1))
+/2 width=2 by lleq_fwd_bind_dx, lreq_succ/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a, I}V.T, l] L →
+                    G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
+/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma
new file mode 100644 (file)
index 0000000..d77771e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpxs.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/lpxs_lpxs.ma".
+include "basic_2A/computation/lsx_alt.ma".
+
+(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
+
+(* Advanced properties ******************************************************)
+
+fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
+                        ∀Y,T. G ⊢ ⬊*[h, g, T, ⫯l] Y →
+                        ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                        G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L2.
+#h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind_alt … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro_alt
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 l)
+  [ #HV @(IHL1 … L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
+  | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
+  ]
+| /3 width=4 by lpxs_pair/
+]
+qed-.
+
+lemma lsx_bind: ∀h,g,a,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
+                ∀T. G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V →
+                G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L.
+/2 width=3 by lsx_bind_lpxs_aux/ qed.
+
+lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
+                     ∀L2,T. G ⊢ ⬊*[h, g, T, l] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                     G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L2.
+#h #g #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_alt … H) -L2
+#L2 #HL2 #IHL2 #HL12 @lsx_intro_alt
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 l)
+  [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
+  | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
+  ]
+| /3 width=1 by/
+]
+qed-.
+
+lemma lsx_flat: ∀h,g,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
+                ∀T. G ⊢ ⬊*[h, g, T, l] L → G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L.
+/2 width=3 by lsx_flat_lpxs/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma
new file mode 100644 (file)
index 0000000..f26ee09
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds.ma
@@ -0,0 +1,48 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/dpredstar_7.ma".
+include "basic_2A/static/da.ma".
+include "basic_2A/computation/cprs.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
+
+definition scpds: ∀h. sd h → nat → relation4 genv lenv term term ≝
+                  λh,g,d2,G,L,T1,T2.
+                  ∃∃T,d1. d2 ≤ d1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 & ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T ➡* T2.
+
+interpretation "stratified decomposed parallel computation (term)"
+   'DPRedStar h g d G L T1 T2 = (scpds h g d G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma sta_cprs_scpds: ∀h,g,G,L,T1,T,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T →
+                      ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 1] T2.
+/2 width=6 by ex4_2_intro/ qed.
+
+lemma lstas_scpds: ∀h,g,G,L,T1,T2,d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 →
+                   ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d2] T2.
+/2 width=6 by ex4_2_intro/ qed.
+
+lemma scpds_strap1: ∀h,g,G,L,T1,T,T2,d.
+                    ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2.
+#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_strap1, ex4_2_intro/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma scpds_fwd_cprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 0] T2 →
+                      ⦃G, L⦄ ⊢ T1 ➡* T2.
+#h #g #G #L #T1 #T2 * /3 width=3 by cprs_strap2, lstas_cpr/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_aaa.ma
new file mode 100644 (file)
index 0000000..3c32574
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_aaa.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/unfold/lstas_aaa.ma".
+include "basic_2A/computation/cpxs_aaa.ma".
+include "basic_2A/computation/scpds.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma scpds_aaa_conf: ∀h,g,G,L,d. Conf3 … (aaa G L) (scpds h g d G L).
+#h #g #G #L #d #A #T #HT #U * /3 width=6 by lstas_aaa_conf, cprs_aaa_conf/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_lift.ma
new file mode 100644 (file)
index 0000000..29e09ca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_lift.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/da_lift.ma".
+include "basic_2A/unfold/lstas_lift.ma".
+include "basic_2A/computation/cprs_lift.ma".
+include "basic_2A/computation/scpds.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
+
+(* Relocation properties ****************************************************)
+
+lemma scpds_lift: ∀h,g,G,d. d_liftable (scpds h g d G).
+#h #g #G #d2 #K #T1 #T2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L #s #l #m
+elim (lift_total T l m)
+/3 width=15 by cprs_lift, da_lift, lstas_lift, ex4_2_intro/
+qed.
+
+lemma scpds_inv_lift1: ∀h,g,G,d. d_deliftable_sn (scpds h g d G).
+#h #g #G #d2 #L #U1 #U2 * #U #d1 #Hd21 #Hd1 #HU1 #HU2 #K #s #l #m #HLK #T1 #HTU1
+lapply (da_inv_lift … Hd1 … HLK … HTU1) -Hd1 #Hd1
+elim (lstas_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
+elim (cprs_inv_lift1 … HU2 … HLK … HTU) -U -L
+/3 width=8 by ex4_2_intro, ex2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_scpds.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_scpds.ma
new file mode 100644 (file)
index 0000000..29abce7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/computation/scpds_scpds.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/unfold/lstas_da.ma".
+include "basic_2A/computation/lprs_cprs.ma".
+include "basic_2A/computation/cpxs_cpxs.ma".
+include "basic_2A/computation/scpds.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
+
+(* Advanced properties ******************************************************)
+
+lemma scpds_strap2: ∀h,g,G,L,T1,T,T2,d1,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1+1 →
+                    ⦃G, L⦄ ⊢ T1 •*[h, 1] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, d] T2 →
+                    ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d+1] T2.
+#h #g #G #L #T1 #T #T2 #d1 #d #Hd1 #HT1 *
+#T0 #d0 #Hd0 #HTd0 #HT0 #HT02
+lapply (lstas_da_conf … HT1 … Hd1) <minus_plus_m_m #HTd1
+lapply (da_mono … HTd0 … HTd1) -HTd0 -HTd1 #H destruct
+lapply (lstas_trans … HT1 … HT0) -T >commutative_plus
+/3 width=6 by le_S_S, ex4_2_intro/
+qed.
+
+lemma scpds_cprs_trans: ∀h,g,G,L,T1,T,T2,d.
+                        ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2.
+#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_trans, ex4_2_intro/
+qed-.
+
+lemma lstas_scpds_trans: ∀h,g,G,L,T1,T,T2,d1,d2,d.
+                         d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 →
+                         ⦃G, L⦄ ⊢ T1 •*[h, d2] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, d] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d2+d] T2.
+#h #g #G #L #T1 #T #T2 #d1 #d2 #d #Hd21 #HTd1 #HT1 * #T0 #d0 #Hd0 #HTd0 #HT0 #HT02
+lapply (lstas_da_conf … HT1 … HTd1) #HTd12
+lapply (da_mono … HTd12 … HTd0) -HTd12 -HTd0 #H destruct
+lapply (le_minus_to_plus_r … Hd21 Hd0) -Hd21 -Hd0
+/3 width=7 by lstas_trans, ex4_2_intro/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma scpds_inv_abst1: ∀h,g,a,G,L,V1,T1,U2,d. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 •*➡*[h, g, d] U2 →
+                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 •*➡*[h, g, d] T2 &
+                                U2 = ⓛ{a}V2.T2.
+#h #g #a #G #L #V1 #T1 #U2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+elim (lstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
+elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct
+/3 width=6 by ex4_2_intro, ex3_2_intro/
+qed-.
+
+lemma scpds_inv_abbr_abst: ∀h,g,a1,a2,G,L,V1,W2,T1,T2,d. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, g, d] ⓛ{a2}W2.T2 →
+                           ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 •*➡*[h, g, d] T & ⬆[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
+#h #g #a1 #a2 #G #L #V1 #W2 #T1 #T2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+elim (lstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
+elim (cprs_inv_abbr1 … H2) -H2 *
+[ #V2 #U2 #HV12 #HU12 #H destruct
+| /3 width=6 by ex4_2_intro, ex3_intro/
+]
+qed-.
+
+lemma scpds_inv_lstas_eq: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2 →
+                          ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d] T → ⦃G, L⦄ ⊢ T ➡* T2.
+#h #g #G #L #T1 #T2 #d2 *
+#T0 #d1 #_ #_ #HT10 #HT02 #T #HT1
+lapply (lstas_mono … HT10 … HT1) #H destruct //
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma scpds_fwd_cpxs: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #d * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem scpds_conf_eq: ∀h,g,G,L,T0,T1,d. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, d] T1 →
+                       ∀T2. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, d] T2 →
+                       ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
+#h #g #G #L #T0 #T1 #d0 * #U1 #d1 #_ #_ #H1 #HUT1 #T2 * #U2 #d2 #_ #_ #H2 #HUT2 -d1 -d2
+lapply (lstas_mono … H1 … H2) #H destruct -h -d0 /2 width=3 by cprs_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc.ma b/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc.ma
new file mode 100644 (file)
index 0000000..a1bb87e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/pconv_4.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
+
+definition cpc: relation4 genv lenv term term ≝
+                λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 ∨ ⦃G, L⦄ ⊢ T2 ➡ T1.
+
+interpretation
+   "context-sensitive parallel conversion (term)"
+   'PConv G L T1 T2 = (cpc G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpc_refl: ∀G,L. reflexive … (cpc G L).
+/2 width=1 by or_intror/ qed.
+
+lemma cpc_sym: ∀G,L. symmetric … (cpc L G).
+#G #L #T1 #T2 * /2 width=1 by or_introl, or_intror/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpc_fwd_cpr: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ∃∃T. ⦃G, L⦄ ⊢ T1 ➡ T & ⦃G, L⦄ ⊢ T2 ➡ T.
+#G #L #T1 #T2 * /2 width=3 by ex2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc_cpc.ma b/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc_cpc.ma
new file mode 100644 (file)
index 0000000..3fe972a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/conversion/cpc_cpc.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/conversion/cpc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
+
+(* Main properties **********************************************************)
+
+theorem cpc_conf: ∀G,L,T0,T1,T2. ⦃G, L⦄ ⊢ T0 ⬌ T1 → ⦃G, L⦄ ⊢ T0 ⬌ T2 →
+                  ∃∃T. ⦃G, L⦄ ⊢ T1 ⬌ T & ⦃G, L⦄ ⊢ T2 ⬌ T.
+/3 width=3 by cpc_sym, ex2_intro/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma
new file mode 100644 (file)
index 0000000..b2fb883
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv.ma
@@ -0,0 +1,161 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lrsubeqv_5.ma".
+include "basic_2A/dynamic/shnv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Note: this is not transitive *)
+inductive lsubsv (h) (g) (G): relation lenv ≝
+| lsubsv_atom: lsubsv h g G (⋆) (⋆)
+| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g G L1 L2 →
+               lsubsv h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubsv_beta: ∀L1,L2,W,V,d1. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, g, d1] → ⦃G, L2⦄ ⊢ W ¡[h, g] →
+               ⦃G, L1⦄ ⊢ V ▪[h, g] d1+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d1 →
+               lsubsv h g G L1 L2 → lsubsv h g G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+  "local environment refinement (stratified native validity)"
+  'LRSubEqV h g G L1 L2 = (lsubsv h g G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃¡[h, g] L2 → L2 = ⋆.
+/2 width=6 by lsubsv_inv_atom1_aux/ qed-.
+
+fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                           ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                           (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                           ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                       ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+                                        G ⊢ K1 ⫃¡[h, g] K2 &
+                                        I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#h #g #G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K1 #X #H destruct /3 width=11 by or_intror, ex8_4_intro/
+]
+qed-.
+
+lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃¡[h, g] L2 →
+                        (∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                        ∃∃K2,W,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                     ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+                                     G ⊢ K1 ⫃¡[h, g] K2 &
+                                     I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsubsv_inv_pair1_aux/ qed-.
+
+fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L2 = ⋆ → L1 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃¡[h, g] ⋆ → L1 = ⋆.
+/2 width=6 by lsubsv_inv_atom2_aux/ qed-.
+
+fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                           ∀I,K2,W. L2 = K2.ⓑ{I}W →
+                           (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                           ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                      ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+                                      G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+#h #g #G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K2 #U #H destruct /3 width=8 by or_intror, ex7_3_intro/
+]
+qed-.
+
+lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃¡[h, g] K2.ⓑ{I}W →
+                        (∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                        ∃∃K1,V,d1. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g, d1] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+                                   ⦃G, K1⦄ ⊢ V ▪[h, g] d1+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d1 &
+                                   G ⊢ K1 ⫃¡[h, g] K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → L1 ⫃ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ⫃¡[h, g] L.
+#h #g #G #L elim L -L /2 width=1 by lsubsv_pair/
+qed.
+
+lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                         ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                           ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
+                           ∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                            ∀K2,s, m. ⬇[s, 0, m] L2 ≡ K2 →
+                            ∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_cpcs.ma
new file mode 100644 (file)
index 0000000..0d0e80a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_cpcs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/equivalence/cpcs_cpcs.ma".
+include "basic_2A/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma lsubsv_cpcs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+                         ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cpcs_trans/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lstas.ma
new file mode 100644 (file)
index 0000000..5837ff2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lstas.ma
@@ -0,0 +1,98 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/equivalence/scpes_cpcs.ma".
+include "basic_2A/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties on nat-iterated static type assignment ************************)
+
+lemma lsubsv_lstas_trans: ∀h,g,G,L2,T,U2,d2. ⦃G, L2⦄ ⊢ T •*[h, d2] U2 →
+                          ∀d1. d2 ≤ d1 → ⦃G, L2⦄ ⊢ T ▪[h, g] d1 →
+                          ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
+                          ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, d2] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L2 #T #U #d2 #H elim H -G -L2 -T -U -d2
+[ /2 width=3 by ex2_intro/
+| #G #L2 #K2 #V #W #U #i #d2 #HLK2 #_ #HWU #IHVW #d1 #Hd21 #Hd1 #L1 #HL12
+  elim (da_inv_lref … Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0
+  lapply (drop_mono … HK0 … HLK2) -HK0 #H destruct
+  elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #Y #H #HLK1
+  elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -HWU -IHVW -HLK1 ]
+  [ #HK12 #H destruct
+    elim (IHVW … Hd21 HV0 … HK12) -K2 -d1 #T #HVT #HTW
+    lapply (drop_fwd_drop2 … HLK1) #H
+    elim (lift_total T 0 (i+1))
+    /3 width=12 by lstas_ldef, cpcs_lift, ex2_intro/
+  | #V0 #d0 #_ #_ #_ #_ #_ #H destruct
+  ]
+| #G #L2 #K2 #V #W #i #HLK2 #_ #IHVW #d1 #_ #Hd1 #L1 #HL12
+  elim (da_inv_lref … Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0 [| #H1 ]
+  lapply (drop_mono … HK0 … HLK2) -HK0 #H2 destruct
+  elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #Y #H #HLK1
+  elim (lsubsv_inv_pair2 … H) -H * #K1
+  [ #HK12 #H destruct
+    elim (IHVW … HV0 … HK12) -K2 /3 width=5 by lstas_zero, ex2_intro/
+  | #V1 #d1 #_ #_ #HV1 #HV #HK12 #_ #H destruct
+    lapply (da_mono … HV0 … HV) -HV #H destruct
+    elim (da_lstas … HV1 0) -HV1 #W1 #HVW1 #_
+    elim (lift_total W1 0 (i+1)) #U1 #HWU1
+    elim (IHVW … HV0 … HK12) -K2 // #X #HVX #_ -W
+    @(ex2_intro … U1) /3 width=6 by lstas_cast, lstas_ldef/ (**) (* full auto too slow *)
+    @cpcs_cprs_sn @(cprs_delta … HLK1 … HWU1)
+    /4 width=2 by cprs_strap1, cpr_cprs, lstas_cpr, cpr_eps/
+  ]
+| #G #L2 #K2 #V #W #U #i #d2 #HLK2 #_ #HWU #IHVW #d1 #Hd21 #Hd1 #L1 #HL12
+  elim (da_inv_lref … Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0 [| #H1 ]
+  lapply (drop_mono … HK0 … HLK2) -HK0 #H2 destruct
+  lapply (le_plus_to_le_r … Hd21) -Hd21 #Hd21
+  elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #Y #H #HLK1
+  elim (lsubsv_inv_pair2 … H) -H * #K1
+  [ #HK12 #H destruct
+    elim (IHVW … Hd21 HV0 … HK12) -K2 -Hd21 #X
+    lapply (drop_fwd_drop2 … HLK1)
+    elim (lift_total X 0 (i+1))
+    /3 width=12 by lstas_succ, cpcs_lift, ex2_intro/
+  | #V1 #d1 #H0 #_ #HV1 #HV #HK12 #_ #H destruct
+    lapply (da_mono … HV0 … HV) -HV #H destruct
+    elim (shnv_inv_cast … H0) -H0 #_ #_ #H
+    lapply (H … Hd21) -H #HVV1
+    elim (IHVW … Hd21 HV0 … HK12) -K2 -Hd21 #X #HVX #HXW
+    elim (da_lstas … HV1 (d2+1)) -HV1 #X1 #HVX1 #_
+    lapply (scpes_inv_lstas_eq … HVV1 … HVX … HVX1) -HVV1 -HVX #HXX1
+    lapply (cpcs_canc_sn … HXX1 … HXW) -X
+    elim (lift_total X1 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK1)
+    /4 width=12 by cpcs_lift, lstas_cast, lstas_ldef, ex2_intro/
+  ]
+| #a #I #G #L2 #V2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+  lapply (da_inv_bind … Hd2) -Hd2 #Hd2
+  elim (IHTU2 … Hd2 (L1.ⓑ{I}V2) …)
+  /3 width=3 by lsubsv_pair, lstas_bind, cpcs_bind_dx, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+  lapply (da_inv_flat … Hd2) -Hd2 #Hd2
+  elim (IHTU2 … Hd2 … HL12) -L2
+  /3 width=5 by lstas_appl, cpcs_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+  lapply (da_inv_flat … Hd2) -Hd2 #Hd2
+  elim (IHTU2 … Hd2 … HL12) -L2
+  /3 width=3 by lstas_cast, ex2_intro/
+]
+qed-.
+
+lemma lsubsv_sta_trans: ∀h,g,G,L2,T,U2. ⦃G, L2⦄ ⊢ T •*[h, 1] U2 →
+                        ∀d. ⦃G, L2⦄ ⊢ T ▪[h, g] d+1 →
+                        ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
+                        ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, 1] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
+/2 width=7 by lsubsv_lstas_trans/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsuba.ma
new file mode 100644 (file)
index 0000000..c157c75
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsuba.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/equivalence/scpes_aaa.ma".
+include "basic_2A/dynamic/snv_aaa.ma".
+include "basic_2A/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Forward lemmas on lenv refinement for atomic arity assignment ************)
+
+lemma lsubsv_fwd_lsuba: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → G ⊢ L1 ⫃⁝ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_pair/
+#L1 #L2 #W #V #d1 #H #_ #_ #_ #_ #IHL12
+elim (shnv_inv_cast … H) -H #HW #HV #H
+lapply (H 0 ?) // -d1 #HWV
+elim (snv_fwd_aaa … HW) -HW #B #HW
+elim (snv_fwd_aaa … HV) -HV #A #HV
+lapply (scpes_aaa_mono … HWV … HW … HV) #H destruct
+/4 width=5 by lsuba_aaa_conf, lsuba_beta, aaa_cast/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsubd.ma
new file mode 100644 (file)
index 0000000..a7d3561
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_lsubd.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsubd.ma".
+include "basic_2A/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Forward lemmas on lenv refinement for degree assignment ******************)
+
+lemma lsubsv_fwd_lsubd: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → G ⊢ L1 ⫃▪[h, g] L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=3 by lsubd_pair, lsubd_beta/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_scpds.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_scpds.ma
new file mode 100644 (file)
index 0000000..c079da4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_scpds.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsubd_da.ma".
+include "basic_2A/dynamic/lsubsv_lsubd.ma".
+include "basic_2A/dynamic/lsubsv_lstas.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties on decomposed extended parallel computation on terms **********)
+
+lemma lsubsv_scpds_trans: ∀h,g,G,L2,T1,T2,d. ⦃G, L2⦄ ⊢ T1 •*➡*[h, g, d] T2 →
+                          ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
+                          ∃∃T. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, d] T & ⦃G, L1⦄ ⊢ T2 ➡* T.
+#h #g #G #L2 #T1 #T2 #d2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L1 #HL12
+lapply (lsubsv_cprs_trans … HL12 … HT2) -HT2 #HT2
+elim (lsubsv_lstas_trans … HT1 … Hd1 … HL12) // #T0 #HT10 #HT0
+lapply (cpcs_cprs_strap1 … HT0 … HT2) -T #HT02
+elim (cpcs_inv_cprs … HT02) -HT02
+/5 width=5 by lsubsv_fwd_lsubd, lsubd_da_trans, ex4_2_intro, ex2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_snv.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_snv.ma
new file mode 100644 (file)
index 0000000..d7be376
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/lsubsv_snv.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/dynamic/lsubsv_scpds.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties concerning stratified native validity *************************)
+
+lemma lsubsv_snv_trans: ∀h,g,G,L2,T. ⦃G, L2⦄ ⊢ T ¡[h, g] →
+                        ∀L1. G ⊢ L1 ⫃¡[h, g] L2 → ⦃G, L1⦄ ⊢ T ¡[h, g].
+#h #g #G #L2 #T #H elim H -G -L2 -T //
+[ #I #G #L2 #K2 #V #i #HLK2 #_ #IHV #L1 #HL12
+  elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+  elim (lsubsv_inv_pair2 … H) -H * #K1
+  [ #HK12 #H destruct /3 width=5 by snv_lref/
+  | #W #d #HVW #_ #_ #_ #_ #H1 #H2 destruct -IHV
+    /3 width=6 by shnv_inv_snv, snv_lref/
+  ]
+| #a #I #G #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 destruct
+  /4 width=1 by snv_bind, lsubsv_pair/
+| #a #G #L2 #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L1 #HL12
+  elim (lsubsv_scpds_trans … HVW0 … HL12) -HVW0 #V0 #HV0 #HWV0
+  elim (lsubsv_scpds_trans … HTU0 … HL12) -HTU0 #X #HT0 #H
+  elim (cprs_inv_abst1 … H) -H #W #T0 #HW0 #_ #H destruct
+  elim (cprs_conf … HWV0 … HW0) -W0
+  /4 width=10 by snv_appl, scpds_cprs_trans, cprs_bind/
+| #G #L2 #U #T #U0 #_ #_ #HU0 #HTU0 #IHU #IHT #L1 #HL12
+  elim (lsubsv_scpds_trans … HTU0 … HL12) -HTU0 #X0 #HTX0 #H1
+  elim (lsubsv_scpds_trans … HU0 … HL12) -HU0 #X #HUX #H2
+  elim (cprs_conf … H1 … H2) -U0 /3 width=5 by snv_cast, scpds_cprs_trans/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/shnv.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/shnv.ma
new file mode 100644 (file)
index 0000000..1c0bf77
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/shnv.ma
@@ -0,0 +1,55 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/nativevalid_6.ma".
+include "basic_2A/equivalence/scpes.ma".
+include "basic_2A/dynamic/snv.ma".
+
+(* STRATIFIED HIGHER NATIVE VALIDITY FOR TERMS ******************************)
+
+inductive shnv (h) (g) (d1) (G) (L): predicate term ≝
+| shnv_cast: ∀U,T. ⦃G, L⦄ ⊢ U ¡[h, g] → ⦃G, L⦄ ⊢ T ¡[h, g] →
+             (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, d2, d2+1] T) →
+             shnv h g d1 G L (ⓝU.T)
+.
+
+interpretation "stratified higher native validity (term)"
+   'NativeValid h g d G L T = (shnv h g d G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact shnv_inv_cast_aux: ∀h,g,G,L,X,d1. ⦃G, L⦄ ⊢ X ¡[h, g, d1] → ∀U,T. X = ⓝU.T →
+                        ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
+                         & (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, d2, d2+1] T).
+#h #g #G #L #X #d1 * -X
+#U #T #HU #HT #HUT #U1 #T1 #H destruct /3 width=1 by and3_intro/
+qed-.
+
+lemma shnv_inv_cast: ∀h,g,G,L,U,T,d1. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g, d1] →
+                     ∧∧ ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g]
+                      & (∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ U •*⬌*[h, g, d2, d2+1] T).
+/2 width=3 by shnv_inv_cast_aux/ qed-.
+
+lemma shnv_inv_snv: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ¡[h, g, d] → ⦃G, L⦄ ⊢ T ¡[h, g].
+#h #g #G #L #T #d * -T
+#U #T #HU #HT #HUT elim (HUT 0) -HUT /2 width=3 by snv_cast/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma snv_shnv_cast: ∀h,g,G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g] → ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g, 0].
+#h #g #G #L #U #T #H elim (snv_inv_cast … H) -H
+#U0 #HU #HT #HU0 #HTU0 @shnv_cast // -HU -HT
+#d #H <(le_n_O_to_eq … H) -d /2 width=3 by scpds_div/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv.ma
new file mode 100644 (file)
index 0000000..0e420b1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv.ma
@@ -0,0 +1,111 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/nativevalid_5.ma".
+include "basic_2A/computation/scpds.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* activate genv *)
+inductive snv (h) (g): relation3 genv lenv term ≝
+| snv_sort: ∀G,L,k. snv h g G L (⋆k)
+| snv_lref: ∀I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → snv h g G K V → snv h g G L (#i)
+| snv_bind: ∀a,I,G,L,V,T. snv h g G L V → snv h g G (L.ⓑ{I}V) T → snv h g G L (ⓑ{a,I}V.T)
+| snv_appl: ∀a,G,L,V,W0,T,U0,d. snv h g G L V → snv h g G L T →
+            ⦃G, L⦄ ⊢ V •*➡*[h, g, 1] W0 → ⦃G, L⦄ ⊢ T •*➡*[h, g, d] ⓛ{a}W0.U0 → snv h g G L (ⓐV.T)
+| snv_cast: ∀G,L,U,T,U0. snv h g G L U → snv h g G L T →
+            ⦃G, L⦄ ⊢ U •*➡*[h, g, 0] U0 → ⦃G, L⦄ ⊢ T •*➡*[h, g, 1] U0 → snv h g G L (ⓝU.T)
+.
+
+interpretation "stratified native validity (term)"
+   'NativeValid h g G L T = (snv h g G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact snv_inv_lref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀i. X = #i →
+                       ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #i #H destruct
+| #I #G #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5 by ex2_3_intro/
+| #a #I #G #L #V #T #_ #_ #i #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #i #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #i #H destruct
+]
+qed-.
+
+lemma snv_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ #i ¡[h, g] →
+                    ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+/2 width=3 by snv_inv_lref_aux/ qed-.
+
+fact snv_inv_gref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀p. X = §p → ⊥.
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #p #H destruct
+| #I #G #L #K #V #i #_ #_ #p #H destruct
+| #a #I #G #L #V #T #_ #_ #p #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #p #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #p #H destruct
+]
+qed-.
+
+lemma snv_inv_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ §p ¡[h, g] → ⊥.
+/2 width=8 by snv_inv_gref_aux/ qed-.
+
+fact snv_inv_bind_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
+                       ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #b #Z #X1 #X2 #H destruct
+| #I #G #L #K #V #i #_ #_ #b #Z #X1 #X2 #H destruct
+| #a #I #G #L #V #T #HV #HT #b #Z #X1 #X2 #H destruct /2 width=1 by conj/
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #b #Z #X1 #X2 #H destruct
+| #G #L #U #T #U0 #_ #_ #_ #_ #b #Z #X1 #X2 #H destruct
+]
+qed-.
+
+lemma snv_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T ¡[h, g] →
+                    ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
+/2 width=4 by snv_inv_bind_aux/ qed-.
+
+fact snv_inv_appl_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀V,T. X = ⓐV.T →
+                       ∃∃a,W0,U0,d. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                                    ⦃G, L⦄ ⊢ V •*➡*[h, g, 1] W0 & ⦃G, L⦄ ⊢ T •*➡*[h, g, d] ⓛ{a}W0.U0.
+#h #g #G #L #X * -L -X
+[ #G #L #k #X1 #X2 #H destruct
+| #I #G #L #K #V #i #_ #_ #X1 #X2 #H destruct
+| #a #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
+| #a #G #L #V #W0 #T #U0 #d #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /2 width=6 by ex4_4_intro/
+| #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
+]
+qed-.
+
+lemma snv_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ¡[h, g] →
+                    ∃∃a,W0,U0,d. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                                 ⦃G, L⦄ ⊢ V •*➡*[h, g, 1] W0 & ⦃G, L⦄ ⊢ T •*➡*[h, g, d] ⓛ{a}W0.U0.
+/2 width=3 by snv_inv_appl_aux/ qed-.
+
+fact snv_inv_cast_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀U,T. X = ⓝU.T →
+                       ∃∃U0. ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                             ⦃G, L⦄ ⊢ U •*➡*[h, g, 0] U0 & ⦃G, L⦄ ⊢ T •*➡*[h, g, 1] U0.
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #X1 #X2 #H destruct
+| #I #G #L #K #V #i #_ #_ #X1 #X2 #H destruct
+| #a #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #X1 #X2 #H destruct
+| #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
+]
+qed-.
+
+lemma snv_inv_cast: ∀h,g,G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g] →
+                    ∃∃U0. ⦃G, L⦄ ⊢ U ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                          ⦃G, L⦄ ⊢ U •*➡*[h, g, 0] U0 & ⦃G, L⦄ ⊢ T •*➡*[h, g, 1] U0.
+/2 width=3 by snv_inv_cast_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_aaa.ma
new file mode 100644 (file)
index 0000000..8c8144b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_aaa.ma
@@ -0,0 +1,50 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/da_aaa.ma".
+include "basic_2A/computation/scpds_aaa.ma".
+include "basic_2A/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Forward lemmas on atomic arity assignment for terms **********************)
+
+lemma snv_fwd_aaa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
+#h #g #G #L #T #H elim H -G -L -T
+[ /2 width=2 by aaa_sort, ex_intro/
+| #I #G #L #K #V #i #HLK #_ * /3 width=6 by aaa_lref, ex_intro/
+| #a * #G #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2 by aaa_abbr, aaa_abst, ex_intro/
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 * #B #HV * #X #HT
+  lapply (scpds_aaa_conf … HV … HVW0) -HVW0 #HW0
+  lapply (scpds_aaa_conf … HT … HTU0) -HTU0 #H
+  elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct
+  lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4 by aaa_appl, ex_intro/
+| #G #L #U #T #U0 #_ #_ #HU0 #HTU0 * #B #HU * #A #HT
+  lapply (scpds_aaa_conf … HU … HU0) -HU0 #HU0
+  lapply (scpds_aaa_conf … HT … HTU0) -HTU0 #H
+  lapply (aaa_mono … H … HU0) -U0 #H destruct /3 width=3 by aaa_cast, ex_intro/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma snv_fwd_da: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃d. ⦃G, L⦄ ⊢ T ▪[h, g] d.
+#h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_da/
+qed-.
+
+lemma snv_fwd_lstas: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+                     ∀d. ∃U. ⦃G, L⦄ ⊢ T •*[h, d] U.
+#h #g #G #L #T #H #d elim (snv_fwd_aaa … H) -H
+#A #HT elim (aaa_lstas h … HT d) -HT /2 width=2 by ex_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_da_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_da_lpr.ma
new file mode 100644 (file)
index 0000000..2c2ecce
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_da_lpr.ma
@@ -0,0 +1,92 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsubd_da.ma".
+include "basic_2A/dynamic/snv_aaa.ma".
+include "basic_2A/dynamic/snv_scpes.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+fact da_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                     (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                     (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                     (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                     ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_da_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #_ #_ #_ #_ #d #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
+  lapply (da_inv_sort … H2) -H2
+  lapply (cpr_inv_sort1 … H3) -H3 #H destruct /2 width=1 by da_sort/
+| #i #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+  elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #H #HX0
+  elim (da_inv_lref … H2) -H2 * #K1 [ #V1 | #W1 #d1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
+  lapply (drop_mono … H … HLK1) -H #H destruct
+  elim (cpr_inv_lref1 … H3) -H3
+  [1,3: #H destruct
+    lapply (fqup_lref … G1 … HLK1)
+    elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+    elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+    /4 width=10 by da_ldef, da_ldec, fqup_fpbg/
+  |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
+    lapply (drop_mono … H … HLK1) -H #H destruct
+    lapply (fqup_lref … G1 … HLK1)
+    elim (lpr_drop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
+    elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #_ #H destruct
+    lapply (drop_fwd_drop2 … HLK2) -V2
+    /4 width=8 by da_lift, fqup_fpbg/
+  ]
+| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+  elim (snv_inv_bind … H1) -H1 #_ #HT1
+  lapply (da_inv_bind … H2) -H2
+  elim (cpr_inv_bind1 … H3) -H3 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct
+    /4 width=9 by da_bind, fqup_fpbg, lpr_pair/
+  | #T2 #HT12 #HT2 #H1 #H2 destruct
+    /4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, drop_drop/
+  ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct
+  elim (snv_inv_appl … H1) -H1 #b1 #W1 #U1 #d1 #HV1 #HT1 #HVW1 #HTU1
+  lapply (da_inv_flat … H2) -H2 #Hd
+  elim (cpr_inv_appl1 … H3) -H3 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fqup_fpbg/
+  | #b #V2 #W #W2 #U #U2 #HV12 #HW2 #HU2 #H1 #H2 destruct
+    elim (snv_inv_bind … HT1) -HT1 #HW #HU
+    lapply (da_inv_bind … Hd) -Hd #Hd
+    elim (scpds_inv_abst1 … HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -d1
+    elim (snv_fwd_da … HV1) #d1 #Hd1
+    elim (snv_fwd_da … HW) #d0 #Hd0
+    lapply (cprs_scpds_div … HW3 … Hd0 … 1 HVW1) -W3 #H
+    elim (da_scpes_aux … IH3 IH2 IH1 … Hd0 … Hd1 … H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
+    <minus_n_O #H destruct >(plus_minus_m_m d1 1) in Hd1; // -H1 #Hd1
+    lapply (IH1 … HV1 … Hd1 … HV12 … HL12) -HV1 -Hd1 -HV12 [ /2 by fqup_fpbg/ ]
+    lapply (IH1 … Hd0 … HW2 … HL12) -Hd0 /2 width=1 by fqup_fpbg/ -HW
+    lapply (IH1 … HU … Hd … HU2 (L2.ⓛW2) ?) -IH1 -HU -Hd -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
+    /4 width=6 by da_bind, lsubd_da_trans, lsubd_beta/
+  | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -d1 -HV1
+    elim (snv_inv_bind … HT1) -HT1 #_
+    lapply (da_inv_bind … Hd) -Hd
+    /5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
+  ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+  elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
+  lapply (da_inv_flat … H2) -H2 #Hd
+  elim (cpr_inv_cast1 … H3) -H3
+  [ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
+  | /3 width=7 by fqup_fpbg/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_fsb.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_fsb.ma
new file mode 100644 (file)
index 0000000..41ff410
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_fsb.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fsb_aaa.ma".
+include "basic_2A/dynamic/snv_aaa.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* forward lemmas on "qrst" strongly normalizing closures *********************)
+
+lemma snv_fwd_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦥[h, g] ⦃G, L, T⦄.
+#h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lift.ma
new file mode 100644 (file)
index 0000000..469484b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lift.ma
@@ -0,0 +1,117 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqus_alt.ma".
+include "basic_2A/computation/scpds_lift.ma".
+include "basic_2A/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Relocation properties ****************************************************)
+
+lemma snv_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, g] → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
+#h #g #G #K #T #H elim H -G -K -T
+[ #G #K #k #L #s #l #m #_ #X #H
+  >(lift_inv_sort1 … H) -X -K -l -m //
+| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #l #m #HLK #X #H
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (drop_trans_le … HLK … HK0) -K /2 width=2 by lt_to_le/ #X #HL0 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #L0 #W #HLK0 #HVW #H destruct
+    /3 width=9 by snv_lref/
+  | lapply (drop_trans_ge … HLK … HK0 ?) -K
+    /3 width=9 by snv_lref, drop_inv_gen/
+  ]
+| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
+  /4 width=5 by snv_bind, drop_skip/
+| #a #G #K #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
+  elim (lift_total W0 l m)
+  elim (lift_total U0 (l+1) m)
+  /4 width=17 by snv_appl, scpds_lift, lift_bind/
+| #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
+  elim (lift_total U0 l m)
+  /3 width=12 by snv_cast, scpds_lift/
+]
+qed.
+
+lemma snv_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                    ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ T ¡[h, g].
+#h #g #G #L #U #H elim H -G -L -U
+[ #G #L #k #K #s #l #m #_ #X #H
+  >(lift_inv_sort2 … H) -X -L -l -m //
+| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #l #m #HLK #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct
+  [ elim (drop_conf_le … HLK … HL0) -L /2 width=2 by lt_to_le/ #X #HK0 #H
+    elim (drop_inv_skip1 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K0 #V #HLK0 #HVW #H destruct
+    /3 width=12 by snv_lref/
+  | lapply (drop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
+  ]
+| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
+  /4 width=5 by snv_bind, drop_skip/
+| #a #G #L #W #W1 #U #U1 #d #_ #_ #HW1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
+  elim (scpds_inv_lift1 … HW1 … HLK … HVW) -HW1 #W0 #HW01 #HVW0
+  elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU0
+  elim (lift_inv_bind2 … H) -H #Y #U0 #HY #HU01 #H destruct
+  lapply (lift_inj … HY … HW01) -HY #H destruct
+  /3 width=6 by snv_appl/
+| #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
+  elim (scpds_inv_lift1 … HWU1 … HLK … HVW) -HWU1 #U0 #HU01 #HVU0
+  elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #HX #HTU0 
+  lapply (lift_inj … HX … HU01) -HX #H destruct
+  /3 width=5 by snv_cast/
+]
+qed-.
+
+(* Properties on subclosure *************************************************)
+
+lemma snv_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I1 #G1 #L1 #V1 #H
+  elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
+  lapply (drop_inv_O2 … H) -H #H destruct //
+|2: *
+|5,6: /3 width=8 by snv_inv_lift/
+]
+[1,3: #a #I #G1 #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
+|2,4: * #G1 #L1 #V1 #T1 #H
+  [1,3: elim (snv_inv_appl … H) -H //
+  |2,4: elim (snv_inv_cast … H) -H //
+  ]
+]
+qed-.
+
+lemma snv_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_inv_gen … H) -H [|*]
+/2 width=5 by snv_fqu_conf/
+qed-.
+
+lemma snv_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqup_strap1, snv_fqu_conf/
+qed-.
+
+lemma snv_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_gen … H) -H [|*]
+/2 width=5 by snv_fqup_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lpr.ma
new file mode 100644 (file)
index 0000000..78f7e59
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lpr.ma
@@ -0,0 +1,119 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/dynamic/snv_lift.ma".
+include "basic_2A/dynamic/snv_aaa.ma".
+include "basic_2A/dynamic/snv_scpes.ma".
+include "basic_2A/dynamic/lsubsv_snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact snv_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                      ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
+  >(cpr_inv_sort1 … H2) -X //
+| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+  elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+  elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+  lapply (fqup_lref … G1 … HLK1) #HKL
+  elim (cpr_inv_lref1 … H2) -H2
+  [ #H destruct -HLK1 /4 width=10 by fqup_fpbg, snv_lref/
+  | * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
+    lapply (drop_mono … H … HLK1) -HLK1 -H #H destruct
+    lapply (drop_fwd_drop2 … HLK2) -HLK2 /4 width=8 by fqup_fpbg, snv_lift/
+  ]
+| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+  elim (cpr_inv_bind1 … H2) -H2 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbg, snv_bind, lpr_pair/
+  | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
+    /4 width=10 by fqup_fpbg, snv_inv_lift, lpr_pair, drop_drop/
+  ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+  elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
+  elim (cpr_inv_appl1 … H2) -H2 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
+    lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+    lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ #HT2
+    elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 -HV12 #XV #HVW2 #HXV
+    elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -HTU1 -HT12 #X #HTU2 #H
+    elim (cprs_inv_abst1 … H) -H #XW #U2 #HXW #_ #H destruct -IH1 -IH3 -IH2 -L1
+    elim (cprs_conf … HXV … HXW) -W1 #W2 #HXV #HXW
+    lapply (scpds_cprs_trans … HVW2 … HXV) -XV
+    lapply (scpds_cprs_trans … (ⓛ{a}W2.U2) … HTU2 ?)
+    /2 width=7 by snv_appl, cprs_bind/
+  | #b #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
+    elim (snv_inv_bind … HT1) -HT1 #HW10 #HT10
+    elim (scpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30 -d0
+    elim (snv_fwd_da … HV1) #d #HV1d
+    elim (snv_fwd_da … HW10) #d0 #HW10d
+    lapply (cprs_scpds_div … HW130 … HW10d … 1 HVW1) -W30 #HVW10
+    elim (da_scpes_aux … IH4 IH1 IH2 … HW10d … HV1d … HVW10) /2 width=1 by fqup_fpbg/
+    #_ #Hd <minus_n_O #H destruct >(plus_minus_m_m d 1) in HV1d; // -Hd #HV1d
+    lapply (scpes_cpr_lpr_aux … IH2 IH3 … HVW10 … HW120 … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HVW10 #HVW20
+    lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1d #HV2d
+    lapply (IH2 … HW10d … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10d #HW20d
+    lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+    lapply (IH1 … HW120 … HL12) /2 width=1 by fqup_fpbg/ -HW10 #HW20
+    lapply (IH1 … HT10 … HT120 … (L2.ⓛW20) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT10 #HT20
+    @snv_bind /2 width=1 by snv_cast_scpes/
+    @(lsubsv_snv_trans … HT20) -HT20
+    @(lsubsv_beta … (d-1)) //
+    @shnv_cast [1,2: // ] #d0 #Hd0
+    lapply (scpes_le_aux … IH4 IH1 IH2 IH3 … HW20d … HV2d … d0 … HVW20) -IH4 -IH3 -IH2 -IH1 -HW20d -HV2d -HVW20
+    /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
+  | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
+    elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
+    elim (scpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
+    elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
+    elim (scpds_cpr_lpr_aux … IH2 IH3 … HVW1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ -HVW1 #XV #HXV0 #HXVW1
+    elim (scpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbg, lpr_pair/ -HTU0 #X #HXT2 #H
+    elim (cprs_inv_abst1 … H) -H #W #U2 #HW3 #_ #H destruct -U3
+    lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbg/ #HW2
+    lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbg/ #HV0
+    lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -L1 #HT2
+    lapply (snv_lift … HV0 (L2.ⓓW2) (Ⓕ) … HV02) /2 width=1 by drop_drop/ -HV0 #HV2
+    elim (lift_total XV 0 1) #XW #HXVW
+    lapply (scpds_lift … HXV0 (L2.ⓓW2) (Ⓕ) … HV02 … HXVW) /2 width=1 by drop_drop/ -V0 #HXWV2
+    lapply (cprs_lift … HXVW1 (L2.ⓓW2) (Ⓕ) … HW13 … HXVW) /2 width=1 by drop_drop/ -W1 -XV #HXW3
+    elim (cprs_conf … HXW3 … HW3) -W3 #W3 #HXW3 #HW3
+    lapply (scpds_cprs_trans … HXWV2 … HXW3) -XW
+    lapply (scpds_cprs_trans … (ⓛ{a}W3.U2) … HXT2 ?) /2 width=1 by cprs_bind/ -W
+    /3 width=6 by snv_appl, snv_bind/
+  ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
+  elim (snv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
+  elim (cpr_inv_cast1 … H2) -H2
+  [ * #W2 #T2 #HW12 #HT12 #H destruct
+    elim (snv_fwd_da … HW1) #d #HW1d
+    lapply (scpds_div … HWU1 … HTU1) -U1 -d #HWT1
+    lapply (scpes_cpr_lpr_aux … IH2 IH3 … HWT1 … HW12 … HT12 … HL12) /2 width=1 by fqup_fpbg/
+    lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbg/
+    lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbg/ -L1
+    /2 width=1 by snv_cast_scpes/
+  | #H -IH3 -IH2 -HW1 -U1
+    lapply (IH1 … H … HL12) /2 width=1 by fqup_fpbg/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas.ma
new file mode 100644 (file)
index 0000000..a60439d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/dynamic/snv_lift.ma".
+include "basic_2A/dynamic/snv_scpes.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on nat-iterated stratified static type assignment for terms ***)
+
+fact snv_lstas_aux: ∀h,g,G0,L0,T0.
+                    (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                    (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                    (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                    (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                    ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_lstas h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #HG0 #HL0 #HT0 #_ #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
+  >(lstas_inv_sort1 … H2) -X //
+| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #T #H2 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HLK0 #HX0
+  elim (da_inv_lref … Hd1) -Hd1 * #K1 [ #V1 | #W1 #d0 ] #HLK1 [ #Hd1 | #Hd0 #H ]
+  lapply (drop_mono … HLK0 … HLK1) -HLK0 #H0 destruct
+  elim (lstas_inv_lref1 … H2) -H2 * #K #Y #X [3,6: #d ] #HLK #HYX [1,2: #HXT #H0 |3,5: #HXT |4,6: #H1 #H2 ]
+  lapply (drop_mono … HLK … HLK1) -HLK #H destruct
+  [ lapply (le_plus_to_le_r … Hd21) -Hd21 #Hd21 |3: -Hd21 ]
+  lapply (fqup_lref … G1 … HLK1) #H
+  lapply (drop_fwd_drop2 … HLK1) /4 width=8 by snv_lift, snv_lref, fqup_fpbg/
+| #p #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+  lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+  elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fqup_fpbg, snv_bind/
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct
+  elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
+  lapply (da_inv_flat … Hd1) -Hd1 #Hd1
+  elim (lstas_inv_appl1 … H2) -H2 #T0 #HT10 #H destruct
+  lapply (IH1 … HT1 … Hd1 … HT10) /2 width=1 by fqup_fpbg/ #HT0
+  lapply (lstas_scpds_aux … IH1 IH4 IH3 IH2 … Hd1 … HT10 … HTU1) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fqup_fpbg/ -T1 -d1 #H
+  elim (scpes_inv_abst2 … H) -H /3 width=6 by snv_appl, scpds_cprs_trans/
+| #U1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_cast … H1) -H1
+  lapply (da_inv_flat … Hd1) -Hd1
+  lapply (lstas_inv_cast1 … H2) -H2 /3 width=8 by fqup_fpbg/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas_lpr.ma
new file mode 100644 (file)
index 0000000..4012342
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_lstas_lpr.ma
@@ -0,0 +1,139 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/dynamic/snv_aaa.ma".
+include "basic_2A/dynamic/snv_scpes.ma".
+include "basic_2A/dynamic/lsubsv_lstas.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on sn parallel reduction for local environments ***************)
+
+fact lstas_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                        ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lstas_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+[ #k #_ #_ #_ #_ #d1 #d2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
+  >(lstas_inv_sort1 … H2) -X2
+  >(cpr_inv_sort1 … H3) -X3 /2 width=3 by ex2_intro/
+| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
+  elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
+  elim (da_inv_lref … Hd1) -Hd1 * #K1 [ #V1 | #W1 #d ] #HLK1 [ #HVd1 | #HWd1 #H destruct ]
+  lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
+  elim (lstas_inv_lref1 … H2) -H2 * #K0 #V0 #X0 [3,6: #d0 ] #HK0 #HVX0 [1,2: #HX02 #H |3,5: #HX02 |4,6: #H1 #H2 ] destruct
+  lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
+  [ lapply (le_plus_to_le_r … Hd21) -Hd21 #Hd21 |3: -Hd21 ]
+  lapply (fqup_lref … G1 … HLK1) #HKV1
+  elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+  elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #W2 | #V2 ] #HK12 [ #HW12 | #HW12 | #HV12 ] #H destruct
+  lapply (drop_fwd_drop2 … HLK2) #H2
+  elim (cpr_inv_lref1 … H3) -H3
+  [1,3,5: #H destruct -HLK1
+  |2,4,6: * #K #V #X #H #HVX #HX3
+          lapply (drop_mono … H … HLK1) -H -HLK1 #H destruct
+  ]
+  [ lapply (IH2 … HWd1 … HW12 … HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
+    elim (da_lstas … H d0) -H
+    elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
+    elim (lift_total V2 0 (i+1))
+    /3 width=12 by cpcs_lift, lstas_succ, ex2_intro/
+  | elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0
+    /3 width=5 by fqup_fpbg, lstas_zero, ex2_intro/
+  | elim (IH1 … HVd1 … HVX0 … HV12 … HK12) -IH1 -HVd1 -HVX0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
+    elim (lift_total W2 0 (i+1))
+    /4 width=12 by cpcs_lift, lstas_ldef, ex2_intro/
+  | elim (IH1 … HVd1 … HVX0 … HVX … HK12) -IH1 -HVd1 -HVX0 -HVX -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -d1 #W2 #HXW2 #HW02
+    elim (lift_total W2 0 (i+1))
+    /3 width=12 by cpcs_lift, lstas_lift, ex2_intro/
+  ]
+| #p #_ #_ #HT0 #H1 destruct -IH4 -IH3 -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_bind … H1) -H1 #_ #HT1
+  lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+  elim (lstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
+  elim (cpr_inv_bind1 … H3) -H3 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct
+    elim (IH1 … Hd1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hd1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
+    /4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
+  | #T3 #HT13 #HXT3 #H1 #H2 destruct
+    elim (IH1 … Hd1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hd1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
+    elim (lstas_inv_lift1 … HTU3 L2 … HXT3) -T3
+    /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, drop_drop, ex2_intro/
+  ]
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
+  elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
+  lapply (da_inv_flat … Hd1) -Hd1 #Hd1
+  elim (lstas_inv_appl1 … H2) -H2 #X #HT1U #H destruct
+  elim (cpr_inv_appl1 … H3) -H3 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct -a -d0 -W1 -U1 -HV1 -IH4 -IH3 -IH2
+    elim (IH1 … Hd1 … HT1U … HT12 … HL12) -IH1 -Hd1 -HT1U
+    /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_appl, ex2_intro/
+  | #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
+    elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
+    lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+    elim (lstas_inv_bind1 … HT1U) -HT1U #U #HT2U #H destruct
+    elim (scpds_inv_abst1 … HTU1) -HTU1 #W0 #U0 #HW20 #_ #H destruct -U0 -d0
+    elim (snv_fwd_da … HW2) #d0 #HW2d
+    lapply (cprs_scpds_div … HW20 … HW2d … HVW1) -W0 #H21
+    elim (snv_fwd_da … HV1) #d #HV1d
+    elim (da_scpes_aux … IH4 IH3 IH2 … HW2d … HV1d … H21) /2 width=1 by fqup_fpbg/ #_ #H
+    <minus_n_O #H0 destruct >(plus_minus_m_m d 1) in HV1d; // -H #HV1d
+    lapply (scpes_cpr_lpr_aux … IH2 IH1 … H21 … HW23 … HV12 … HL12) -H21 /2 width=1 by fqup_fpbg/ #H32
+    lapply (IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
+    lapply (IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
+    lapply (IH2 … HW2d … HW23 … HL12) /2 width=1 by fqup_fpbg/ -HW2 -HW2d #HW3d
+    lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV1d #HV2d
+    elim (IH1 … Hd1 … HT2U … HT23 (L2.ⓛW3)) -HT2U /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
+    elim (lsubsv_lstas_trans … g … HTU3 … Hd21 … (L2.ⓓⓝW3.V2)) -HTU3
+    [ #U4 #HT3U4 #HU43 -IH1 -IH2 -IH3 -IH4 -d -d1 -HW3 -HV2 -HT2
+      @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
+      @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
+      @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
+      /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
+    | -U3
+      @(lsubsv_beta … (d-1)) /3 width=7 by fqup_fpbg/
+      @shnv_cast [1,2: // ] #d0 #Hd0
+      lapply (scpes_le_aux … IH4 IH3 IH2 IH1 … HW3d … HV2d … d0 … H32) -IH4 -IH3 -IH2 -IH1 -HW3d -HV2d -H32
+      /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
+    | -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbg, lpr_pair/
+    ]
+  | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -d0 -W1 -HV1 -IH4 -IH3 -IH2
+    elim (snv_inv_bind … HT1) -HT1 #_ #HT0
+    lapply (da_inv_bind … Hd1) -Hd1 #Hd1
+    elim (lstas_inv_bind1 … HT1U) -HT1U #U0 #HTU0 #H destruct
+    elim (IH1 … Hd1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hd1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
+    lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
+    lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
+    lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
+    lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
+    lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
+    /4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
+  ]
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
+  elim (snv_inv_cast … H1) -H1 #U1 #_ #HT1 #_ #_ -U1
+  lapply (da_inv_flat … Hd1) -Hd1 #Hd1
+  lapply (lstas_inv_cast1 … H2) -H2 #HTU1
+  elim (cpr_inv_cast1 … H3) -H3
+  [ * #U2 #T2 #_ #HT12 #H destruct
+    elim (IH1 … Hd1 … HTU1 … HT12 … HL12) -IH1 -Hd1 -HTU1 -HL12
+    /3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
+  | #HT1X3 elim (IH1 … Hd1 … HTU1 … HT1X3 … HL12) -IH1 -Hd1 -HTU1 -HL12
+    /2 width=3 by fqup_fpbg, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_preserve.ma
new file mode 100644 (file)
index 0000000..829e001
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_preserve.ma
@@ -0,0 +1,94 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fsb_aaa.ma".
+include "basic_2A/dynamic/snv_da_lpr.ma".
+include "basic_2A/dynamic/snv_lstas.ma".
+include "basic_2A/dynamic/snv_lstas_lpr.ma".
+include "basic_2A/dynamic/snv_lpr.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Main preservation properties *********************************************)
+
+lemma snv_preserve: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+                    ∧∧ IH_da_cpr_lpr h g G L T
+                     & IH_snv_cpr_lpr h g G L T
+                     & IH_snv_lstas h g G L T
+                     & IH_lstas_cpr_lpr h g G L T.
+#h #g #G #L #T #HT elim (snv_fwd_aaa … HT) -HT
+#A #HT @(aaa_ind_fpbg h g … HT) -G -L -T -A
+#G #L #T #A #_ #IH -A @and4_intro
+[ letin aux ≝ da_cpr_lpr_aux | letin aux ≝ snv_cpr_lpr_aux
+| letin aux ≝ snv_lstas_aux | letin aux ≝ lstas_cpr_lpr_aux
+]
+@(aux … G L T) // #G0 #L0 #T0 #H elim (IH … H) -IH -H //
+qed-.
+
+theorem da_cpr_lpr: ∀h,g,G,L,T. IH_da_cpr_lpr h g G L T.
+#h #g #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
+qed-.
+
+theorem snv_cpr_lpr: ∀h,g,G,L,T. IH_snv_cpr_lpr h g G L T.
+#h #g #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/
+qed-.
+
+theorem snv_lstas: ∀h,g,G,L,T. IH_snv_lstas h g G L T.
+#h #g #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/
+qed-.
+
+theorem lstas_cpr_lpr: ∀h,g,G,L,T. IH_lstas_cpr_lpr h g G L T.
+#h #g #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/
+qed-.
+
+(* Advanced preservation properties *****************************************)
+
+lemma snv_cprs_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                    ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G #L1 #T1 #HT1 #T2 #H
+@(cprs_ind … H) -T2 /3 width=5 by snv_cpr_lpr/
+qed-.
+
+lemma da_cprs_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                   ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d →
+                   ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] d.
+#h #g #G #L1 #T1 #HT1 #d #Hd #T2 #H
+@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/
+qed-.
+
+lemma lstas_cprs_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                      ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 →
+                      ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+                      ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                      ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (lstas_cpr_lpr … g … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2
+[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
+qed-.
+
+lemma lstas_cpcs_lpr: ∀h,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                      ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 →
+                      ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+                      ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, g] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 →
+                      ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1
+elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2
+lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_scpes.ma b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_scpes.ma
new file mode 100644 (file)
index 0000000..40f8b7b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/dynamic/snv_scpes.ma
@@ -0,0 +1,198 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbg_fpbs.ma".
+include "basic_2A/equivalence/scpes_cpcs.ma".
+include "basic_2A/equivalence/scpes_scpes.ma".
+include "basic_2A/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Inductive premises for the preservation results **************************)
+
+definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                           λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                           ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+
+definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                          λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                          ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d →
+                          ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                          ⦃G, L2⦄ ⊢ T2 ▪[h, g] d.
+
+definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                             λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                             ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 →
+                             ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+                             ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                             ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+
+definition IH_snv_lstas: ∀h:sh. sd h → relation3 genv lenv term ≝
+                         λh,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+                         ∀d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, g] d1 →
+                         ∀U. ⦃G, L⦄ ⊢ T •*[h, d2] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+
+(* Properties for the preservation results **********************************)
+
+fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                       ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
+@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
+qed-.
+
+fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                      ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                      ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d →
+                      ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] d.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
+qed-.
+
+fact da_scpds_lpr_aux: ∀h,g,G0,L0,T0.
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                       ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀d1. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 →
+                       ∀T2,d2. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, d2] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                       d2 ≤ d1 ∧ ⦃G, L2⦄ ⊢ T2 ▪[h, g] d1-d2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12
+lapply (da_mono … H … Hd1) -H #H destruct
+lapply (lstas_da_conf … HT1 … Hd1) #Hd12
+lapply (da_cprs_lpr_aux … IH2 IH1 … Hd12 … HT2 … HL12) -IH2 -IH1 -HT2 -HL12
+/3 width=8 by fpbg_fpbs_trans, lstas_fpbs, conj/
+qed-.
+
+fact da_scpes_aux: ∀h,g,G0,L0,T0.
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                   ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+                   ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+                   ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, g] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, g] d12 →
+                   ∀d21,d22. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21, d22] T2 →
+                   ∧∧ d21 ≤ d11 & d22 ≤ d12 & d11 - d21 = d12 - d22.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2
+elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd11 … HT1 … L) -Hd11 -HT1 //
+elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd12 … HT2 … L) -Hd12 -HT2 //
+/3 width=7 by da_mono, and3_intro/
+qed-.
+
+fact lstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                         (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                         (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                         (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                         ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                         ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 →
+                         ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 →
+                         ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                         ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (IH1 … Hd21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
+[2: /3 width=12 by da_cprs_lpr_aux/
+|3: /3 width=10 by snv_cprs_lpr_aux/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
+] -G0 -L0 -T0 -T1 -T -d1
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
+qed-.
+
+fact scpds_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                        ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                        ∀U1,d. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, d] U1 →
+                        ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                        ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g, d] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
+lapply (IH2 … H01 … HTd1 … HT12 … HL12) -L0 -T0 // -T1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/
+qed-.
+
+fact scpes_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                        (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                        ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                        ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+                        ∀d1,d2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2 →
+                        ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                        ⦃G, L2⦄ ⊢ U1 •*⬌*[h, g, d1, d2] U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
+elim (scpds_cpr_lpr_aux … IH2 IH1 … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
+elim (scpds_cpr_lpr_aux … IH2 IH1 … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
+elim (cprs_conf … H1 … H2) -T0
+/3 width=5 by scpds_div, scpds_cprs_trans/
+qed-.
+
+fact lstas_scpds_aux: ∀h,g,G0,L0,T0.
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                      ∀G,L,T. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+                      ∀d,d1. d1 ≤ d → ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀T1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 →
+                      ∀T2,d2. ⦃G, L⦄ ⊢ T •*➡*[h, g, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d2-d1, d1-d2] T2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2
+lapply (da_mono … H … HTd) -H #H destruct
+lapply (lstas_da_conf … HT1 … HTd) #HTd1
+lapply (lstas_da_conf … HTX … HTd) #HXd
+lapply (da_cprs_lpr_aux … IH3 IH2 … HXd … HXT2 L ?)
+[1,2,3: /3 width=8 by fpbg_fpbs_trans, lstas_fpbs/ ] #HTd2
+elim (le_or_ge d1 d2) #Hd12 >(eq_minus_O … Hd12)
+[ elim (da_lstas … HTd2 0) #X2 #HTX2 #_ -IH4 -IH3 -IH2 -IH1 -H0 -HT -HTd -HXd
+  /5 width=6 by lstas_scpds, scpds_div, cprs_strap1, lstas_cpr, lstas_conf_le, monotonic_le_minus_l, ex4_2_intro/
+| elim (da_lstas … HTd1 0) #X1 #HTX1 #_
+  lapply (lstas_conf_le … HTX … HT1) // #HXT1 -HT1
+  elim (lstas_cprs_lpr_aux … IH3 IH2 IH1 … HXd … HXT1 … HXT2 L) -IH3 -IH2 -IH1 -HXd -HXT1 -HXT2
+  /4 width=8 by cpcs_scpes, cpcs_cpr_conf, fpbg_fpbs_trans, lstas_fpbs, lstas_cpr, monotonic_le_minus_l/
+]
+qed-.
+
+fact scpes_le_aux: ∀h,g,G0,L0,T0.
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) →
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) →
+                   ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+                   ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+                   ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, g] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, g] d12 →
+                   ∀d21,d22,d. d21 + d ≤ d11 → d22 + d ≤ d12 →
+                   ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21, d22] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21+d, d22+d] T2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20
+elim (da_lstas … Hd11 (d21+d)) #X1 #HTX1 #_
+elim (da_lstas … Hd12 (d22+d)) #X2 #HTX2 #_
+lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd11 … HTX1 … HT10) -HT10
+[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_m_m_commutative #HX1 ]
+lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd12 … HTX2 … HT20) -HT20
+[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_m_m_commutative #HX2 ]
+lapply (lstas_scpes_trans … Hd11 … HTX1 … HX1) [ // ] -Hd11 -HTX1 -HX1 -H1 #H1
+lapply (lstas_scpes_trans … Hd12 … HTX2 … HX2) [ // ] -Hd12 -HTX2 -HX2 -H2 #H2
+/2 width=4 by scpes_canc_dx/ (**) (* full auto fails *)
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma snv_cast_scpes: ∀h,g,G,L,U,T. ⦃G, L⦄ ⊢ U ¡[h, g] →  ⦃G, L⦄ ⊢ T ¡[h, g] →
+                      ⦃G, L⦄ ⊢ U •*⬌*[h, g, 0, 1] T → ⦃G, L⦄ ⊢ ⓝU.T ¡[h, g].
+#h #g #G #L #U #T #HU #HT * /2 width=3 by snv_cast/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma
new file mode 100644 (file)
index 0000000..ffae53a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/pconvstar_4.ma".
+include "basic_2A/conversion/cpc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+definition cpcs: relation4 genv lenv term term ≝
+           λG. LTC … (cpc G).
+
+interpretation "context-sensitive parallel equivalence (term)"
+   'PConvStar G L T1 T2 = (cpcs G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpcs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T2.
+normalize /3 width=6 by TC_star_ind/
+qed-.
+
+lemma cpcs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T1.
+normalize /3 width=6 by TC_star_ind_dx/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: pc3_refl *)
+lemma cpcs_refl: ∀G,L. reflexive … (cpcs G L).
+/2 width=1 by inj/ qed.
+
+(* Basic_1: was: pc3_s *)
+lemma cpcs_sym: ∀G,L. symmetric … (cpcs G L).
+normalize /3 width=1 by cpc_sym, TC_symmetric/
+qed-.
+
+lemma cpc_cpcs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/2 width=1 by inj/ qed.
+
+lemma cpcs_strap1: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+normalize /2 width=3 by step/
+qed-.
+
+lemma cpcs_strap2: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+normalize /2 width=3 by TC_strap/
+qed-.
+
+(* Basic_1: was: pc3_pr2_r *)
+lemma cpr_cpcs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=1 by cpc_cpcs, or_introl/ qed.
+
+(* Basic_1: was: pc3_pr2_x *)
+lemma cpr_cpcs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=1 by cpc_cpcs, or_intror/ qed.
+
+lemma cpcs_cpr_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_strap1, or_introl/ qed-.
+
+(* Basic_1: was: pc3_pr2_u *)
+lemma cpcs_cpr_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_strap2, or_introl/ qed-.
+
+lemma cpcs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_strap1, or_intror/ qed-.
+
+lemma cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpr_cpcs_dx, cpcs_strap1, or_intror/ qed-.
+
+(* Basic_1: was: pc3_pr2_u2 *)
+lemma cpcs_cpr_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_strap2, or_intror/ qed-.
+
+(* Basic_1: removed theorems 9:
+            clear_pc3_trans pc3_ind_left
+            pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
+            pc3_pr2_fqubst0 pc3_pr2_fqubst0_back pc3_fqubst0
+            pc3_gen_abst pc3_gen_abst_shift
+*)
+(* Basic_1: removed local theorems 6:
+            pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
+            pc3_wcpr0_t_aux
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_aaa.ma
new file mode 100644 (file)
index 0000000..90d87f6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_aaa.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cpxs_aaa.ma".
+include "basic_2A/equivalence/cpcs_cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Main inversion lemmas about atomic arity assignment on terms *************)
+
+theorem cpcs_aaa_mono: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                       ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
+                       A1 = A2.
+#G #L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+lapply (cprs_aaa_conf … HA1 … HT1) -T1 #HA1
+lapply (cprs_aaa_conf … HA2 … HT2) -T2 #HA2
+lapply (aaa_mono … HA1 … HA2) -L -T //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cpcs.ma
new file mode 100644 (file)
index 0000000..cdad296
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cpcs.ma
@@ -0,0 +1,212 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/lprs_cprs.ma".
+include "basic_2A/conversion/cpc_cpc.ma".
+include "basic_2A/equivalence/cpcs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
+#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
+[ /3 width=3 by ex2_intro/
+| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
+  [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
+  | /3 width=5 by cprs_strap2, ex2_intro/
+  ]
+]
+qed-.
+
+(* Basic_1: was: pc3_gen_sort *)
+lemma cpcs_inv_sort: ∀G,L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
+#G #L #k1 #k2 #H elim (cpcs_inv_cprs … H) -H
+#T #H1 >(cprs_inv_sort1 … H1) -T #H2
+lapply (cprs_inv_sort1 … H2) -L #H destruct //
+qed-.
+
+lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+#a #G #L #W1 #T1 #T #H
+elim (cpcs_inv_cprs … H) -H #X #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+/3 width=6 by cprs_bind, ex2_2_intro/
+qed-.
+
+lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+/3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
+
+(* Basic_1: was: pc3_gen_sort_abst *)
+lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+#a #G #L #W #T #k #H
+elim (cpcs_inv_cprs … H) -H #X #H1
+>(cprs_inv_sort1 … H1) -X #H2
+elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: pc3_gen_lift *)
+lemma cpcs_inv_lift: ∀G,L,K,s,l,m. ⬇[s, l, m] L ≡ K →
+                     ∀T1,U1. ⬆[l, m] T1 ≡ U1 → ∀T2,U2. ⬆[l, m] T2 ≡ U2 →
+                     ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
+#G #L #K #s #l #m #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
+elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
+elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
+elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
+>(lift_inj … HXU … HTU) -X -U -l -m /2 width=3 by cprs_div/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lpr_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                      ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, lpr_cprs_trans/
+qed-.
+
+lemma lprs_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                       ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, lprs_cprs_trans/
+qed-.
+
+lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cpr_cprs_div/
+qed-.
+
+lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_cpr_div/
+qed-.
+
+lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_div/
+qed-.
+
+lemma lprs_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was: pc3_wcpr0_t *)
+(* Basic_1: note: pc3_wcpr0_t should be renamed *)
+(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
+lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
+
+(* Basic_1: was only: pc3_pr0_pr2_t *)
+(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
+lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                    ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+/3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
+
+(* Basic_1: was only: pc3_thin_dx *)
+lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
+elim (cpcs_inv_cprs … HV12) -HV12
+elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_flat, cprs_div/
+qed.
+
+lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                            ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
+
+lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
+                    ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
+#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
+/3 width=5 by cprs_div, cprs_bind/
+qed.
+
+lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
+                        ∀L2. L2 ⫃ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, lsubr_cprs_trans/
+qed-.
+
+(* Basic_1: was: pc3_lift *)
+lemma cpcs_lift: ∀G,L,K,s,l,m. ⬇[s, l, m] L ≡ K →
+                 ∀T1,U1. ⬆[l, m] T1 ≡ U1 → ∀T2,U2. ⬆[l, m] T2 ≡ U2 →
+                 ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
+#G #L #K #s #l #m #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+elim (lift_total T l m) /3 width=12 by cprs_div, cprs_lift/
+qed.
+
+lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+                  ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
+#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
+
+(* More inversion lemmas ****************************************************)
+
+(* Note: there must be a proof suitable for llpr *)
+lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
+elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
+lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
+lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
+/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
+qed-.
+
+lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
+#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was pc3_t *)
+theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
+
+theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
+
+lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+                  ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+                  ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
+
+(* Basic_1: was: pc3_wcpr0 *)
+lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cprs.ma
new file mode 100644 (file)
index 0000000..d66896b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/cpcs_cprs.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/cprs.ma".
+include "basic_2A/equivalence/cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+(* Properties about context sensitive computation on terms ******************)
+
+(* Basic_1: was: pc3_pr3_r *)
+lemma cpcs_cprs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T2 #H @(cprs_ind … H) -T2
+/3 width=3 by cpcs_cpr_strap1, cpcs_strap1, cpc_cpcs/
+qed.
+
+(* Basic_1: was: pc3_pr3_x *)
+lemma cpcs_cprs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡* T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2
+/3 width=3 by cpcs_cpr_div, cpcs_strap1, cpcs_cprs_dx/
+qed.
+
+lemma cpcs_cprs_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /2 width=3 by cpcs_cpr_strap1/
+qed-.
+
+lemma cpcs_cprs_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_strap2/
+qed-.
+
+lemma cpcs_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_div/
+qed-.
+
+(* Basic_1: was: pc3_pr3_conf *)
+lemma cpcs_cprs_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /2 width=3 by cpcs_cpr_conf/
+qed-.
+
+(* Basic_1: was: pc3_pr3_t *)
+(* Basic_1: note: pc3_pr3_t should be renamed *)
+lemma cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2
+/2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/
+qed.
+
+lemma cprs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=5 by cpr_cprs, cprs_div/ qed-.
+
+lemma cpr_cprs_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpr_cprs, cprs_div/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes.ma
new file mode 100644 (file)
index 0000000..111eef9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes.ma
@@ -0,0 +1,37 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/dpconvstar_8.ma".
+include "basic_2A/computation/scpds.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
+
+definition scpes: ∀h. sd h → nat → nat → relation4 genv lenv term term ≝
+                  λh,g,d1,d2,G,L,T1,T2.
+                  ∃∃T. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d1] T & ⦃G, L⦄ ⊢ T2 •*➡*[h, g, d2] T.
+
+interpretation "stratified decomposed parallel equivalence (term)"
+   'DPConvStar h g d1 d2 G L T1 T2 = (scpes h g d1 d2 G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma scpds_div: ∀h,g,G,L,T1,T2,T,d1,d2.
+                 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d1] T → ⦃G, L⦄ ⊢ T2 •*➡*[h, g, d2] T →
+                 ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2.
+/2 width=3 by ex2_intro/ qed.
+
+lemma scpes_sym: ∀h,g,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2 →
+                 ⦃G, L⦄ ⊢ T2 •*⬌*[h, g, d2, d1] T1.
+#h #g #G #L #T1 #T2 #L1 #d2 * /2 width=3 by scpds_div/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_aaa.ma
new file mode 100644 (file)
index 0000000..3034f5b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_aaa.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/scpds_aaa.ma".
+include "basic_2A/equivalence/scpes.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL EQUIVALENCE FOR TERMS ***********************)
+
+(* Main inversion lemmas about atomic arity assignment on terms *************)
+
+theorem scpes_aaa_mono: ∀h,g,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2 →
+                        ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
+                        A1 = A2.
+#h #g #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
+lapply (scpds_aaa_conf … HA1 … HT1) -T1 #HA1
+lapply (scpds_aaa_conf … HA2 … HT2) -T2 #HA2
+lapply (aaa_mono … HA1 … HA2) -L -T //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_cpcs.ma
new file mode 100644 (file)
index 0000000..4e3af3c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_cpcs.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/scpds_scpds.ma".
+include "basic_2A/equivalence/cpcs_cpcs.ma".
+include "basic_2A/equivalence/scpes.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
+
+(* Inversion lemmas on parallel equivalence for terms ***********************)
+
+lemma scpes_inv_lstas_eq: ∀h,g,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2 →
+                          ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, d1] U1 →
+                          ∀U2. ⦃G, L⦄ ⊢ T2 •*[h, d2] U2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #U1 #HTU1 #U2 #HTU2
+/3 width=8 by scpds_inv_lstas_eq, cprs_div/
+qed-.
+
+(* Properties on parallel equivalence for terms *****************************)
+
+lemma cpcs_scpes: ∀h,g,G,L,T1,d11. ⦃G, L⦄ ⊢ T1 ▪[h, g] d11 →
+                  ∀U1,d12. d12 ≤ d11 → ⦃G, L⦄ ⊢ T1 •*[h, d12] U1 →
+                  ∀T2,d21. ⦃G, L⦄ ⊢ T2 ▪[h, g] d21 →
+                  ∀U2,d22. d22 ≤ d21 → ⦃G, L⦄ ⊢ T2 •*[h, d22] U2 →
+                  ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d12, d22] T2.
+#h #g #G #L #T1 #d11 #HT1 #U1 #d12 #Hd121 #HTU1 #T2 #d21 #HT2 #U2 #d22 #Hd221 #HTU2 #HU12
+elim (cpcs_inv_cprs … HU12) -HU12 /3 width=6 by scpds_div, ex4_2_intro/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_scpes.ma b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_scpes.ma
new file mode 100644 (file)
index 0000000..c48ddd0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/equivalence/scpes_scpes.ma
@@ -0,0 +1,69 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/scpds_scpds.ma".
+include "basic_2A/equivalence/scpes.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma scpes_inv_abst2: ∀h,g,a,G,L,T1,T2,W2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] ⓛ{a}W2.T2 →
+                       ∃∃W,T. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d1] ⓛ{a}W.T & ⦃G, L⦄ ⊢ W2 ➡* W & 
+                              ⦃G, L.ⓛW2⦄ ⊢ T2 •*➡*[h, g, d2] T.
+#h #g #a #G #L #T1 #T2 #W2 #d1 #d2 * #T0 #HT10 #H
+elim (scpds_inv_abst1 … H) -H #W #T #HW2 #HT2 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma scpes_refl: ∀h,g,G,L,T,d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, g] d1 →
+                  ⦃G, L⦄ ⊢ T •*⬌*[h, g, d2, d2] T.
+#h #g #G #L #T #d1 #d2 #Hd21 #Hd1
+elim (da_lstas … Hd1 … d2) #U #HTU #_
+/3 width=3 by scpds_div, lstas_scpds/
+qed.
+
+lemma lstas_scpes_trans: ∀h,g,G,L,T1,d0,d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d0 → d1 ≤ d0 →
+                         ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                         ∀T2,d,d2. ⦃G, L⦄ ⊢ T •*⬌*[h,g,d,d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,g,d1+d,d2] T2.
+#h #g #G #L #T1 #d0 #d1 #Hd0 #Hd10 #T #HT1 #T2 #d #d2 *
+/3 width=3 by scpds_div, lstas_scpds_trans/ qed-.
+
+(* Properties on parallel computation for terms *****************************)
+
+lemma cprs_scpds_div: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T →
+                      ∀d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d →
+                      ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*➡*[h, g, d2] T →
+                      ⦃G, L⦄⊢ T1 •*⬌*[h, g, 0, d2] T2.
+#h #g #G #L #T1 #T #HT1 #d #Hd elim (da_lstas … Hd 0)
+#X1 #HTX1 #_ elim (cprs_strip … HT1 X1) -HT1
+/3 width=5 by scpds_strap1, scpds_div, lstas_cpr, ex4_2_intro/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem scpes_trans: ∀h,g,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d] T →
+                     ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, g, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2.
+#h #g #G #L #T1 #T #d1 #d * #X1 #HT1X1 #HTX1 #T2 #d2 * #X2 #HTX2 #HT2X2
+elim (scpds_conf_eq … HTX1 … HTX2) -T -d /3 width=5 by scpds_cprs_trans, scpds_div/
+qed-.
+
+theorem scpes_canc_sn: ∀h,g,G,L,T,T1,d,d1. ⦃G, L⦄ ⊢ T •*⬌*[h, g, d, d1] T1 →
+                       ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, g, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2.
+/3 width=4 by scpes_trans, scpes_sym/ qed-.
+
+theorem scpes_canc_dx: ∀h,g,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d] T →
+                       ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*⬌*[h, g, d2, d] T → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2.
+/3 width=4 by scpes_trans, scpes_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_cpr_omega.ma b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_cpr_omega.ma
new file mode 100644 (file)
index 0000000..5ddc266
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_cpr_omega.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpr.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* A reduction cycle in two steps: the term Omega ***************************)
+
+definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
+
+definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
+
+definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
+
+(* Basic properties *********************************************************)
+
+lemma Delta_lift: ∀W1,W2,l,m. ⬆[l, m] W1 ≡ W2 →
+                  ⬆[l, m] (Delta W1) ≡ (Delta W2).
+/4 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
+
+(* Main properties **********************************************************)
+
+theorem cpr_Omega_12: ∀G,L,W. ⦃G, L⦄ ⊢ Omega1 W ➡ Omega2 W.
+/2 width=1 by cpr_beta/ qed.
+
+theorem cpr_Omega_21: ∀G,L,W. ⦃G, L⦄ ⊢ Omega2 W ➡ Omega1 W.
+#G #L #W1 elim (lift_total W1 0 1) #W2 #HW12
+@(cpr_zeta … (Omega1 W2)) /3 width=1 by Delta_lift, lift_flat/
+@cpr_flat @(cpr_delta … (Delta W1) ? 0)
+[3,5,8,10: /2 width=2 by Delta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_fpbg_refl.ma b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_fpbg_refl.ma
new file mode 100644 (file)
index 0000000..3d8d971
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_fpbg_refl.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/computation/fpbg_fpbs.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
+
+definition ApplDelta: term → nat → term ≝ λW,k. +ⓛW.ⓐ⋆k.ⓐ#0.#0.
+
+definition ApplOmega1: term → nat → term ≝ λW,k. ⓐ(ApplDelta W k).(ApplDelta W k).
+
+definition ApplOmega2: term → nat → term ≝ λW,k. +ⓓⓝW.(ApplDelta W k).ⓐ⋆k.ⓐ#0.#0.
+
+definition ApplOmega3: term → nat → term ≝ λW,k. ⓐ⋆k.(ApplOmega1 W k).
+
+(* Basic properties *********************************************************)
+
+lemma ApplDelta_lift: ∀W1,W2,k,l,m. ⬆[l, m] W1 ≡ W2 →
+                      ⬆[l, m] (ApplDelta W1 k) ≡ (ApplDelta W2 k).
+/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
+
+lemma cpr_ApplOmega_12: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡ ApplOmega2 W k.
+/2 width=1 by cpr_beta/ qed.
+
+lemma cpr_ApplOmega_23: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega2 W k ➡ ApplOmega3 W k.
+#G #L #W1 #k elim (lift_total W1 0 1) #W2 #HW12
+@(cpr_zeta … (ApplOmega3 W2 k)) /4 width=1 by ApplDelta_lift, lift_flat/
+@cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 k) ? 0)
+[3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
+qed.
+
+lemma cpxs_ApplOmega_13: ∀h,g,G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡*[h,g] ApplOmega3 W k.
+/4 width=3 by cpxs_strap1, cpr_cpx/ qed.
+
+lemma fqup_ApplOmega_13: ∀G,L,W,k. ⦃G, L, ApplOmega3 W k⦄ ⊐+ ⦃G, L, ApplOmega1 W k⦄.
+/2 width=1 by/ qed.
+
+(* Main properties **********************************************************)
+
+theorem fpbg_refl: ∀h,g,G,L,W,k. ⦃G, L, ApplOmega1 W k⦄ >≡[h,g] ⦃G, L, ApplOmega1 W k⦄.
+/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_snv_eta.ma b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_snv_eta.ma
new file mode 100644 (file)
index 0000000..65abf63
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_snv_eta.ma
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/dynamic/snv.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Extended validy (basic?_2) vs. restricted validity (basic_1) *************)
+
+(* extended validity of a closure, last arg of snv_appl > 1 *)
+lemma snv_extended: ∀h,g,a,G,L,k. ⦃G, L.ⓛ⋆k.ⓛⓛ{a}⋆k.⋆k.ⓛ#0⦄ ⊢ ⓐ#2.#0 ¡[h, g].
+#h #g #a #G #L #k elim (deg_total h g k)
+#d #Hd @(snv_appl … a … (⋆k) … (⋆k) (0+1+1))
+[ /4 width=5 by snv_lref, drop_drop_lt/
+| /4 width=13 by snv_bind, snv_lref/
+| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
+| @(lstas_scpds … (d+1+1))
+  /5 width=11 by lstas_bind, lstas_succ, da_bind, da_ldec, da_sort, lift_bind/
+]
+qed.
+
+(* restricted validity of the η-expanded closure, last arg of snv_appl = 1 **)
+lemma snv_restricted: ∀h,g,a,G,L,k. ⦃G, L.ⓛ⋆k.ⓛⓛ{a}⋆k.⋆k.ⓛⓛ{a}⋆k.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ¡[h, g].
+#h #g #a #G #L #k elim (deg_total h g k)
+#d #Hd @(snv_appl … a … (⋆k) … (ⓐ#0.#2) (0+1))
+[ /4 width=5 by snv_lref, drop_drop_lt/
+| @snv_lref [4: // |1,2,3: skip ]
+  @snv_bind //
+  @(snv_appl … a … (⋆k) … (⋆k) (0+1))
+  [ @snv_lref [4: // |1,2,3: skip ] //
+  | @snv_lref [4: /2 width=1 by drop_drop_lt/ |1,2,3: skip ] @snv_bind //
+  | @(lstas_scpds … (d+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
+  | @(lstas_scpds … (d+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
+    @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ] /3 width=1 by da_sort, da_bind/
+  ]
+| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
+| @(lstas_scpds … (d+1+1)) //
+  [ @da_ldec [3: // |1,2: skip ]
+    @da_bind @da_flat @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ]
+    /3 width=1 by da_sort, da_bind/
+  | @lstas_succ [4: // |1,2: skip ]
+    [2: @lstas_bind | skip ]
+    [2: @lstas_appl | skip ]
+    [2: @lstas_zero
+        [4: /2 width=1 by drop_drop_lt/ |5: /2 width=2 by lstas_bind/ |*: skip ]
+    |1: skip ]
+    /4 width=2 by lift_flat, lift_bind, lift_lref_ge_minus, lift_lref_lt/
+  ]
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_sta_ldec.ma b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_sta_ldec.ma
new file mode 100644 (file)
index 0000000..c680c74
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/examples/ex_sta_ldec.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/unfold/lstas.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Static type assignment (iterated vs primitive): the declared variable ****)
+
+(* basic_1: we have "L.ⓛⓝ⋆k1.⋆k2⦄ ⊢ #0 • ⓝ⋆k1.⋆k2". *)
+theorem sta_ldec: ∀h,G,L,k1,k2. ⦃G, L.ⓛⓝ⋆k1.⋆k2⦄ ⊢ #0 •*[h, 1] ⋆k2.
+/3 width=6 by lstas_sort, lstas_succ, lstas_cast, drop_pair/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma
new file mode 100644 (file)
index 0000000..b980bc1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/aarity.ma
@@ -0,0 +1,73 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
+ * Initial invocation: - Patience on me to gain peace and perfection! -
+ *)
+
+include "ground_2A/lib/star.ma".
+include "basic_2A/notation/constructors/item0_0.ma".
+include "basic_2A/notation/constructors/snitem2_2.ma".
+
+(* ATOMIC ARITY *************************************************************)
+
+inductive aarity: Type[0] ≝
+  | AAtom: aarity                   (* atomic aarity construction *)
+  | APair: aarity → aarity → aarity (* binary aarity construction *)
+.
+
+interpretation "atomic arity construction (atomic)"
+   'Item0 = AAtom.
+
+interpretation "atomic arity construction (binary)"
+   'SnItem2 A1 A2 = (APair A1 A2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_apair_apair_aux: ∀A1,A2,B1,B2. ②B1.A1 = ②B2.A2 → B1 = B2 ∧ A1 = A2.
+#A1 #A2 #B1 #B2 #H destruct /2 width=1 by conj/
+qed-.
+
+lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥.
+#A #B elim B -B
+[ #H destruct
+| #Y #X #IHY #_ #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥.
+#B #A elim A -A
+[ #H destruct
+| #Y #X #_ #IHX #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma eq_aarity_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
+#A1 elim A1 -A1
+[ #A2 elim A2 -A2 /2 width=1 by or_introl/
+  #B2 #A2 #_ #_ @or_intror #H destruct
+| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
+  [ -IHB1 -IHA1 @or_intror #H destruct
+  | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
+    [ #H destruct elim (IHA1 A2) -IHA1
+      [ #H destruct /2 width=1 by or_introl/
+      | #HA12 @or_intror #H destruct /2 width=1 by/
+      ]
+    | -IHA1 #HB12 @or_intror #H destruct /2 width=1 by/
+    ]
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_restricted_weight.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_restricted_weight.ma
new file mode 100644 (file)
index 0000000..49fd20d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_restricted_weight.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/functions/weight_2.ma".
+include "basic_2A/grammar/lenv_weight.ma".
+
+(* WEIGHT OF A RESTRICTED CLOSURE *******************************************)
+
+definition rfw: lenv → term → ? ≝ λL,T. ♯{L} + ♯{T}.
+
+interpretation "weight (restricted closure)" 'Weight L T = (rfw L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: flt_shift *)
+lemma rfw_shift: ∀a,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{a,I}V.T}.
+normalize //
+qed.
+
+lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?%%); //
+qed.
+
+lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?%%); //
+qed.
+
+lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+(* Basic_1: removed theorems 7:
+            flt_thead_sx flt_thead_dx flt_trans
+            flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_weight.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_weight.ma
new file mode 100644 (file)
index 0000000..72631a7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/cl_weight.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/functions/weight_3.ma".
+include "basic_2A/grammar/lenv_weight.ma".
+include "basic_2A/grammar/genv.ma".
+
+(* WEIGHT OF A CLOSURE ******************************************************)
+
+(* activate genv *)
+definition fw: genv → lenv → term → ? ≝ λG,L,T. ♯{L} + ♯{T}.
+
+interpretation "weight (closure)" 'Weight G L T = (fw G L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: flt_shift *)
+lemma fw_shift: ∀a,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{a,I}V.T}.
+normalize //
+qed.
+
+lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
+qed.
+
+lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
+qed.
+
+lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+(* Basic_1: removed theorems 7:
+            flt_thead_sx flt_thead_dx flt_trans
+            flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma
new file mode 100644 (file)
index 0000000..235b741
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/genv.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/lib/list.ma".
+include "basic_2A/notation/constructors/star_0.ma".
+include "basic_2A/notation/constructors/dxbind2_3.ma".
+include "basic_2A/notation/constructors/dxabbr_2.ma".
+include "basic_2A/notation/constructors/dxabst_2.ma".
+include "basic_2A/grammar/term.ma".
+
+(* GLOBAL ENVIRONMENTS ******************************************************)
+
+(* global environments *)
+definition genv ≝ list2 bind2 term.
+
+interpretation "sort (global environment)"
+   'Star = (nil2 bind2 term).
+
+interpretation "global environment binding construction (binary)"
+   'DxBind2 L I T = (cons2 bind2 term I T L).
+
+interpretation "abbreviation (global environment)"
+   'DxAbbr L T = (cons2 bind2 term Abbr T L).
+
+interpretation "abstraction (global environment)"
+   'DxAbst L T = (cons2 bind2 term Abst T L).
+
+(* Basic properties *********************************************************)
+
+axiom eq_genv_dec: ∀G1,G2:genv. Decidable (G1 = G2).

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma
new file mode 100644 (file)
index 0000000..88d5cf5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/item.ma
@@ -0,0 +1,88 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/lib/bool.ma".
+include "ground_2A/lib/arith.ma".
+
+(* ITEMS ********************************************************************)
+
+(* atomic items *)
+inductive item0: Type[0] ≝
+   | Sort: nat → item0 (* sort: starting at 0 *)
+   | LRef: nat → item0 (* reference by index: starting at 0 *)
+   | GRef: nat → item0 (* reference by position: starting at 0 *)
+.
+
+(* binary binding items *)
+inductive bind2: Type[0] ≝
+  | Abbr: bind2 (* abbreviation *)
+  | Abst: bind2 (* abstraction *)
+.
+
+(* binary non-binding items *)
+inductive flat2: Type[0] ≝
+  | Appl: flat2 (* application *)
+  | Cast: flat2 (* explicit type annotation *)
+.
+
+(* binary items *)
+inductive item2: Type[0] ≝
+  | Bind2: bool → bind2 → item2 (* polarized binding item *)
+  | Flat2: flat2 → item2        (* non-binding item *)
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_sort_sort_aux: ∀k1,k2. Sort k1 = Sort k2 → k1 = k2.
+#k1 #k2 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma eq_item0_dec: ∀I1,I2:item0. Decidable (I1 = I2).
+* #i1 * #i2 [2,3,4,6,7,8: @or_intror #H destruct ]
+elim (eq_nat_dec i1 i2) /2 width=1 by or_introl/
+#Hni12 @or_intror #H destruct /2 width=1 by/ 
+qed-.
+
+(* Basic_1: was: bind_dec *)
+lemma eq_bind2_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
+* * /2 width=1 by or_introl/
+@or_intror #H destruct
+qed-.
+
+(* Basic_1: was: flat_dec *)
+lemma eq_flat2_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
+* * /2 width=1 by or_introl/
+@or_intror #H destruct
+qed-.
+
+(* Basic_1: was: kind_dec *)
+lemma eq_item2_dec: ∀I1,I2:item2. Decidable (I1 = I2).
+* [ #a1 ] #I1 * [1,3: #a2 ] #I2
+[2,3: @or_intror #H destruct
+| elim (eq_bool_dec a1 a2) #Ha
+  [ elim (eq_bind2_dec I1 I2) /2 width=1 by or_introl/ #HI ]
+  @or_intror #H destruct /2 width=1 by/
+| elim (eq_flat2_dec I1 I2) /2 width=1 by or_introl/ #HI
+  @or_intror #H destruct /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: removed theorems 21:
+            s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc
+            s_arith0 s_arith1
+            r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1
+            not_abbr_abst bind_dec_not
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv.ma
new file mode 100644 (file)
index 0000000..5de6035
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv.ma
@@ -0,0 +1,68 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/constructors/star_0.ma".
+include "basic_2A/notation/constructors/dxbind2_3.ma".
+include "basic_2A/notation/constructors/dxabbr_2.ma".
+include "basic_2A/notation/constructors/dxabst_2.ma".
+include "basic_2A/grammar/term.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+(* local environments *)
+inductive lenv: Type[0] ≝
+| LAtom: lenv                       (* empty *)
+| LPair: lenv → bind2 → term → lenv (* binary binding construction *)
+.
+
+interpretation "sort (local environment)"
+   'Star = LAtom.
+
+interpretation "local environment binding construction (binary)"
+   'DxBind2 L I T = (LPair L I T).
+
+interpretation "abbreviation (local environment)"
+   'DxAbbr L T = (LPair L Abbr T).
+
+interpretation "abstraction (local environment)"
+   'DxAbst L T = (LPair L Abst T).
+
+(* Basic properties *********************************************************)
+
+lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ] * [2,4: #L2 #I2 #V2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_bind2_dec I1 I2) #HI
+  [ elim (eq_term_dec V1 V2) #HV
+    [ elim (IHL1 L2) -IHL1 /2 width=1 by or_introl/ #HL ]
+  ]
+  @or_intror #H destruct /2 width=1 by/
+| /2 width=1 by or_introl/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_lpair_lpair_aux: ∀I1,I2,L1,L2,V1,V2. L1.ⓑ{I1}V1 = L2.ⓑ{I2}V2 →
+                               ∧∧L1 = L2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma discr_lpair_x_xy: ∀I,V,L. L = L.ⓑ{I}V → ⊥.
+#I #V #L elim L -L
+[ #H destruct
+| #L #J #W #IHL #H
+  elim (destruct_lpair_lpair_aux … H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma
new file mode 100644 (file)
index 0000000..25273f9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_append.ma
@@ -0,0 +1,131 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/notation/functions/append_2.ma".
+include "basic_2A/notation/functions/snbind2_3.ma".
+include "basic_2A/notation/functions/snabbr_2.ma".
+include "basic_2A/notation/functions/snabst_2.ma".
+include "basic_2A/grammar/lenv_length.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+let rec append L K on K ≝ match K with
+[ LAtom       ⇒ L
+| LPair K I V ⇒ (append L K). ⓑ{I} V
+].
+
+interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
+
+interpretation "local environment tail binding construction (binary)"
+   'SnBind2 I T L = (append (LPair LAtom I T) L).
+
+interpretation "tail abbreviation (local environment)"
+   'SnAbbr T L = (append (LPair LAtom Abbr T) L).
+
+interpretation "tail abstraction (local environment)"
+   'SnAbst L T = (append (LPair LAtom Abst T) L).
+
+definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
+                            ∀K,T1,T2. R K T1 T2 → ∀L. R (L @@ K) T1 T2.
+
+(* Basic properties *********************************************************)
+
+lemma append_atom_sn: ∀L. ⋆ @@ L = L.
+#L elim L -L normalize //
+qed.
+
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 normalize //
+qed.
+
+lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
+#L1 #L2 elim L2 -L2 normalize //
+qed.
+
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = |L| + 1.
+#I #L #V >append_length //
+qed.
+
+(* Basic_1: was just: chead_ctail *)
+lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
+#L elim L -L /2 width=5 by ex2_3_intro/
+#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
+#J #K #W #H #_ >H -H >ltail_length
+@(ex2_3_intro … J (K.ⓑ{I}V) W) //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
+                     L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * normalize /2 width=1 by conj/
+  #K2 #I2 #V2 #L1 #L2 #_ <plus_n_Sm #H destruct
+| #K1 #I1 #V1 #IH * normalize
+  [ #L1 #L2 #_ <plus_n_Sm #H destruct
+  | #K2 #I2 #V2 #L1 #L2 #H1 #H2
+    elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+    elim (IH … H1) -IH -H1 /2 width=1 by conj/
+  ]
+]
+qed-.
+
+(* Note: lemma 750 *)
+lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
+                     L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * normalize /2 width=1 by conj/
+  #K2 #I2 #V2 #L1 #L2 #H1 #H2 destruct
+  normalize in H2; >append_length in H2; #H
+  elim (plus_xySz_x_false … H)
+| #K1 #I1 #V1 #IH * normalize
+  [ #L1 #L2 #H1 #H2 destruct
+    normalize in H2; >append_length in H2; #H
+    elim (plus_xySz_x_false … (sym_eq … H))
+  | #K2 #I2 #V2 #L1 #L2 #H1 #H2
+    elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+    elim (IH … H1) -IH -H1 /2 width=1 by conj/
+  ]
+]
+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
+#L #K #H elim (append_inj_dx … (⋆) … H) //
+qed-.
+
+lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+qed-.
+
+lemma length_inv_pos_dx_ltail: ∀L,l. |L| = l + 1 →
+                               ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_pos_dx … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+lemma length_inv_pos_sn_ltail: ∀L,l. l + 1 = |L| →
+                               ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_pos_sn … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+                    R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+                    ∀L. R L.
+#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
+#L #I #V normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma
new file mode 100644 (file)
index 0000000..cefc9b0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_length.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/lenv.ma".
+
+(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
+
+let rec length L ≝ match L with
+[ LAtom       ⇒ 0
+| LPair L _ _ ⇒ length L + 1
+].
+
+interpretation "length (local environment)" 'card L = (length L).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
+* // #L #I #V normalize <plus_n_Sm #H destruct
+qed-.
+
+lemma length_inv_zero_sn: ∀L. 0 = |L| → L = ⋆.
+* // #L #I #V normalize <plus_n_Sm #H destruct
+qed-.
+
+lemma length_inv_pos_dx: ∀l,L. |L| = l + 1 →
+                         ∃∃I,K,V. |K| = l & L = K. ⓑ{I}V.
+#l *
+[ normalize <plus_n_Sm #H destruct
+| #K #I #V #H
+  lapply (injective_plus_l … H) -H /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma length_inv_pos_sn: ∀l,L. l + 1 = |L| →
+                         ∃∃I,K,V. l = |K| & L = K. ⓑ{I}V.
+#l *
+[ normalize <plus_n_Sm #H destruct
+| #K #I #V #H
+  lapply (injective_plus_l … H) -H /2 width=5 by ex2_3_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_weight.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_weight.ma
new file mode 100644 (file)
index 0000000..598a9e3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lenv_weight.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/term_weight.ma".
+include "basic_2A/grammar/lenv.ma".
+
+(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************)
+
+let rec lw L ≝ match L with
+[ LAtom       ⇒ 0
+| LPair L _ V ⇒ lw L + ♯{V}
+].
+
+interpretation "weight (local environment)" 'Weight L = (lw L).
+
+(* Basic properties *********************************************************)
+
+lemma lw_pair: ∀I,L,V. ♯{L} < ♯{L.ⓑ{I}V}.
+/3 width=1 by lt_plus_to_minus_r, monotonic_lt_plus_r/ qed.
+
+(* Basic_1: removed theorems 4: clt_cong clt_head clt_thead clt_wf_ind *)
+(* Basic_1: removed local theorems 1: clt_wf__q_ind *)
+(* Basic_1: note: clt_thead should be renamed clt_ctail *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma
new file mode 100644 (file)
index 0000000..820eaa4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq.ma
@@ -0,0 +1,192 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_lt.ma".
+include "basic_2A/notation/relations/midiso_4.ma".
+include "basic_2A/grammar/lenv_length.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
+
+inductive lreq: relation4 ynat ynat lenv lenv ≝
+| lreq_atom: ∀l,m. lreq l m (⋆) (⋆)
+| lreq_zero: ∀I1,I2,L1,L2,V1,V2.
+             lreq 0 0 L1 L2 → lreq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lreq_pair: ∀I,L1,L2,V,m. lreq 0 m L1 L2 →
+             lreq 0 (⫯m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lreq_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
+             lreq l m L1 L2 → lreq (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+.
+
+interpretation
+  "equivalence (local environment)"
+  'MidIso l m L1 L2 = (lreq l m L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lreq_pair_lt: ∀I,L1,L2,V,m. L1 ⩬[0, ⫰m] L2 → 0 < m →
+                    L1.ⓑ{I}V ⩬[0, m] L2.ⓑ{I}V.
+#I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lreq_pair/
+qed.
+
+lemma lreq_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[⫰l, m] L2 → 0 < l →
+                    L1.ⓑ{I1}V1 ⩬[l, m] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lreq_succ/
+qed.
+
+lemma lreq_pair_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 →
+                     ∀I,V. L1.ⓑ{I}V ⩬[0, ∞] L2.ⓑ{I}V.
+#L1 #L2 #HL12 #I #V lapply (lreq_pair I … V … HL12) -HL12 //
+qed.
+
+lemma lreq_refl: ∀L,l,m. L ⩬[l, m] L.
+#L elim L -L //
+#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
+#Hl destruct /2 width=1 by lreq_succ/
+#m elim (ynat_cases … m) [| * #x ]
+#Hm destruct /2 width=1 by lreq_zero, lreq_pair/
+qed.
+
+lemma lreq_O2: ∀L1,L2,l. |L1| = |L2| → L1 ⩬[l, yinj 0] L2.
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
+* // [1,3: #L2 #I2 #V2 ] #l normalize
+[1,3: <plus_n_Sm #H destruct ]
+#H lapply (injective_plus_l … H) -H #HL12
+elim (ynat_cases l) /3 width=1 by lreq_zero/
+* /3 width=1 by lreq_succ/
+qed.
+
+lemma lreq_sym: ∀l,m. symmetric … (lreq l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+/2 width=1 by lreq_zero, lreq_pair, lreq_succ/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lreq_inv_atom1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
+]
+qed-.
+
+lemma lreq_inv_atom1: ∀L2,l,m. ⋆ ⩬[l, m] L2 → L2 = ⋆.
+/2 width=5 by lreq_inv_atom1_aux/ qed-.
+
+fact lreq_inv_zero1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+                         ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
+                         ∃∃J2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J1 #K1 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+  /2 width=5 by ex2_3_intro/
+| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lreq_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⩬[0, 0] L2 →
+                      ∃∃I2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by lreq_inv_zero1_aux/ qed-.
+
+fact lreq_inv_pair1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+                         ∀J,K1,W. L1 = K1.ⓑ{J}W → l = 0 → 0 < m →
+                         ∃∃K2. K1 ⩬[0, ⫰m] K2 & L2 = K2.ⓑ{J}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J #K1 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
+  elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #m #HL12 #J #K1 #W #H #_ #_ destruct
+  /2 width=3 by ex2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J #K1 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lreq_inv_pair1: ∀I,K1,L2,V,m. K1.ⓑ{I}V ⩬[0, m] L2 → 0 < m →
+                      ∃∃K2. K1 ⩬[0, ⫰m] K2 & L2 = K2.ⓑ{I}V.
+/2 width=6 by lreq_inv_pair1_aux/ qed-.
+
+lemma lreq_inv_pair: ∀I1,I2,L1,L2,V1,V2,m. L1.ⓑ{I1}V1 ⩬[0, m] L2.ⓑ{I2}V2 → 0 < m →
+                    ∧∧ L1 ⩬[0, ⫰m] L2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 … H) -H //
+#Y #HL12 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact lreq_inv_succ1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+                         ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
+                         ∃∃J2,K2,W2. K1 ⩬[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J1 #K1 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+  elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
+  /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lreq_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⩬[l, m] L2 → 0 < l →
+                      ∃∃I2,K2,V2. K1 ⩬[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by lreq_inv_succ1_aux/ qed-.
+
+lemma lreq_inv_atom2: ∀L1,l,m. L1 ⩬[l, m] ⋆ → L1 = ⋆.
+/3 width=3 by lreq_inv_atom1, lreq_sym/
+qed-.
+
+lemma lreq_inv_succ: ∀I1,I2,L1,L2,V1,V2,l,m. L1.ⓑ{I1}V1 ⩬[l, m] L2.ⓑ{I2}V2 → 0 < l →
+                     L1 ⩬[⫰l, m] L2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … H) -H //
+#Z #Y #X #HL12 #H destruct //
+qed-.
+
+lemma lreq_inv_zero2: ∀I2,K2,L1,V2. L1 ⩬[0, 0] K2.ⓑ{I2}V2 →
+                      ∃∃I1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+#I2 #K2 #L1 #V2 #H elim (lreq_inv_zero1 … (lreq_sym … H)) -H 
+/3 width=5 by lreq_sym, ex2_3_intro/
+qed-.
+
+lemma lreq_inv_pair2: ∀I,K2,L1,V,m. L1 ⩬[0, m] K2.ⓑ{I}V → 0 < m →
+                      ∃∃K1. K1 ⩬[0, ⫰m] K2 & L1 = K1.ⓑ{I}V.
+#I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 … (lreq_sym … H)) -H
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
+
+lemma lreq_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.ⓑ{I2}V2 → 0 < l →
+                      ∃∃I1,K1,V1. K1 ⩬[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
+#I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … (lreq_sym … H)) -H 
+/3 width=5 by lreq_sym, ex2_3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lreq_fwd_length: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → |L1| = |L2|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact lreq_inv_O_Y_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → l = 0 → m = ∞ → L1 = L2.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
+| /4 width=1 by eq_f3, ysucc_inv_Y_dx/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #_ #H elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lreq_inv_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 → L1 = L2.
+/2 width=5 by lreq_inv_O_Y_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma
new file mode 100644 (file)
index 0000000..700b87d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/lreq_lreq.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+include "basic_2A/grammar/lreq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
+
+(* Main properties **********************************************************)
+
+theorem lreq_trans: ∀l,m. Transitive … (lreq l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #l #m #X #H lapply (lreq_inv_atom1 … H) -H
+  #H destruct //
+| #I1 #I #L1 #L #V1 #V #_ #IHL1 #X #H elim (lreq_inv_zero1 … H) -H
+  #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_zero/
+| #I #L1 #L #V #m #_ #IHL1 #X #H elim (lreq_inv_pair1 … H) -H //
+  #L2 #HL2 #H destruct /3 width=1 by lreq_pair/
+| #I1 #I #L1 #L #V1 #V #l #m #_ #IHL1 #X #H elim (lreq_inv_succ1 … H) -H //
+  #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_succ/
+]
+qed-.
+
+theorem lreq_canc_sn: ∀l,m,L,L1,L2. L ⩬[l, m] L1 → L ⩬[l, m] L2 → L1 ⩬[l, m] L2.
+/3 width=3 by lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_canc_dx: ∀l,m,L,L1,L2. L1 ⩬[l, m] L → L2 ⩬[l, m] L → L1 ⩬[l, m] L2.
+/3 width=3 by lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_join: ∀L1,L2,l,i. L1 ⩬[l, i] L2 →
+                   ∀m. L1 ⩬[i+l, m] L2 → L1 ⩬[l, i+m] L2.
+#L1 #L2 #l #i #H elim H -L1 -L2 -l -i //
+[ #I #L1 #L2 #V #m #_ #IHL12 #m #H
+  lapply (lreq_inv_succ … H ?) -H // >ypred_succ /3 width=1 by lreq_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #m #H
+  lapply (lreq_inv_succ … H ?) -H // >yplus_succ2 >ypred_succ /3 width=1 by lreq_succ/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term.ma
new file mode 100644 (file)
index 0000000..7dfa163
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term.ma
@@ -0,0 +1,153 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/constructors/item0_1.ma".
+include "basic_2A/notation/constructors/snitem2_3.ma".
+include "basic_2A/notation/constructors/snbind2_4.ma".
+include "basic_2A/notation/constructors/snbind2pos_3.ma".
+include "basic_2A/notation/constructors/snbind2neg_3.ma".
+include "basic_2A/notation/constructors/snflat2_3.ma".
+include "basic_2A/notation/constructors/star_1.ma".
+include "basic_2A/notation/constructors/lref_1.ma".
+include "basic_2A/notation/constructors/gref_1.ma".
+include "basic_2A/notation/constructors/snabbr_3.ma".
+include "basic_2A/notation/constructors/snabbrpos_2.ma".
+include "basic_2A/notation/constructors/snabbrneg_2.ma".
+include "basic_2A/notation/constructors/snabst_3.ma".
+include "basic_2A/notation/constructors/snabstpos_2.ma".
+include "basic_2A/notation/constructors/snabstneg_2.ma".
+include "basic_2A/notation/constructors/snappl_2.ma".
+include "basic_2A/notation/constructors/sncast_2.ma".
+include "basic_2A/grammar/item.ma".
+
+(* TERMS ********************************************************************)
+
+(* terms *)
+inductive term: Type[0] ≝
+  | TAtom: item0 → term               (* atomic item construction *)
+  | TPair: item2 → term → term → term (* binary item construction *)
+.
+
+interpretation "term construction (atomic)"
+   'Item0 I = (TAtom I).
+
+interpretation "term construction (binary)"
+   'SnItem2 I T1 T2 = (TPair I T1 T2).
+
+interpretation "term binding construction (binary)"
+   'SnBind2 a I T1 T2 = (TPair (Bind2 a I) T1 T2).
+
+interpretation "term positive binding construction (binary)"
+   'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
+
+interpretation "term negative binding construction (binary)"
+   'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
+
+interpretation "term flat construction (binary)"
+   'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
+
+interpretation "sort (term)"
+   'Star k = (TAtom (Sort k)).
+
+interpretation "local reference (term)"
+   'LRef i = (TAtom (LRef i)).
+
+interpretation "global reference (term)"
+   'GRef p = (TAtom (GRef p)).
+
+interpretation "abbreviation (term)"
+   'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2).
+
+interpretation "positive abbreviation (term)"
+   'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
+
+interpretation "negative abbreviation (term)"
+   'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
+
+interpretation "abstraction (term)"
+   'SnAbst a T1 T2 = (TPair (Bind2 a Abst) T1 T2).
+
+interpretation "positive abstraction (term)"
+   'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
+
+interpretation "negative abstraction (term)"
+   'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
+
+interpretation "application (term)"
+   'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+
+interpretation "native type annotation (term)"
+   'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: term_dec *)
+lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
+#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_item2_dec I1 I2) #HI
+  [ elim (IHV1 V2) -IHV1 #HV
+    [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
+  ]
+  @or_intror #H destruct /2 width=1 by/
+| elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
+  @or_intror #H destruct /2 width=1 by/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
+#I1 #I2 #H destruct //
+qed-.
+
+fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
+                               ∧∧T1 = T2 & I1 = I2 & V1 = V2.
+#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥.
+#I #T #V elim V -V
+[ #J #H destruct
+| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+(* Basic_1: was: thead_x_y_y *)
+lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → ⊥.
+#I #V #T elim T -T
+[ #J #H destruct
+| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
+                             (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+                             (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1 by/
+qed-.
+
+lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
+                             (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+                             (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: removed theorems 3:
+            not_void_abst not_abbr_void not_abst_void
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_simple.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_simple.ma
new file mode 100644 (file)
index 0000000..434aea6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_simple.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/simple_1.ma".
+include "basic_2A/grammar/term.ma".
+
+(* SIMPLE (NEUTRAL) TERMS ***************************************************)
+
+inductive simple: predicate term ≝
+   | simple_atom: ∀I. simple (⓪{I})
+   | simple_flat: ∀I,V,T. simple (ⓕ{I} V. T)
+.
+
+interpretation "simple (term)" 'Simple T = (simple T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact simple_inv_bind_aux: ∀T. 𝐒⦃T⦄ → ∀a,J,W,U. T = ⓑ{a,J} W. U → ⊥.
+#T * -T
+[ #I #a #J #W #U #H destruct
+| #I #V #T #a #J #W #U #H destruct
+]
+qed.
+
+lemma simple_inv_bind: ∀a,I,V,T. 𝐒⦃ⓑ{a,I} V. T⦄ → ⊥.
+/2 width=7 by simple_inv_bind_aux/ qed-.
+
+lemma simple_inv_pair: ∀I,V,T.  𝐒⦃②{I}V.T⦄ → ∃J. I = Flat2 J.
+* /2 width=2 by ex_intro/ #a #I #V #T #H
+elim (simple_inv_bind … H)
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma
new file mode 100644 (file)
index 0000000..f6b1633
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_vector.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/lib/list.ma".
+include "basic_2A/notation/functions/snapplvector_2.ma".
+include "basic_2A/grammar/term_simple.ma".
+
+(* TERMS ********************************************************************)
+
+let rec applv Vs T on Vs ≝
+  match Vs with
+  [ nil        ⇒ T
+  | cons hd tl ⇒ ⓐhd. (applv tl T)
+  ].
+
+interpretation "application to vector (term)"
+   'SnApplVector Vs T = (applv Vs T).
+
+(* properties concerning simple terms ***************************************)
+
+lemma applv_simple: ∀T,Vs.  𝐒⦃T⦄ → 𝐒⦃ⒶVs.T⦄.
+#T * //
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_weight.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_weight.ma
new file mode 100644 (file)
index 0000000..0bedc4e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/term_weight.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/functions/weight_1.ma".
+include "basic_2A/grammar/term.ma".
+
+(* WEIGHT OF A TERM *********************************************************)
+
+let rec tw T ≝ match T with
+[ TAtom _     ⇒ 1
+| TPair _ V T ⇒ tw V + tw T + 1
+].
+
+interpretation "weight (term)" 'Weight T = (tw T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: tweight_lt *)
+lemma tw_pos: ∀T. 1 ≤ ♯{T}.
+#T elim T -T //
+qed.
+
+(* Basic_1: removed theorems 11:
+            wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O
+            weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma
new file mode 100644 (file)
index 0000000..80311fe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts.ma
@@ -0,0 +1,108 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/topiso_2.ma".
+include "basic_2A/grammar/term_simple.ma".
+
+(* SAME TOP TERM STRUCTURE **************************************************)
+
+inductive tsts: relation term ≝
+   | tsts_atom: ∀I. tsts (⓪{I}) (⓪{I})
+   | tsts_pair: ∀I,V1,V2,T1,T2. tsts (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation "same top structure (term)" 'TopIso T1 T2 = (tsts T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tsts_inv_atom1_aux: ∀T1,T2. T1 ≂ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I}.
+#T1 #T2 * -T1 -T2 //
+#J #V1 #V2 #T1 #T2 #I #H destruct
+qed-.
+
+(* Basic_1: was: iso_gen_sort iso_gen_lref *)
+lemma tsts_inv_atom1: ∀I,T2. ⓪{I} ≂ T2 → T2 = ⓪{I}.
+/2 width=3 by tsts_inv_atom1_aux/ qed-.
+
+fact tsts_inv_pair1_aux: ∀T1,T2. T1 ≂ T2 → ∀I,W1,U1. T1 = ②{I}W1.U1 →
+                         ∃∃W2,U2. T2 = ②{I}W2. U2.
+#T1 #T2 * -T1 -T2
+[ #J #I #W1 #U1 #H destruct
+| #J #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_1: was: iso_gen_head *)
+lemma tsts_inv_pair1: ∀I,W1,U1,T2. ②{I}W1.U1 ≂ T2 →
+                      ∃∃W2,U2. T2 = ②{I}W2. U2.
+/2 width=5 by tsts_inv_pair1_aux/ qed-.
+
+fact tsts_inv_atom2_aux: ∀T1,T2. T1 ≂ T2 → ∀I. T2 = ⓪{I} → T1 = ⓪{I}.
+#T1 #T2 * -T1 -T2 //
+#J #V1 #V2 #T1 #T2 #I #H destruct
+qed-.
+
+lemma tsts_inv_atom2: ∀I,T1. T1 ≂ ⓪{I} → T1 = ⓪{I}.
+/2 width=3 by tsts_inv_atom2_aux/ qed-.
+
+fact tsts_inv_pair2_aux: ∀T1,T2. T1 ≂ T2 → ∀I,W2,U2. T2 = ②{I}W2.U2 →
+                         ∃∃W1,U1. T1 = ②{I}W1.U1.
+#T1 #T2 * -T1 -T2
+[ #J #I #W2 #U2 #H destruct
+| #J #V1 #V2 #T1 #T2 #I #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+lemma tsts_inv_pair2: ∀I,T1,W2,U2. T1 ≂ ②{I}W2.U2 →
+                      ∃∃W1,U1. T1 = ②{I}W1.U1.
+/2 width=5 by tsts_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+lemma tsts_refl: reflexive … tsts.
+#T elim T -T //
+qed.
+
+lemma tsts_sym: symmetric … tsts.
+#T1 #T2 #H elim H -T1 -T2 //
+qed-.
+
+lemma tsts_dec: ∀T1,T2. Decidable (T1 ≂ T2).
+* #I1 [2: #V1 #T1 ] * #I2 [2,4: #V2 #T2 ]
+[ elim (eq_item2_dec I1 I2) #HI12
+  [ destruct /2 width=1 by tsts_pair, or_introl/
+  | @or_intror #H
+    elim (tsts_inv_pair1 … H) -H #V #T #H destruct /2 width=1 by/
+  ]
+| @or_intror #H
+  lapply (tsts_inv_atom1 … H) -H #H destruct
+| @or_intror #H
+  lapply (tsts_inv_atom2 … H) -H #H destruct
+| elim (eq_item0_dec I1 I2) #HI12
+  [ destruct /2 width=1 by or_introl/
+  | @or_intror #H
+    lapply (tsts_inv_atom2 … H) -H #H destruct /2 width=1 by/
+  ]
+]
+qed.
+
+lemma simple_tsts_repl_dx: ∀T1,T2. T1 ≂ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 * -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #H
+elim (simple_inv_pair … H) -H #J #H destruct //
+qed-.
+
+lemma simple_tsts_repl_sn: ∀T1,T2. T1 ≂ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+/3 width=3 by simple_tsts_repl_dx, tsts_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_tsts.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_tsts.ma
new file mode 100644 (file)
index 0000000..50f8291
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_tsts.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/tsts.ma".
+
+(* SAME TOP TERM STRUCTURE **************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: iso_trans *)
+theorem tsts_trans: Transitive … tsts.
+#T1 #T * -T1 -T //
+#I #V1 #V #T1 #T #X #H
+elim (tsts_inv_pair1 … H) -H #V2 #T2 #H destruct //
+qed-.
+
+theorem tsts_canc_sn: ∀T,T1. T ≂ T1 → ∀T2. T ≂ T2 → T1 ≂ T2.
+/3 width=3 by tsts_trans, tsts_sym/ qed-.
+
+theorem tsts_canc_dx: ∀T1,T. T1 ≂ T → ∀T2. T2 ≂ T → T1 ≂ T2.
+/3 width=3 by tsts_trans, tsts_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_vector.ma
new file mode 100644 (file)
index 0000000..c53a56d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/grammar/tsts_vector.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/term_vector.ma".
+include "basic_2A/grammar/tsts.ma".
+
+(* SAME TOP TERM STRUCTURE **************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
+lemma tsts_inv_bind_applv_simple: ∀a,I,Vs,V2,T1,T2. ⒶVs.T1 ≂ ⓑ{a,I} V2. T2 →
+                                  𝐒⦃T1⦄ → ⊥.
+#a #I #Vs #V2 #T1 #T2 #H
+elim (tsts_inv_pair2 … H) -H #V0 #T0
+elim Vs -Vs normalize
+[ #H destruct #H /2 width=5 by simple_inv_bind/
+| #V #Vs #_ #H destruct
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma
new file mode 100644 (file)
index 0000000..6f3f953
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys.ma
@@ -0,0 +1,166 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/psubststar_6.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+definition cpys: ynat → ynat → relation4 genv lenv term term ≝
+                 λl,m,G. LTC … (cpy l m G).
+
+interpretation "context-sensitive extended multiple substritution (term)"
+   'PSubstStar G L T1 l m T2 = (cpys l m G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpys_ind: ∀G,L,T1,l,m. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T2.
+#G #L #T1 #l #m #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cpys_ind_dx: ∀G,L,T2,l,m. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R T1.
+#G #L #T2 #l #m #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cpy_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+/2 width=1 by inj/ qed.
+
+lemma cpys_strap1: ∀G,L,T1,T,T2,l,m.
+                   ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by step/ qed-.
+
+lemma cpys_strap2: ∀G,L,T1,T,T2,l,m.
+                   ⦃G, L⦄ ⊢ T1 ▶[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+lemma lsuby_cpys_trans: ∀G,l,m. lsub_trans … (cpys l m G) (lsuby l m).
+/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
+qed-.
+
+lemma cpys_refl: ∀G,L,l,m. reflexive … (cpys l m G L).
+/2 width=1 by cpy_cpys/ qed.
+
+lemma cpys_bind: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] ⓑ{a,I}V2.T2.
+#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
+[ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
+| /3 width=5 by cpys_strap1, cpy_bind/
+]
+qed.
+
+lemma cpys_flat: ∀G,L,V1,V2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+                 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] ⓕ{I}V2.T2.
+#G #L #V1 #V2 #l #m #HV12 @(cpys_ind … HV12) -V2
+[ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
+| /3 width=5 by cpys_strap1, cpy_flat/
+qed.
+
+lemma cpys_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T2 →
+                 ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+                 ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(cpys_ind … H) -T2
+/3 width=7 by cpys_strap1, cpy_weak/
+qed-.
+
+lemma cpys_weak_top: ∀G,L,T1,T2,l,m.
+                     ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
+/3 width=4 by cpys_strap1, cpy_weak_top/
+qed-.
+
+lemma cpys_weak_full: ∀G,L,T1,T2,l,m.
+                      ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2
+/3 width=5 by cpys_strap1, cpy_weak_full/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                   ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+                   l ≤ lt → l + m ≤ lt + mt →
+                   ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
+  elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
+/2 width=3 by transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶*[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_sort1 … HT2) -HT2 //
+qed-.
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶*[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_gref1 … HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[l, m] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 &
+                               ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 &
+                               U2 = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+  elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
+  /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[l, m] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 & ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 &
+                               U2 = ⓕ{I}V2.T2.
+#I #G #L #V1 #T1 #U2 #l #m #H @(cpys_ind … H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+  elim (cpy_inv_flat1 … HU2) -HU2
+  /3 width=5 by cpys_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 0] T2 → T1 = T2.
+#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀l,m:nat.
+                         ⦃G, L⦄ ⊢ U1 ▶*[l, m] U2 → ∀T1. ⬆[l, m] T1 ≡ U1 → U1 = U2.
+#G #L #U1 #U2 #l #m #H #T1 #HTU1 @(cpys_ind … H) -U2
+/2 width=7 by cpy_inv_lift1_eq/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma
new file mode 100644 (file)
index 0000000..7d1274a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_alt.ma
@@ -0,0 +1,102 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/psubststaralt_6.ma".
+include "basic_2A/multiple/cpys_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* alternative definition of cpys *)
+inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
+| cpysa_atom : ∀I,G,L,l,m. cpysa l m G L (⓪{I}) (⓪{I})
+| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l+m →
+               ⬇[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(l+m-i)) G K V1 V2 →
+               ⬆[0, i+1] V2 ≡ W2 → cpysa l m G L (#i) W2
+| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+               cpysa l m G L V1 V2 → cpysa (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
+               cpysa l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpysa_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+               cpysa l m G L V1 V2 → cpysa l m G L T1 T2 →
+               cpysa l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+   "context-sensitive extended multiple substritution (term) alternative"
+   'PSubstStarAlt G L T1 l m T2 = (cpysa l m G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpysa_trans: ∀G,l,m. lsub_trans … (cpysa l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
+[ //
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
+| /4 width=1 by lsuby_succ, cpysa_bind/
+| /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpysa_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶▶*[l, m] T.
+#G #T elim T -T //
+#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
+qed.
+
+lemma cpysa_cpy_trans: ∀G,L,T1,T,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T →
+                       ∀T2. ⦃G, L⦄ ⊢ T ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
+#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m
+[ #I #G #L #l #m #X #H
+  elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
+| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H
+  lapply (drop_fwd_drop2 … HLK) #H0LK
+  lapply (cpy_weak … H 0 (l+m) ? ?) -H // #H
+  elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
+  /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
+| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
+  elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+  /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #IHV1 #IHT1 #X #H
+  elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpys_cpysa: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2.
+/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶▶*[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+                    (∀I,G,L,l,m. R l m G L (⓪{I}) (⓪{I})) →
+                    (∀I,G,L,K,V1,V2,W2,i,l,m. l ≤ yinj i → i < l + m →
+                     ⬇[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 →
+                     ⬆[O, i+1] V2 ≡ W2 → R O (⫰(l+m-i)) G K V1 V2 → R l m G L (#i) W2
+                    ) →
+                    (∀a,I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+                     ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯l, m] T2 → R l m G L V1 V2 →
+                     R (⫯l) m G (L.ⓑ{I}V1) T1 T2 → R l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+                    ) →
+                    (∀I,G,L,V1,V2,T1,T2,l,m. ⦃G, L⦄ ⊢ V1 ▶*[l, m] V2 →
+                     ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L V1 V2 →
+                     R l m G L T1 T2 → R l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+                    ) →
+                    ∀l,m,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 → R l m G L T1 T2.
+#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -l -m
+/3 width=8 by cpysa_inv_cpys/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma
new file mode 100644 (file)
index 0000000..a01999d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_cpys.ma
@@ -0,0 +1,117 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/cpy_cpy.ma".
+include "basic_2A/multiple/cpys_alt.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2.
+#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+                     ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+                      ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+                      (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+                      ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+                        ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+                        ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+                        ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+                        ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+                     ∀i. l ≤ i → i ≤ l + m →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2.
+#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+  elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02
+  elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+  >ymax_pre_sn_comm //
+]
+qed-.
+
+lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                         ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                         l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
+                               ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+                      ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 →
+                      ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
+
+theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+                       ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 →
+                       (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+                       ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
+
+theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m.
+                       ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 →
+                       ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by trans_TC/ qed-.
+
+theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+                         ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+                         ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
+
+theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 →
+                         ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2.
+#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2
+  elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ]
+  [ lapply (cpys_weak_full … HW2) -HW2 #HW2
+    lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
+    [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
+    #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
+  | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2
+    elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
+    /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm
+    #X #_ #H elim (lift_inv_lref2_be … H) -H //
+  ]
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
+  #V #T #HV2 #HT2 #H destruct
+  lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
+| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
+  #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma
new file mode 100644 (file)
index 0000000..d3f292a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/cpys_lift.ma
@@ -0,0 +1,226 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/cpy_lift.ma".
+include "basic_2A/multiple/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_subst: ∀I,G,L,K,V,U1,i,l,m.
+                  l ≤ yinj i → i < l + m →
+                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(l+m-i)] U1 →
+                  ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, m] U2.
+#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #HLK #H @(cpys_ind … H) -U1
+[ /3 width=5 by cpy_cpys, cpy_subst/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+  elim (lift_total U 0 (i+1)) #U0 #HU0
+  lapply (IHU … HU0) -IHU #H
+  lapply (drop_fwd_drop2 … HLK) -HLK #HLK
+  lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+  lapply (cpy_weak … HU02 l m ? ?) -HU02
+  [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+  >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+]
+qed.
+
+lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,l.
+                     l ≤ yinj i →
+                     ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
+                     ∀U2. ⬆[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[l, ∞] U2.
+#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12
+@(cpys_subst … HLK … HU12) >yminus_Y_inj //
+qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+lemma cpys_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶*[l, m] T2 →
+                      T2 = ⓪{I} ∨
+                      ∃∃J,K,V1,V2,i. l ≤ yinj i & i < l + m &
+                                    ⬇[i] L ≡ K.ⓑ{J}V1 &
+                                     ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 &
+                                     ⬆[O, i+1] V2 ≡ T2 &
+                                     I = LRef i.
+#I #G #L #T2 #l #m #H @(cpys_ind … H) -T2
+[ /2 width=1 by or_introl/
+| #T #T2 #_ #HT2 *
+  [ #H destruct
+    elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
+  | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI
+    lapply (drop_fwd_drop2 … HLK) #H
+    elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+    [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+    /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+  ]
+]
+qed-.
+
+lemma cpys_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
+                      T2 = #i ∨
+                      ∃∃I,K,V1,V2. l ≤ i & i < l + m &
+                                   ⬇[i] L ≡ K.ⓑ{I}V1 &
+                                   ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2 &
+                                   ⬆[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
+qed-.
+
+lemma cpys_inv_lref1_Y2: ∀G,L,T2,i,l. ⦃G, L⦄ ⊢ #i ▶*[l, ∞] T2 →
+                         T2 = #i ∨
+                         ∃∃I,K,V1,V2. l ≤ i & ⬇[i] L ≡ K.ⓑ{I}V1 &
+                                      ⦃G, K⦄ ⊢ V1 ▶*[0, ∞] V2 & ⬆[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #l #H elim (cpys_inv_lref1 … H) -H /2 width=1 by or_introl/
+* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
+qed-.
+
+lemma cpys_inv_lref1_drop: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶*[l, m] T2 →
+                            ∀I,K,V1. ⬇[i] L ≡ K.ⓑ{I}V1 →
+                            ∀V2. ⬆[O, i+1] V2 ≡ T2 →
+                            ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(l+m-i)] V2
+                             & l ≤ i
+                             & i < l + m.
+#G #L #T2 #i #l #m #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
+[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
+| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2
+  lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
+  lapply (drop_mono … HLY … HLK) -L #H destruct
+  /2 width=1 by and3_intro/
+]
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma cpys_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+                    ∀L,U1,s,l,m. lt + mt ≤ yinj l → ⬇[s, l, m] L ≡ K →
+                    ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
+                    ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+  elim (lift_total T l m) #U #HTU
+  lapply (IHT … HTU) -IHT #HU1
+  lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+                    ∀L,U1,s,l,m. lt ≤ yinj l → l ≤ lt + mt →
+                    ⬇[s, l, m] L ≡ K → ⬆[l, m] T1 ≡ U1 →
+                    ∀U2. ⬆[l, m] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+  elim (lift_total T l m) #U #HTU
+  lapply (IHT … HTU) -IHT #HU1
+  lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 →
+                    ∀L,U1,s,l,m. yinj l ≤ lt → ⬇[s, l, m] L ≡ K →
+                    ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 →
+                    ⦃G, L⦄ ⊢ U1 ▶*[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #HLK #HTU1 @(cpys_ind … H) -T2
+[ #U2 #H >(lift_mono … HTU1 … H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+  elim (lift_total T l m) #U #HTU
+  lapply (IHT … HTU) -IHT #HU1
+  lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+(* Inversion lemmas for relocation ******************************************)
+
+lemma cpys_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                         ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                         lt + mt ≤ l →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                         ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                         lt ≤ l → yinj l + m ≤ lt + mt →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, mt - m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                         ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                         yinj l + m ≤ lt →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt - m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                            ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                            l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (yinj l + m)] T2 &
+                                 ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                            ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                            lt ≤ l → lt + mt ≤ yinj l + m →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+                            ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                            lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(cpys_ind … H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+  elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,l,m. ⦃G, L⦄ ⊢ W1 ▶*[l, m] W2 →
+                            ∀K,V1,i. ⬇[i+1] L ≡ K → ⬆[O, i+1] V1 ≡ W1 → 
+                            l ≤ yinj i → i < l + m →
+                            ∃∃V2.  ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(l+m-i)] V2 & ⬆[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm
+elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma
new file mode 100644 (file)
index 0000000..a86afce
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops.ma
@@ -0,0 +1,122 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rdropstar_3.ma".
+include "basic_2A/notation/relations/rdropstar_4.ma".
+include "basic_2A/substitution/drop.ma".
+include "basic_2A/multiple/mr2_minus.ma".
+include "basic_2A/multiple/lifts_vector.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+inductive drops (s:bool): list2 nat nat → relation lenv ≝
+| drops_nil : ∀L. drops s (◊) L L
+| drops_cons: ∀L1,L,L2,cs,l,m.
+              drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2
+.
+
+interpretation "iterated slicing (local environment) abstract"
+   'RDropStar s cs T1 T2 = (drops s cs T1 T2).
+(*
+interpretation "iterated slicing (local environment) general"
+   'RDropStar des T1 T2 = (drops true des T1 T2).
+*)
+
+definition d_liftable1: relation2 lenv term → predicate bool ≝
+                        λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K →
+                        ∀U. ⬆[l, m] T ≡ U → R L U.
+
+definition d_liftables1: relation2 lenv term → predicate bool ≝
+                         λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
+                         ∀T,U. ⬆*[cs] T ≡ U → R K T → R L U.
+
+definition d_liftables1_all: relation2 lenv term → predicate bool ≝
+                             λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
+                             ∀Ts,Us. ⬆*[cs] Ts ≡ Us → 
+                             all … (R K) Ts → all … (R L) Us.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2.
+#L1 #L2 #s #cs * -L1 -L2 -cs //
+#L1 #L #L2 #l #m #cs #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: drop1_gen_pnil *)
+lemma drops_inv_nil: ∀L1,L2,s. ⬇*[s, ◊] L1 ≡ L2 → L1 = L2.
+/2 width=4 by drops_inv_nil_aux/ qed-.
+
+fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
+                         ∀l,m,tl. cs = {l, m} @ tl →
+                         ∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
+#L1 #L2 #s #cs * -L1 -L2 -cs
+[ #L #l #m #tl #H destruct
+| #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
+  /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop1_gen_pcons *)
+lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
+                      ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
+/2 width=3 by drops_inv_cons_aux/ qed-.
+
+lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 →
+                       ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
+                       ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 &
+                                     ⬇*[s, cs1] K1 ≡ K2 &
+                                     ⬆*[cs1] V2 ≡ V1 &
+                                     L1 = K1. ⓑ{I} V1.
+#I #s #cs #cs2 #i #H elim H -cs -cs2 -i
+[ #i #L1 #K2 #V2 #H
+  >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
+| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
+  elim (drops_inv_cons … H) -H #L #HL1 #H
+  elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
+  elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
+  @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
+  normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
+| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
+  elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
+  /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: drop1_skip_bind *)
+lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 →
+                  ∀I. ⬇*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #s #cs #H elim H -L1 -L2 -cs
+[ #L #V1 #V2 #HV12 #I
+  >(lifts_inv_nil … HV12) -HV12 //
+| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I
+  elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
+].
+qed.
+
+lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s.
+#R #s #HR #L #K #cs #H elim H -L -K -cs
+[ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
+  elim (lifts_inv_cons … H) -H /3 width=10 by/
+]
+qed.
+
+lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s.
+#R #s #HR #L #K #cs #HLK #Ts #Us #H elim H -Ts -Us normalize //
+#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
+qed.
+
+(* Basic_1: removed theorems 1: drop1_getl_trans *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma
new file mode 100644 (file)
index 0000000..2e938ac
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drop.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/multiple/drops.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+lemma drops_drop_trans: ∀L1,L,cs. ⬇*[Ⓕ, cs] L1 ≡ L → ∀L2,i. ⬇[i] L ≡ L2 →
+                        ∃∃L0,cs0,i0. ⬇[i0] L1 ≡ L0 & ⬇*[Ⓕ, cs0] L0 ≡ L2 &
+                                      @⦃i, cs⦄ ≡ i0 & cs ▭ i ≡ cs0.
+#L1 #L #cs #H elim H -L1 -L -cs
+[ /2 width=7 by drops_nil, minuss_nil, at_nil, ex4_3_intro/
+| #L1 #L0 #L #cs #l #m #_ #HL0 #IHL0 #L2 #i #HL2
+  elim (lt_or_ge i l) #Hil
+  [ elim (drop_trans_le … HL0 … HL2) -L /2 width=2 by lt_to_le/
+    #L #HL0 #HL2 elim (IHL0 … HL0) -L0 /3 width=7 by drops_cons, minuss_lt, at_lt, ex4_3_intro/
+  | lapply (drop_trans_ge … HL0 … HL2 ?) -L // #HL02
+    elim (IHL0 … HL02) -L0 /3 width=7 by minuss_ge, at_ge, ex4_3_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma
new file mode 100644 (file)
index 0000000..03473dc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/drops_drops.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/drops_drop.ma".
+
+(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop1_trans *)
+theorem drops_trans: ∀L,L2,s,cs2. ⬇*[s, cs2] L ≡ L2 → ∀L1,cs1. ⬇*[s, cs1] L1 ≡ L →
+                     ⬇*[s, cs2 @@ cs1] L1 ≡ L2.
+#L #L2 #s #cs2 #H elim H -L -L2 -cs2 /3 width=3 by drops_cons/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq.ma
new file mode 100644 (file)
index 0000000..cea4b65
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lazyeq_7.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR CLOSURES ********************************************)
+
+inductive fleq (l) (G) (L1) (T): relation3 genv lenv term ≝
+| fleq_intro: ∀L2. L1 ≡[T, l] L2 → fleq l G L1 T G L2 T
+.
+
+interpretation
+   "lazy equivalence (closure)"
+   'LazyEq l G1 L1 T1 G2 L2 T2 = (fleq l G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fleq_refl: ∀l. tri_reflexive … (fleq l).
+/2 width=1 by fleq_intro/ qed.
+
+lemma fleq_sym: ∀l. tri_symmetric … (fleq l).
+#l #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fleq_inv_gen: ∀G1,G2,L1,L2,T1,T2,l. ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄ →
+                    ∧∧ G1 = G2 & L1 ≡[T1, l] L2 & T1 = T2.
+#G1 #G2 #L1 #L2 #T1 #T2 #l * -G2 -L2 -T2 /2 width=1 by and3_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq_fleq.ma
new file mode 100644 (file)
index 0000000..de886f6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fleq_fleq.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_lleq.ma".
+include "basic_2A/multiple/fleq.ma".
+
+(* LAZY EQUIVALENCE FOR CLOSURES  *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem fleq_trans: ∀l. tri_transitive … (fleq l).
+#l #G1 #G #L1 #L #T1 #T * -G -L -T
+#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/3 width=3 by lleq_trans, fleq_intro/
+qed-.
+
+theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,l.
+                      ⦃G, L, T⦄ ≡[l] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[l] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ qed-.
+
+theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,l.
+                      ⦃G1, L1, T1⦄ ≡[l] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[l] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup.ma
new file mode 100644 (file)
index 0000000..764ba38
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup.ma
@@ -0,0 +1,109 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/suptermplus_6.ma".
+include "basic_2A/substitution/fqu.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu.
+
+interpretation "plus-iterated structural successor (closure)"
+   'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2.
+                   ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2.
+                   ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fqup_drop: ∀G1,G2,L1,K1,K2,T1,T2,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 →
+                  ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #m #HLK1 #HTU1 #HT12 elim (eq_or_gt … m) #H destruct
+[ >(drop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 //
+| /3 width=5 by fqup_strap2, fqu_drop_lt/
+]
+qed-.
+
+lemma fqup_lref: ∀I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
+/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_drop/ qed.
+
+lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄.
+/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
+
+lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊐+ ⦃G, L.ⓑ{I}V, T⦄.
+/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
+
+lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+ ⦃G, L, T⦄.
+/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
+
+lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+ ⦃G, L, V2⦄.
+/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
+
+lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I1}V1, T⦄.
+/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
+
+lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊐+ ⦃G, L.ⓑ{I2}V2, T⦄.
+/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 ….
+                (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
+qed-.
+
+lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
+                   (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G1 L1 T1) →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
+                   ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=3 by fqu_fwd_fw, transitive_lt/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqup_wf_ind: ∀R:relation3 …. (
+                      ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                      R G1 L1 T1
+                   ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
+qed-.
+
+lemma fqup_wf_ind_eq: ∀R:relation3 …. (
+                         ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                         ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
+                      ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup_fqup.ma
new file mode 100644 (file)
index 0000000..642126f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqup_fqup.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqup_trans: tri_transitive … fqup.
+/2 width=5 by tri_TC_transitive/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus.ma
new file mode 100644 (file)
index 0000000..d99b4b9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus.ma
@@ -0,0 +1,83 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/suptermstar_6.ma".
+include "basic_2A/substitution/fquq.ma".
+include "basic_2A/multiple/fqup.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
+
+interpretation "star-iterated structural successor (closure)"
+   'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
+                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
+qed-.
+
+lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fqus_refl: tri_reflexive … fqus.
+/2 width=1 by tri_inj/ qed.
+
+lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
+
+lemma fqus_drop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ →
+                  ∀L1,U1,m. ⬇[m] L1 ≡ K1 → ⬆[0, m] T1 ≡ U1 →
+                  ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄.
+#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
+qed-.
+
+lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
+/3 width=3 by fquq_fwd_fw, transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
+#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_alt.ma
new file mode 100644 (file)
index 0000000..e9f707d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_alt.ma
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/fquq_alt.ma".
+include "basic_2A/multiple/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
+#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2
+[ /3 width=5 by fqup_strap1, or_introl/
+| * #HG #HL #HT destruct /2 width=1 by or_introl/
+| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/
+| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1
+[ /2 width=5 by fqup_strap1/
+| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2
+[ /2 width=5 by fqup_strap2/
+| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
+/2 width=5 by fqus_strap1_fqu, fqup_strap1/
+qed-.
+
+lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
+/3 width=5 by fqus_strap2_fqu, fqup_strap2/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_fqus.ma
new file mode 100644 (file)
index 0000000..b14bd75
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/fqus_fqus.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqus_trans: tri_transitive … fqus.
+/2 width=5 by tri_TC_transitive/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma
new file mode 100644 (file)
index 0000000..7a26437
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees.ma
@@ -0,0 +1,163 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+include "basic_2A/notation/relations/freestar_4.ma".
+include "basic_2A/substitution/lift_neg.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+inductive frees: relation4 ynat lenv term nat ≝
+| frees_eq: ∀L,U,l,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees l L U i
+| frees_be: ∀I,L,K,U,W,l,i,j. l ≤ yinj j → j < i →
+            (∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
+            frees 0 K W (i-j-1) → frees l L U i.
+
+interpretation
+   "context-sensitive free variables (term)"
+   'FreeStar L i l U = (frees l L U i).
+
+definition frees_trans: predicate (relation3 lenv term term) ≝
+                        λR. ∀L,U1,U2,i. R L U1 U2 → L ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L ⊢ i ϵ 𝐅*[0]⦃U1⦄.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma frees_inv: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                 (∀T. ⬆[i, 1] T ≡ U → ⊥) ∨
+                 ∃∃I,K,W,j. l ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
+                            ⬇[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_sort: ∀L,l,i,k. L ⊢ i ϵ 𝐅*[l]⦃⋆k⦄ → ⊥.
+#L #l #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_gref: ∀L,l,i,p. L ⊢ i ϵ 𝐅*[l]⦃§p⦄ → ⊥.
+#L #l #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_lref: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
+                      j = i ∨
+                      ∃∃I,K,W. l ≤ yinj j & j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #x #i #H elim (frees_inv … H) -H
+[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
+| * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
+  >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → j = i.
+#L #l #j #i #H #Hj elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 … HLK) -I
+#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_skip: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → j = i.
+#L #l #j #i #H #Hjl elim (frees_inv_lref … H) -H //
+* #I #K #W #Hlj elim (ylt_yle_false … Hlj) -Hlj //
+qed-. 
+
+lemma frees_inv_lref_ge: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → j = i.
+#L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -l /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_lt: ∀L,l,j,i.L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → j < i →
+                         ∃∃I,K,W. l ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
+[ #H elim (lt_refl_false j) //
+| * /2 width=5 by ex3_3_intro/
+]
+qed-.
+
+lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ .
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_bind … HnX) -HnX
+  /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
+  [ /4 width=9 by frees_be, or_introl/
+  | #HnT @or_intror @(frees_be … HnT) -HnT
+    [4,5,6: /2 width=1 by drop_drop, yle_succ, lt_minus_to_plus/
+    |7: >minus_plus_plus_l //
+    |*: skip
+    ]
+  ]
+]
+qed-.
+
+lemma frees_inv_flat: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[l]⦃U⦄ .
+#J #L #V #U #l #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_flat … HnX) -HnX
+  /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
+  /4 width=9 by frees_be, or_intror, or_introl/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma frees_lref_eq: ∀L,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
+/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
+
+lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
+                     K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
+/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
+
+lemma frees_bind_sn: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
+qed.
+
+lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
+[ /4 width=9 by frees_eq, nlift_bind_dx/
+| * #I #K #W #j #Hlj #Hji #HnU #HLK #HW
+  elim (yle_inv_succ1 … Hlj) -Hlj <yminus_SO2 #Hyj #H
+  lapply (ylt_O … H) -H #Hj
+  >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
+  /4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, lt_plus_to_minus/
+]
+qed.
+
+lemma frees_flat_sn: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
+qed.
+
+lemma frees_flat_dx: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
+qed.
+
+lemma frees_weak: ∀L,U,l1,i. L ⊢ i ϵ 𝐅*[l1]⦃U⦄ →
+                  ∀l2. l2 ≤ l1 → L ⊢ i ϵ 𝐅*[l2]⦃U⦄.
+#L #U #l1 #i #H elim H -L -U -l1 -i
+/3 width=9 by frees_be, frees_eq, yle_trans/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
+                        L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ .
+#a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
+/3 width=3 by frees_weak, or_intror, or_introl/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_append.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_append.ma
new file mode 100644 (file)
index 0000000..4b0ffce
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_append.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_append.ma".
+include "basic_2A/multiple/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties on append for local environments ******************************)
+
+lemma frees_append: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → i ≤ |L2| →
+                    ∀L1. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
+lapply (drop_fwd_length_minus2 … HLK2) normalize #H0
+lapply (drop_O1_append_sn_le … HLK2 … L1) -HLK2
+[ -I -L1 -K2 -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
+| #HLK2 @(frees_be … HnU HLK2) // -HnU -HLK2 @IHW -IHW
+  >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
+]
+qed.
+
+(* Inversion lemmas on append for local environments ************************)
+
+fact frees_inv_append_aux: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ → ∀L1,L2. L = L1 @@ L2 →
+                           i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/
+#Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
+elim (drop_O1_lt (Ⓕ) L2 j) [2: -Z -Y -L1 -X -U -l /2 width=3 by lt_to_le_to_lt/ ]
+#I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 … HLK2) normalize #H0
+lapply (drop_O1_inv_append1_le … HLY … HLK2) -HLY
+[ -Z -I -Y -K2 -L1 -X -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
+| normalize #H destruct
+  @(frees_be … HnU HLK2) -HnU -HLK2 // @IHW -IHW //
+  >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
+]
+qed-.
+
+lemma frees_inv_append: ∀L1,L2,U,l,i. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                        i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+/2 width=4 by frees_inv_append_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lift.ma
new file mode 100644 (file)
index 0000000..8a4fe40
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lift.ma
@@ -0,0 +1,160 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/multiple/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma frees_dec: ∀L,U,l,i. Decidable (frees l L U i).
+#L #U @(f2_ind … rfw … L U) -L -U
+#x #IH #L * *
+[ -IH /3 width=5 by frees_inv_sort, or_intror/
+| #j #Hx #l #i elim (lt_or_eq_or_gt i j) #Hji
+  [ -x @or_intror #H elim (lt_refl_false i)
+    lapply (frees_inv_lref_ge … H ?) -L -l /2 width=1 by lt_to_le/
+  | -x /2 width=1 by or_introl/
+  | elim (ylt_split j l) #Hli
+    [ -x @or_intror #H elim (lt_refl_false i)
+      lapply (frees_inv_lref_skip … H ?) -L //
+    | elim (lt_or_ge j (|L|)) #Hj
+      [ elim (drop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
+        elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
+        @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -l
+        lapply (drop_mono … HLY … HLK) -L #H destruct /2 width=1 by/  
+      | -x @or_intror #H elim (lt_refl_false i)
+        lapply (frees_inv_lref_free … H ?) -l //
+      ]
+    ]
+  ]
+| -IH /3 width=5 by frees_inv_gref, or_intror/
+| #a #I #W #U #Hx #l #i destruct
+  elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
+  elim (IH (L.ⓑ{I}W) U … (⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+  @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
+| #I #W #U #Hx #l #i destruct
+  elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
+  elim (IH L U … l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
+  @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
+]
+qed-.
+
+lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
+               (K ⊢ i-l-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯l]⦃U⦄.
+#L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
+* #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
+lapply (yle_inv_inj … Hlj) -Hlj #Hlj
+elim (le_to_or_lt_eq … Hlj) -Hlj
+[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
+| -Hji -HnU #H destruct
+  lapply (drop_mono … HLK0 … HLK) #H destruct -I
+  elim HnW0 -L -U -HnW0 //
+]
+qed.
+
+(* Note: lemma 1250 *)
+lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
+                       L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
+/4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
+qed.
+
+(* Properties on relocation *************************************************)
+
+lemma frees_lift_ge: ∀K,T,l,i. K ⊢ i ϵ𝐅*[l]⦃T⦄ →
+                     ∀L,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
+                     ∀U. ⬆[l0, m0] T ≡ U → l0 ≤ i →
+                     L ⊢ i+m0 ϵ 𝐅*[l]⦃U⦄.
+#K #T #l #i #H elim H -K -T -l -i
+[ #K #T #l #i #HnT #L #s #l0 #m0 #_ #U #HTU #Hl0i -K -s
+  @frees_eq #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
+| #I #K #K0 #T #V #l #i #j #Hlj #Hji #HnT #HK0 #HV #IHV #L #s #l0 #m0 #HLK #U #HTU #Hl0i
+  elim (lt_or_ge j l0) #H1
+  [ elim (drop_trans_lt … HLK … HK0) // -K #L0 #W #HL0 #HLK0 #HVW
+    @(frees_be … HL0) -HL0 -HV
+    /3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
+    [ #X #HXU >(plus_minus_m_m l0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
+      elim (lift_div_le … HXU … HTU ?) -U /2 width=2 by monotonic_pred/
+    | >minus_plus <plus_minus // <minus_plus
+      /3 width=5 by monotonic_le_minus_l2/
+    ]
+  | lapply (drop_trans_ge … HLK … HK0 ?) // -K #HLK0
+    lapply (drop_inv_gen … HLK0) >commutative_plus -HLK0 #HLK0
+    @(frees_be … HLK0) -HLK0 -IHV
+    /2 width=1 by yle_plus_dx1_trans, lt_minus_to_plus/
+    #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
+  ]
+]
+qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma frees_inv_lift_be: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                         ∀K,s,l0,m0. ⬇[s, l0, m0+1] L ≡ K →
+                         ∀T. ⬆[l0, m0+1] T ≡ U → l0 ≤ i → i ≤ l0 + m0 → ⊥.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #_ #T #HTU #Hl0i #Hilm0
+  elim (lift_split … HTU i m0) -HTU /2 width=2 by/
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hl0i #Hilm0
+  elim (lt_or_ge j l0) #H1
+  [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
+    @(IHW … HKL0 … HVW)
+    [ /2 width=1 by monotonic_le_minus_l2/
+    | >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
+    ]
+  | elim (lift_split … HTU j m0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
+  ]
+]
+qed-.
+
+lemma frees_inv_lift_ge: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                         ∀K,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
+                         ∀T. ⬆[l0, m0] T ≡ U → l0 + m0 ≤ i →
+                         K ⊢ i-m0 ϵ𝐅*[l-yinj m0]⦃T⦄.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
+  elim (le_inv_plus_l … Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
+  elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hlm0i
+  elim (lt_or_ge j l0) #H1
+  [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
+    elim (le_inv_plus_l … Hlm0i) #H0 #Hm0i
+    @(frees_be … H) -H
+    [ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
+    | /2 width=3 by lt_to_le_to_lt/
+    | #X #HXT elim (lift_trans_ge … HXT … HTU) -T /2 width=2 by/
+    | lapply (IHW … HKL0 … HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
+      >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
+    ]
+  | elim (lt_or_ge j (l0+m0)) [ >commutative_plus |] #H2
+    [ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
+      elim (lift_split … HTU j (m0-1)) -HTU //
+      [ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
+        #X #_ #H elim (HnU … H)
+      | >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
+      ]
+    | lapply (drop_conf_ge … HLK … HLK0 ?) // -L #HK0
+      elim (le_inv_plus_l … H2) -H2 #H2 #Hm0j
+      @(frees_be … HK0)
+      [ /2 width=1 by monotonic_yle_minus_dx/
+      | /2 width=1 by monotonic_lt_minus_l/
+      | #X #HXT elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
+      | >arith_b1 /2 width=5 by/
+      ]
+    ]
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lreq.ma
new file mode 100644 (file)
index 0000000..c2f4be0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lreq.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_lreq.ma".
+include "basic_2A/multiple/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_frees_trans: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                        ∀L1. L1 ⩬[l, ∞] L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+elim (lreq_drop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
+lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /3 width=9 by frees_be/
+qed-.
+
+lemma frees_lreq_conf: ∀L1,U,l,i. L1 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                       ∀L2. L1 ⩬[l, ∞] L2 → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+/3 width=3 by lreq_sym, lreq_frees_trans/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts.ma
new file mode 100644 (file)
index 0000000..407a8d8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts.ma
@@ -0,0 +1,150 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rliftstar_3.ma".
+include "basic_2A/substitution/lift.ma".
+include "basic_2A/multiple/mr2_plus.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+inductive lifts: list2 nat nat → relation term ≝
+| lifts_nil : ∀T. lifts (◊) T T
+| lifts_cons: ∀T1,T,T2,cs,l,m.
+              ⬆[l,m] T1 ≡ T → lifts cs T T2 → lifts ({l, m} @ cs) T1 T2
+.
+
+interpretation "generic relocation (term)"
+   'RLiftStar cs T1 T2 = (lifts cs T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_nil_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → cs = ◊ → T1 = T2.
+#T1 #T2 #cs * -T1 -T2 -cs //
+#T1 #T #T2 #l #m #cs #_ #_ #H destruct
+qed-.
+
+lemma lifts_inv_nil: ∀T1,T2. ⬆*[◊] T1 ≡ T2 → T1 = T2.
+/2 width=3 by lifts_inv_nil_aux/ qed-.
+
+fact lifts_inv_cons_aux: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 →
+                         ∀l,m,tl. cs = {l, m} @ tl →
+                         ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[tl] T ≡ T2.
+#T1 #T2 #cs * -T1 -T2 -cs
+[ #T #l #m #tl #H destruct
+| #T1 #T #T2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
+  /2 width=3 by ex2_intro/
+qed-.
+
+lemma lifts_inv_cons: ∀T1,T2,l,m,cs. ⬆*[{l, m} @ cs] T1 ≡ T2 →
+                      ∃∃T. ⬆[l, m] T1 ≡ T & ⬆*[cs] T ≡ T2.
+/2 width=3 by lifts_inv_cons_aux/ qed-.
+
+(* Basic_1: was: lift1_sort *)
+lemma lifts_inv_sort1: ∀T2,k,cs. ⬆*[cs] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #cs elim cs -cs
+[ #H <(lifts_inv_nil … H) -H //
+| #l #m #cs #IH #H
+  elim (lifts_inv_cons … H) -H #X #H
+  >(lift_inv_sort1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+lemma lifts_inv_lref1: ∀T2,cs,i1. ⬆*[cs] #i1 ≡ T2 →
+                       ∃∃i2. @⦃i1, cs⦄ ≡ i2 & T2 = #i2.
+#T2 #cs elim cs -cs
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3 by at_nil, ex2_intro/
+| #l #m #cs #IH #i1 #H
+  elim (lifts_inv_cons … H) -H #X #H1 #H2
+  elim (lift_inv_lref1 … H1) -H1 * #Hli1 #H destruct
+  elim (IH … H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
+]
+qed-.
+
+lemma lifts_inv_gref1: ∀T2,p,cs. ⬆*[cs] §p ≡ T2 → T2 = §p.
+#T2 #p #cs elim cs -cs
+[ #H <(lifts_inv_nil … H) -H //
+| #l #m #cs #IH #H
+  elim (lifts_inv_cons … H) -H #X #H
+  >(lift_inv_gref1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+lemma lifts_inv_bind1: ∀a,I,T2,cs,V1,U1. ⬆*[cs] ⓑ{a,I} V1. U1 ≡ T2 →
+                       ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs + 1] U1 ≡ U2 &
+                                T2 = ⓑ{a,I} V2. U2.
+#a #I #T2 #cs elim cs -cs
+[ #V1 #U1 #H
+  <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #l #m #cs #IHcs #V1 #U1 #H
+  elim (lifts_inv_cons … H) -H #X #H #HT2
+  elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
+  elim (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+  /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+lemma lifts_inv_flat1: ∀I,T2,cs,V1,U1. ⬆*[cs] ⓕ{I} V1. U1 ≡ T2 →
+                       ∃∃V2,U2. ⬆*[cs] V1 ≡ V2 & ⬆*[cs] U1 ≡ U2 &
+                                T2 = ⓕ{I} V2. U2.
+#I #T2 #cs elim cs -cs
+[ #V1 #U1 #H
+  <(lifts_inv_nil … H) -H /2 width=5 by ex3_2_intro, lifts_nil/
+| #l #m #cs #IHcs #V1 #U1 #H
+  elim (lifts_inv_cons … H) -H #X #H #HT2
+  elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
+  elim (IHcs … HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+  /3 width=5 by ex3_2_intro, lifts_cons/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lifts_simple_dx: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_dx/
+qed-.
+
+lemma lifts_simple_sn: ∀T1,T2,cs. ⬆*[cs] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#T1 #T2 #cs #H elim H -T1 -T2 -cs /3 width=5 by lift_simple_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_bind: ∀a,I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+                  ∀T1. ⬆*[cs + 1] T1 ≡ T2 →
+                  ⬆*[cs] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H
+  elim (lifts_inv_cons … H) -H /3 width=3 by lift_bind, lifts_cons/
+]
+qed.
+
+lemma lifts_flat: ∀I,T2,V1,V2,cs. ⬆*[cs] V1 ≡ V2 →
+                  ∀T1. ⬆*[cs] T1 ≡ T2 →
+                  ⬆*[cs] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #T2 #V1 #V2 #cs #H elim H -V1 -V2 -cs
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #cs #l #m #HV1 #_ #IHV #T1 #H
+  elim (lifts_inv_cons … H) -H /3 width=3 by lift_flat, lifts_cons/
+]
+qed.
+
+lemma lifts_total: ∀cs,T1. ∃T2. ⬆*[cs] T1 ≡ T2.
+#cs elim cs -cs /2 width=2 by lifts_nil, ex_intro/
+#l #m #cs #IH #T1 elim (lift_total T1 l m)
+#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_intro/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift.ma
new file mode 100644 (file)
index 0000000..52f1a63
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift.ma
@@ -0,0 +1,59 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_lift.ma".
+include "basic_2A/multiple/mr2_minus.ma".
+include "basic_2A/multiple/lifts.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+(* Properties concerning basic term relocation ******************************)
+
+(* Basic_1: was: lift1_xhg (right to left) *)
+lemma lifts_lift_trans_le: ∀T1,T,cs. ⬆*[cs] T1 ≡ T → ∀T2. ⬆[0, 1] T ≡ T2 →
+                           ∃∃T0. ⬆[0, 1] T1 ≡ T0 & ⬆*[cs + 1] T0 ≡ T2.
+#T1 #T #cs #H elim H -T1 -T -cs
+[ /2 width=3 by lifts_nil, ex2_intro/
+| #T1 #T3 #T #cs #l #m #HT13 #_ #IHT13 #T2 #HT2
+  elim (IHT13 … HT2) -T #T #HT3 #HT2
+  elim (lift_trans_le … HT13 … HT3) -T3 /3 width=5 by lifts_cons, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: lift1_free (right to left) *)
+lemma lifts_lift_trans: ∀cs,i,i0. @⦃i, cs⦄ ≡ i0 →
+                        ∀cs0. cs + 1 ▭ i + 1 ≡ cs0 + 1 →
+                        ∀T1,T0. ⬆*[cs0] T1 ≡ T0 →
+                        ∀T2. ⬆[O, i0 + 1] T0 ≡ T2 →
+                        ∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[cs] T ≡ T2.
+#cs elim cs -cs normalize
+[ #i #x #H1 #cs0 #H2 #T1 #T0 #HT10 #T2
+  <(at_inv_nil … H1) -x #HT02
+  lapply (minuss_inv_nil1 … H2) -H2 #H
+  >(pluss_inv_nil2 … H) in HT10; -cs0 #H
+  >(lifts_inv_nil … H) -T1 /2 width=3 by lifts_nil, ex2_intro/
+| #l #m #cs #IHcs #i #i0 #H1 #cs0 #H2 #T1 #T0 #HT10 #T2 #HT02
+  elim (at_inv_cons … H1) -H1 * #Hil #Hi0
+  [ elim (minuss_inv_cons1_lt … H2) -H2 [2: /2 width=1 by lt_minus_to_plus/ ] #cs1 #Hcs1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
+    elim (pluss_inv_cons2 … H) -H #cs2 #H1 #H2 destruct
+    elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
+    elim (IHcs … Hi0 … Hcs1 … HT0 … HT02) -IHcs -Hi0 -Hcs1 -T0 #T0 #HT0 #HT02
+    elim (lift_trans_le … HT1 … HT0) -T /2 width=1 by/ #T #HT1 <plus_minus_m_m /3 width=5 by lifts_cons, ex2_intro/
+  | >commutative_plus in Hi0; #Hi0
+    lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1 by le_S_S/ ] <associative_plus #Hcs0
+    elim (IHcs … Hi0 … Hcs0 … HT10 … HT02) -IHcs -Hi0 -Hcs0 -T0 #T0 #HT0 #HT02
+    elim (lift_split … HT0 l (i+1)) -HT0 /3 width=5 by lifts_cons, le_S, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift_vector.ma
new file mode 100644 (file)
index 0000000..7f53f2b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lift_vector.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_lift_vector.ma".
+include "basic_2A/multiple/lifts_lift.ma".
+include "basic_2A/multiple/lifts_vector.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lifts1_xhg (right to left) *)
+lemma liftsv_liftv_trans_le: ∀T1s,Ts,cs. ⬆*[cs] T1s ≡ Ts →
+                             ∀T2s:list term. ⬆[0, 1] Ts ≡ T2s →
+                             ∃∃T0s. ⬆[0, 1] T1s ≡ T0s & ⬆*[cs + 1] T0s ≡ T2s.
+#T1s #Ts #cs #H elim H -T1s -Ts
+[ #T1s #H
+  >(liftv_inv_nil1 … H) -T1s /2 width=3 by liftsv_nil, liftv_nil, ex2_intro/
+| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H
+  elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct
+  elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s
+  elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5 by liftsv_cons, liftv_cons, ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lifts.ma
new file mode 100644 (file)
index 0000000..5d27e9d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_lifts.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lifts_lift.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lift1_lift1 (left to right) *)
+theorem lifts_trans: ∀T1,T,cs1. ⬆*[cs1] T1 ≡ T → ∀T2:term. ∀cs2. ⬆*[cs2] T ≡ T2 →
+                     ⬆*[cs1 @@ cs2] T1 ≡ T2.
+#T1 #T #cs1 #H elim H -T1 -T -cs1 /3 width=3 by lifts_cons/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_vector.ma
new file mode 100644 (file)
index 0000000..5d7cc98
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lifts_vector.ma
@@ -0,0 +1,53 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_vector.ma".
+include "basic_2A/multiple/lifts.ma".
+
+(* GENERIC TERM VECTOR RELOCATION *******************************************)
+
+inductive liftsv (cs:list2 nat nat) : relation (list term) ≝
+| liftsv_nil : liftsv cs (◊) (◊)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+               ⬆*[cs] T1 ≡ T2 → liftsv cs T1s T2s →
+               liftsv cs (T1 @ T1s) (T2 @ T2s)
+.
+
+interpretation "generic relocation (vector)"
+   'RLiftStar cs T1s T2s = (liftsv cs T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: lifts1_flat (left to right) *)
+lemma lifts_inv_applv1: ∀V1s,U1,T2,cs. ⬆*[cs] Ⓐ V1s. U1 ≡ T2 →
+                        ∃∃V2s,U2. ⬆*[cs] V1s ≡ V2s & ⬆*[cs] U1 ≡ U2 &
+                                  T2 = Ⓐ V2s. U2.
+#V1s elim V1s -V1s normalize
+[ #T1 #T2 #cs #HT12  
+  @ex3_2_intro [3,4: // |1,2: skip | // ] (**) (* explicit constructor *)
+| #V1 #V1s #IHV1s #T1 #X #cs #H
+  elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
+  elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
+  @(ex3_2_intro) [4: // |3: /2 width=2 by liftsv_cons/ |1,2: skip | // ] (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: lifts1_flat (right to left) *)
+lemma lifts_applv: ∀V1s,V2s,cs. ⬆*[cs] V1s ≡ V2s →
+                   ∀T1,T2. ⬆*[cs] T1 ≡ T2 →
+                   ⬆*[cs] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
+#V1s #V2s #cs #H elim H -V1s -V2s /3 width=1 by lifts_flat/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq.ma
new file mode 100644 (file)
index 0000000..f8c737d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq.ma
@@ -0,0 +1,160 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lazyeq_4.ma".
+include "basic_2A/multiple/llpx_sn.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2.
+
+definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
+
+interpretation
+   "lazy equivalence (local environment)"
+   'LazyEq T l L1 L2 = (lleq l T L1 L2).
+
+definition lleq_transitive: predicate (relation3 lenv term term) ≝
+           λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
+                   ∀L1,L2,l,k. |L1| = |L2| → R l (⋆k) L1 L2
+                ) → (
+                   ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2
+                ) → (
+                   ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
+                   ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
+                   K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2
+                ) → (
+                   ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2
+                ) → (
+                   ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2
+                ) → (
+                   ∀a,I,L1,L2,V,T,l.
+                   L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V →
+                   R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2
+                ) → (
+                   ∀I,L1,L2,V,T,l.
+                   L1 ≡[V, l]L2 → L1 ≡[T, l] L2 →
+                   R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2
+                ) →
+                ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/
+qed-.
+
+lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 →
+                     L1 ≡[V, l] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind/ qed-.
+
+lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 →
+                     L1 ≡[V, l] L2 ∧ L1 ≡[T, l] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: ∀L1,L2,T,l. L1 ≡[T, l] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
+                     ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                      | yinj i < l
+                      | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V &
+                                     ⬇[i] L2 ≡ K2.ⓑ{I}V &
+                                      K1 ≡[V, yinj 0] K2 & l ≤ yinj i.
+#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/
+* /3 width=7 by or3_intro2, ex4_4_intro/
+qed-.
+
+lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 →
+                         ∃K2. ⬇[i] L2 ≡ K2.
+/2 width=7 by llpx_sn_fwd_drop_sn/ qed-.
+
+lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 →
+                         ∃K1. ⬇[i] L1 ≡ K1.
+/2 width=7 by llpx_sn_fwd_drop_dx/ qed-.
+
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,l.
+                        L1 ≡[ⓑ{a,I}V.T, l] L2 → L1 ≡[V, l] L2.
+/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
+
+lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l.
+                        L1 ≡[ⓑ{a,I}V.T, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l.
+                        L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[V, l] L2.
+/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,l.
+                        L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[T, l] L2.
+/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,l,k. |L1| = |L2| → L1 ≡[⋆k, l] L2.
+/2 width=1 by llpx_sn_sort/ qed.
+
+lemma lleq_skip: ∀L1,L2,l,i. yinj i < l → |L1| = |L2| → L1 ≡[#i, l] L2.
+/2 width=1 by llpx_sn_skip/ qed.
+
+lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
+                 ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
+                 K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma lleq_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, l] L2.
+/2 width=1 by llpx_sn_free/ qed.
+
+lemma lleq_gref: ∀L1,L2,l,p. |L1| = |L2| → L1 ≡[§p, l] L2.
+/2 width=1 by llpx_sn_gref/ qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,l.
+                 L1 ≡[V, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V →
+                 L1 ≡[ⓑ{a,I}V.T, l] L2.
+/2 width=1 by llpx_sn_bind/ qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,l.
+                 L1 ≡[V, l] L2 → L1 ≡[T, l] L2 → L1 ≡[ⓕ{I}V.T, l] L2.
+/2 width=1 by llpx_sn_flat/ qed.
+
+lemma lleq_refl: ∀l,T. reflexive … (lleq l T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
+/2 width=1 by llpx_sn_Y/ qed.
+
+lemma lleq_sym: ∀l,T. symmetric … (lleq l T).
+#l #T #L1 #L2 #H @(lleq_ind … H) -l -T -L1 -L2
+/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
+qed-.
+
+lemma lleq_ge_up: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
+                  ∀T,l,m. ⬆[l, m] T ≡ U →
+                  lt ≤ l + m → L1 ≡[U, l] L2.
+/2 width=6 by llpx_sn_ge_up/ qed-.
+
+lemma lleq_ge: ∀L1,L2,T,l1. L1 ≡[T, l1] L2 → ∀l2. l1 ≤ l2 → L1 ≡[T, l2] L2.
+/2 width=3 by llpx_sn_ge/ qed-.
+
+lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+                   L1 ≡[ⓑ{a,I}V.T, 0] L2.
+/2 width=1 by llpx_sn_bind_O/ qed-.
+
+(* Advanceded properties on lazy pointwise extensions ************************)
+
+lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) →
+                     ∀L1,L2,T,l. L1 ≡[T, l] L2 → llpx_sn R l T L1 L2.
+/2 width=3 by llpx_sn_co/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt.ma
new file mode 100644 (file)
index 0000000..97061d9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_alt.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Alternative definition (not recursive) ***********************************)
+
+theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| →
+                        (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ →
+                           ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           I1 = I2 ∧ V1 = V2
+                        ) → L1 ≡[T, l] L2.
+#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
+qed.
+
+theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 →
+                      |L1| = |L2| ∧
+                      ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ →
+                      ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                      I1 = I2 ∧ V1 = V2.
+#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+@(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt_rec.ma
new file mode 100644 (file)
index 0000000..92a5ce6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt_rec.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_alt_rec.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Alternative definition (recursive) ***************************************)
+
+theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| →
+                          (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                             ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                             ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2
+                          ) → L1 ≡[T, l] L2.
+#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed.
+
+theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv.
+                        (∀L1,L2,T,l. |L1| = |L2| → (
+                           ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                           ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2 & S 0 V1 K1 K2
+                        ) → S l T L1 L2) →
+                        ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2.
+#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
+qed-.
+
+theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 →
+                        |L1| = |L2| ∧
+                        ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                        ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                        ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
+#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_drop.ma
new file mode 100644 (file)
index 0000000..5bf3ddf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_drop.ma
@@ -0,0 +1,150 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_drop.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+                        ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W.
+/2 width=7 by llpx_sn_bind_repl_O/ qed-.
+
+lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2).
+/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
+
+lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
+                          ∀L1,L2,T,l. L1 ≡[T, l] L2 →
+                          ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L.
+#R #HR #L1 #L2 #T #l #H @(lleq_ind … H) -L1 -L2 -T -l
+[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
+|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
+| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
+  /3 width=11 by llpx_sn_lref/
+| #a #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
+  /3 width=1 by llpx_sn_bind/
+| #I #L1 #L2 #V #T #l #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
+                         ∀L1,L2,T,l. L1 ≡[T, l] L2 →
+                         ∀L. llpx_sn R l T L1 L → llpx_sn R l T L2 L.
+/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lleq_inv_lref_ge_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
+                           ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
+                           ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2.
+#L1 #L2 #l #i #H #Hli #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
+                           ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
+                           ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2.
+#L1 #L2 #l #i #H #Hli #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_bi: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
+                           ∀I1,I2,K1,K2,V1,V2.
+                           ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           ∧∧ I1 = I2 & K1 ≡[V1, 0] K2 & V1 = V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma lleq_inv_lref_ge: ∀L1,L2,l,i. L1 ≡[#i, l] L2 → l ≤ i →
+                        ∀I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
+                        K1 ≡[V, 0] K2.
+#L1 #L2 #l #i #HL12 #Hli #I #K1 #K2 #V #HLK1 #HLK2
+elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
+qed-.
+
+lemma lleq_inv_S: ∀L1,L2,T,l. L1 ≡[T, l+1] L2 →
+                  ∀I,K1,K2,V. ⬇[l] L1 ≡ K1.ⓑ{I}V → ⬇[l] L2 ≡ K2.ⓑ{I}V →
+                  K1 ≡[V, 0] K2 → L1 ≡[T, l] L2.
+/2 width=9 by llpx_sn_inv_S/ qed-.
+
+lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
+                       L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
+                        ∀I,K2,V. ⬇[i] L2 ≡ K2.ⓑ{I}V →
+                        i < l ∨
+                        ∃∃K1. ⬇[i] L1 ≡ K1.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
+#L1 #L2 #l #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_lref_sn: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
+                        ∀I,K1,V. ⬇[i] L1 ≡ K1.ⓑ{I}V →
+                        i < l ∨
+                        ∃∃K2. ⬇[i] L2 ≡ K2.ⓑ{I}V & K1 ≡[V, 0] K2 & l ≤ i.
+#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
+                          L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma lleq_lift_le: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 →
+                    ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                    ∀U. ⬆[l, m] T ≡ U → lt ≤ l → L1 ≡[U, lt] L2.
+/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
+
+lemma lleq_lift_ge: ∀K1,K2,T,lt. K1 ≡[T, lt] K2 →
+                    ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                    ∀U. ⬆[l, m] T ≡ U → l ≤ lt → L1 ≡[U, lt+m] L2.
+/2 width=9 by llpx_sn_lift_ge/ qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lleq_inv_lift_le: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
+                        ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                        ∀T. ⬆[l, m] T ≡ U → lt ≤ l → K1 ≡[T, lt] K2.
+/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
+
+lemma lleq_inv_lift_be: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
+                        ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                        ∀T. ⬆[l, m] T ≡ U → l ≤ lt → lt ≤ yinj l + m → K1 ≡[T, l] K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma lleq_inv_lift_ge: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
+                        ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                        ∀T. ⬆[l, m] T ≡ U → yinj l + m ≤ lt → K1 ≡[T, lt-m] K2.
+/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,l. (L1 ≡[ⓑ{a,I}V.T, l] L2 → ⊥) →
+                      (L1 ≡[V, l] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
+
+lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) →
+                      (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥).
+/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
+
+lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
+                        (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_fqus.ma
new file mode 100644 (file)
index 0000000..55c2dbc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_fqus.ma
@@ -0,0 +1,75 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqus_alt.ma".
+include "basic_2A/multiple/lleq_drop.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ →
+                      ∀L1. L1 ≡[T, 0] L2 →
+                      ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
+  #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1
+  #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+  [ elim (lleq_inv_bind … H)
+  | elim (lleq_inv_flat … H)
+  ] -H
+  /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
+  #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
+  /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #m #HLK2 #HTU #L1 #HL12
+  elim (drop_O1_le (Ⓕ) (m+1) L1)
+  [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+  | lapply (drop_fwd_length_le2 … HLK2) -K2
+    lapply (lleq_fwd_length … HL12) -T -U //
+  ]
+]
+qed-.
+
+lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
+[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
+  /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
+  #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
+  /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
+[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lleq.ma
new file mode 100644 (file)
index 0000000..e1de914
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lleq.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_drop.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Main properties **********************************************************)
+
+theorem lleq_trans: ∀l,T. Transitive … (lleq l T).
+/2 width=3 by lleq_llpx_sn_trans/ qed-.
+
+theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Advanced properies on negated lazy equivalence *****************************)
+
+(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *) 
+lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L →
+                        ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥).
+/3 width=3 by lleq_canc_sn/ qed-.
+
+lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L →
+                      ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥).
+/3 width=3 by lleq_trans/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_llor.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_llor.ma
new file mode 100644 (file)
index 0000000..665a3a1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_llor.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llor.ma".
+include "basic_2A/multiple/llpx_sn_frees.ma".
+include "basic_2A/multiple/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on pointwise union for local environments **********************)
+
+lemma llpx_sn_llor_dx: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                       ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L2 ≡[T, l] L.
+#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2
+lapply (llpx_sn_frees_trans … H1R H2R … H1) -H1R -H2R #HR
+elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
+elim H2 -H2 #_ #HL1 #IH2
+@lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
+lapply (drop_fwd_length_lt2 … HLK) #HiL
+elim (drop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
+elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
+[ elim (ylt_yle_false … H) -H //
+| elim H -H /2 width=1 by/
+]
+qed.
+
+lemma llpx_sn_llor_dx_sym: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                           ∀L1,L2,T,l. llpx_sn R l T L1 L2 → ∀L. L1 ⋓[T, l] L2 ≡ L → L ≡[T, l] L2.
+/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lreq.ma
new file mode 100644 (file)
index 0000000..155c515
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_lreq.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_lreq.ma".
+include "basic_2A/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_lleq_trans: ∀L2,L,T,l. L2 ≡[T, l] L →
+                       ∀L1. L1 ⩬[l, ∞] L2 → L1 ≡[T, l] L.
+/2 width=3 by lreq_llpx_sn_trans/ qed-.
+
+lemma lleq_lreq_trans: ∀L,L1,T,l. L ≡[T, l] L1 →
+                       ∀L2. L1 ⩬[l, ∞] L2 → L ≡[T, l] L2.
+/2 width=3 by llpx_sn_lreq_trans/ qed-.
+
+lemma lleq_lreq_repl: ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀K1. K1 ⩬[l, ∞] L1 →
+                      ∀K2. L2 ⩬[l, ∞] K2 → K1 ≡[T, l] K2.
+/2 width=5 by llpx_sn_lreq_repl/ qed-.
+
+lemma lleq_bind_repl_SO: ∀I1,I2,L1,L2,V1,V2,T. L1.ⓑ{I1}V1 ≡[T, 0] L2.ⓑ{I2}V2 →
+                         ∀J1,J2,W1,W2. L1.ⓑ{J1}W1 ≡[T, 1] L2.ⓑ{J2}W2.
+/2 width=5 by llpx_sn_bind_repl_SO/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor.ma
new file mode 100644 (file)
index 0000000..3263499
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lazyor_5.ma".
+include "basic_2A/multiple/frees.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition llor: ynat → relation4 term lenv lenv lenv ≝ λl,T,L2,L1,L.
+                 ∧∧ |L1| = |L2| & |L1| = |L|
+                  & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
+                       ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ⬇[i] L ≡ K.ⓑ{I}V → ∨∨
+                       (∧∧ yinj i < l & I1 = I & V1 = V) |
+                       (∧∧ (L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → ⊥) & I1 = I & V1 = V) |
+                       (∧∧ l ≤ yinj i & L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ & I2 = I & V2 = V)
+                    ).
+
+interpretation
+   "lazy union (local environment)"
+   'LazyOr L1 T l L2 L = (llor l T L2 L1 L).
+
+(* Basic properties *********************************************************)
+
+(* Note: this can be proved by llor_skip *)
+lemma llor_atom: ∀T,l. ⋆ ⋓[T, l] ⋆ ≡ ⋆.
+#T #l @and3_intro //
+#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
+elim (drop_inv_atom1 … HLK1) -HLK1 #H destruct
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_alt.ma
new file mode 100644 (file)
index 0000000..ffea814
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_alt.ma
@@ -0,0 +1,74 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/frees_append.ma".
+include "basic_2A/multiple/llor.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Alternative definition ***************************************************)
+
+lemma llor_tail_frees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → l ≤ yinj (|L1|) →
+                       ∀I1,W1. ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ →
+                       ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I2}W2.L.
+#L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2
+@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
+#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
+lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
+lapply (lt_plus_SO_to_le … H) -H #H
+elim (le_to_or_lt_eq … H) -H #H
+[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
+  elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
+  elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
+  lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
+  elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
+  [ /3 width=1 by and3_intro, or3_intro0/
+  | /6 width=2 by frees_inv_append, lt_to_le, or3_intro1, and3_intro/
+  | /5 width=1 by frees_append, lt_to_le, or3_intro2, and4_intro/
+  ]
+| -IH -HLK1 destruct
+  lapply (drop_O1_inv_append1_le … HLK2 … (⋆) ?) // -HLK2 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
+  /3 width=1 by or3_intro2, and4_intro/
+]
+qed.
+
+lemma llor_tail_cofrees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L →
+                         ∀I1,W1. (ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → ⊥) →
+                         ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I1}W1.L.
+#L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2
+@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
+#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
+lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
+lapply (lt_plus_SO_to_le … H) -H #H
+elim (le_to_or_lt_eq … H) -H #H
+[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
+  elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
+  elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
+  lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
+  elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
+  [ /3 width=1 by and3_intro, or3_intro0/
+  | /6 width=2 by frees_inv_append, lt_to_le, or3_intro1, and3_intro/
+  | /5 width=1 by frees_append, lt_to_le, or3_intro2, and4_intro/
+  ]
+| -IH -HLK2 destruct
+  lapply (drop_O1_inv_append1_le … HLK1 … (⋆) ?) // -HLK1 normalize #H destruct
+  lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
+  /4 width=1 by or3_intro1, and3_intro/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_drop.ma
new file mode 100644 (file)
index 0000000..8d0de6f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_drop.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/frees_lift.ma".
+include "basic_2A/multiple/llor_alt.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma llor_skip: ∀L1,L2,U,l. |L1| = |L2| → yinj (|L1|) ≤ l → L1 ⋓[U, l] L2 ≡ L1.
+#L1 #L2 #U #l #HL12 #Hl @and3_intro // -HL12
+#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2
+lapply (drop_mono … HLK … HLK1) -HLK #H destruct
+lapply (drop_fwd_length_lt2 … HLK1) -K1
+/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/
+qed.
+
+(* Note: lemma 1400 concludes the "big tree" theorem *)
+lemma llor_total: ∀L1,L2,T,l. |L1| = |L2| → ∃L. L1 ⋓[T, l] L2 ≡ L.
+#L1 @(lenv_ind_alt … L1) -L1
+[ #L2 #T #l #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/
+| #I1 #L1 #V1 #IHL1 #Y #T #l >ltail_length #H
+  elim (length_inv_pos_sn_ltail … H) -H #I2 #L2 #V2 #HL12 #H destruct
+  elim (ylt_split l (|ⓑ{I1}V1.L1|))
+  [ elim (frees_dec (ⓑ{I1}V1.L1) T l (|L1|)) #HnU
+    elim (IHL1 L2 T l) // -IHL1 -HL12
+    [ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/ 
+    | /4 width=2 by llor_tail_cofrees, ex_intro/
+    ]
+  | -IHL1 /4 width=2 by llor_skip, plus_minus_m_m, ex_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn.ma
new file mode 100644 (file)
index 0000000..9eb82fa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn.ma
@@ -0,0 +1,209 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_sort: ∀L1,L2,l,k. |L1| = |L2| → llpx_sn R l (⋆k) L1 L2
+| llpx_sn_skip: ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn R l (#i) L1 L2
+| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i →
+                ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+                llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2
+| llpx_sn_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R l (#i) L1 L2
+| llpx_sn_gref: ∀L1,L2,l,p. |L1| = |L2| → llpx_sn R l (§p) L1 L2
+| llpx_sn_bind: ∀a,I,L1,L2,V,T,l.
+                llpx_sn R l V L1 L2 → llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+                llpx_sn R l (ⓑ{a,I}V.T) L1 L2
+| llpx_sn_flat: ∀I,L1,L2,V,T,l.
+                llpx_sn R l V L1 L2 → llpx_sn R l T L1 L2 → llpx_sn R l (ⓕ{I}V.T) L1 L2
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
+                           ∀a,I,V,T. X = ⓑ{a,I}V.T →
+                           llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #p #_ #b #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #l #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
+| #I #L1 #L2 #V #T #l #_ #_ #b #J #W #U #H destruct
+]
+qed-.
+
+lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
+                        llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
+
+fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
+                           ∀I,V,T. X = ⓕ{I}V.T →
+                           llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2.
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #l #p #_ #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #l #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #V #T #l #HV #HT #J #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
+                        llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2.
+/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_fwd_length: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l //
+#I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #HLK1 #HLK2 #_ #_ #HK12
+lapply (drop_fwd_length … HLK1) -HLK1
+lapply (drop_fwd_length … HLK2) -HLK2
+normalize //
+qed-.
+
+lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
+                            ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2.
+#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/
+qed-.
+
+lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
+                            ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1.
+#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
+#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/
+qed-.
+
+fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → ∀i. X = #i →
+                           ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                            | yinj i < l
+                            | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
+                                               ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
+                                               llpx_sn R (yinj 0) V1 K1 K2 &
+                                               R K1 V1 V2 & l ≤ yinj i.
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #j #H destruct
+| #L1 #L2 #l #i #_ #Hil #j #H destruct /2 width=1 by or3_intro1/
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
+  /3 width=9 by or3_intro2, ex5_5_intro/
+| #L1 #L2 #l #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
+| #L1 #L2 #l #p #_ #j #H destruct
+| #a #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
+| #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
+]
+qed-.
+
+lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
+                        ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                         | yinj i < l
+                         | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
+                                            ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
+                                            llpx_sn R (yinj 0) V1 K1 K2 &
+                                            R K1 V1 V2 & l ≤ yinj i.
+/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
+
+lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
+                           llpx_sn R l V L1 L2.
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
+                           llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
+                           llpx_sn R l V L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
+                           llpx_sn R l T L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H //
+qed-.
+
+lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (②{I}V.T) L1 L2 →
+                           llpx_sn R l V L1 L2.
+#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,l. llpx_sn R l T L L.
+#R #HR #T #L @(f2_ind … rfw … L T) -L -T
+#x #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
+#i #Hx elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
+#HiL #l elim (ylt_split i l) /2 width=1 by llpx_sn_skip/
+elim (drop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
+qed-.
+
+lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#x #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
+#a #I #V1 #T1 #Hx #L2 #HL12
+@llpx_sn_bind /2 width=1 by/ (**) (* explicit constructor *)
+@IH -IH // normalize /2 width=1 by eq_f2/
+qed-.
+
+lemma llpx_sn_ge_up: ∀R,L1,L2,U,lt. llpx_sn R lt U L1 L2 → ∀T,l,m. ⬆[l, m] T ≡ U →
+                     lt ≤ l + m → llpx_sn R l U L1 L2.
+#R #L1 #L2 #U #lt #H elim H -L1 -L2 -U -lt
+[ #L1 #L2 #lt #k #HL12 #X #l #m #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #lt #i #HL12 #Hilt #X #l #m #H #Hltlm
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
+  elim (ylt_yle_false … Hilt) -Hilt
+  @(yle_trans … Hltlm) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
+| #I #L1 #L2 #K1 #K2 #W1 #W2 #lt #i #Hlti #HLK1 #HLK2 #HW1 #HW12 #_ #X #l #m #H #_
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
+    lapply (drop_fwd_length … HLK1) lapply (drop_fwd_length … HLK2)
+    normalize in ⊢ (%→%→?); -I -W1 -W2 -lt /3 width=1 by llpx_sn_skip, ylt_inj/
+  | /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
+  ]
+| /2 width=1 by llpx_sn_free/
+| #L1 #L2 #lt #p #HL12 #X #l #m #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct
+  elim (lift_inv_bind2 … H) -H #V #T #HVW >commutative_plus #HTU #H destruct 
+  @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
+  @(IHT … HTU) /2 width=1 by yle_succ/
+| #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct
+  elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
+  /3 width=4 by llpx_sn_flat/
+]
+qed-.
+
+(**) (* the minor premise comes first *)
+lemma llpx_sn_ge: ∀R,L1,L2,T,l1,l2. l1 ≤ l2 →
+                  llpx_sn R l1 T L1 L2 → llpx_sn R l2 T L1 L2.
+#R #L1 #L2 #T #l1 #l2 * -l1 -l2 (**) (* destructed yle *)
+/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
+qed-.
+
+lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
+                      llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+                      llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
+/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
+
+lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+                  ∀L1,L2,T,l. llpx_sn R1 l T L1 L2 → llpx_sn R2 l T L1 L2.
+#R1 #R2 #HR12 #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
+/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt.ma
new file mode 100644 (file)
index 0000000..5a2116d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/frees.ma".
+include "basic_2A/multiple/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (not recursive) *)
+definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝
+                        λR,l,T,L1,L2. |L1| = |L2| ∧
+                        (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ →
+                           ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           I1 = I2 ∧ R K1 V1 V2
+                        ).
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,l. llpx_sn R l T L1 L2 → llpx_sn_alt R l T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#x #IHx #L1 #U #Hx #L2 #l #H elim (llpx_sn_inv_alt_r … H) -H
+#HL12 #IHU @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv … H) -H
+[ -x #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
+| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 #HnW10 destruct
+  lapply (drop_fwd_drop2 … HLK10) #H
+  lapply (drop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
+  elim (drop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by drop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
+  lapply (drop_fwd_drop2 … HLK20) #H
+  lapply (drop_conf_ge … H … HLK2 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK20
+  elim (IHx K10 W10 … K20 0) -IHx -HL12 /3 width=6 by drop_fwd_rfw/
+  elim (IHU … HnU HLK10 HLK20) -IHU -HnU -HLK10 -HLK20 //
+]
+qed.
+
+theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,l. llpx_sn_alt R l T L1 L2 → llpx_sn R l T L1 L2.
+#R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
+#x #IHx #L1 #U #Hx #L2 #l * #HL12 #IHU @llpx_sn_intro_alt_r //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnU #HLK1 #HLK2 destruct
+elim (IHU … HLK1 HLK2) /3 width=2 by frees_eq/
+#H #HV12 @and3_intro // @IHx -IHx /3 width=6 by drop_fwd_rfw/
+lapply (drop_fwd_drop2 … HLK1) #H1
+lapply (drop_fwd_drop2 … HLK2) -HLK2 #H2
+@conj [ @(drop_fwd_length_eq1 … H1 H2) // ] -HL12
+#Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #_
+>(minus_plus_m_m j (i+1)) in ⊢ (%→?); >commutative_plus <minus_plus
+#HnV1 #HKY1 #HKY2 (**) (* full auto too slow *)
+lapply (drop_trans_ge … H1 … HKY1 ?) -H1 -HKY1 // #HLY1
+lapply (drop_trans_ge … H2 … HKY2 ?) -H2 -HKY2 // #HLY2
+/4 width=14 by frees_be, yle_plus_dx2_trans, yle_succ_dx/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt_rec.ma
new file mode 100644 (file)
index 0000000..82d8628
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_alt_rec.ma
@@ -0,0 +1,250 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_neg.ma".
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* alternative definition of llpx_sn (recursive) *)
+inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
+| llpx_sn_alt_r_intro: ∀L1,L2,T,l.
+                       (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                          ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+                       ) →
+                       (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                          ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
+                       ) → |L1| = |L2| → llpx_sn_alt_r R l T L1 L2
+.
+
+(* Compact definition of llpx_sn_alt_r **************************************)
+
+lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| →
+                               (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                                 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                                 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
+                               ) → llpx_sn_alt_r R l T L1 L2.
+#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+                             (∀L1,L2,T,l. |L1| = |L2| → (
+                                ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                                ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                                ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
+                             ) → S l T L1 L2) →
+                             ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2.
+#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
+#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 →
+                             |L1| = |L2| ∧
+                             ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                               ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                             ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
+#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 →
+                              llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
+qed-.
+
+lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 →
+                              llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+  /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hli) -Hli * #Hli #Hi
+  lapply (drop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+  lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+  elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+  @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H //
+qed-.
+
+lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,l,i. llpx_sn_alt_r R l (#i) L1 L2 →
+                              ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+                               | yinj i < l
+                               | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
+                                                  ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
+                                                  llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
+                                                  R K1 V1 V2 & l ≤ yinj i.
+#R #L1 #L2 #l #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i l) /3 width=1 by or3_intro1/
+#Hli #HL1 elim (drop_O1_lt (Ⓕ) … HL1)
+#I1 #K1 #V1 #HLK1 elim (drop_O1_lt (Ⓕ) L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma llpx_sn_alt_r_sort: ∀R,L1,L2,l,k. |L1| = |L2| → llpx_sn_alt_r R l (⋆k) L1 L2.
+#R #L1 #L2 #l #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
+qed.
+
+lemma llpx_sn_alt_r_gref: ∀R,L1,L2,l,p. |L1| = |L2| → llpx_sn_alt_r R l (§p) L1 L2.
+#R #L1 #L2 #l #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
+qed.
+
+lemma llpx_sn_alt_r_skip: ∀R,L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn_alt_r R l (#i) L1 L2.
+#R #L1 #L2 #l #i #HL12 #Hil @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #Hlj #H elim (H (#i)) -H
+/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
+qed.
+
+lemma llpx_sn_alt_r_free: ∀R,L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
+                          llpx_sn_alt_r R l (#i) L1 L2.
+#R #L1 #L2 #l #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
+lapply (drop_fwd_length_lt2 … HLK1) -HLK1
+/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
+qed.
+
+lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i →
+                          ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+                          llpx_sn_alt_r R 0 V1 K1 K2 → R K1 V1 V2 →
+                          llpx_sn_alt_r R l (#i) L1 L2.
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
+[ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
+  @(drop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hlj #H #HLY1 #HLY2
+  elim (lt_or_eq_or_gt i j) #Hij destruct
+  [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+  | lapply (drop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+    lapply (drop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+  | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+  ]
+]
+qed.
+
+lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,l.
+                          llpx_sn_alt_r R l V L1 L2 → llpx_sn_alt_r R l T L1 L2 →
+                          llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2.
+#R #I #L1 #L2 #V #T #l #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
+
+lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,l.
+                          llpx_sn_alt_r R l V L1 L2 →
+                          llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
+                          llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2.
+#R #a #I #L1 #L2 #V #T #l #HV #HT
+elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by drop_drop, yle_succ, and3_intro/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → llpx_sn_alt_r R l T L1 L2.
+#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
+/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,l. llpx_sn_alt_r R l T L1 L2 → llpx_sn R l T L1 L2.
+#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #x #IH #L1 * *
+[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
+| #i #Hx #L2 #l #H lapply (llpx_sn_alt_r_fwd_length … H)
+  #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
+  [ * /2 width=1 by llpx_sn_free/
+  | /2 width=1 by llpx_sn_skip/
+  | * /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
+  ]
+| #a #I #V #T #Hx #L2 #l #H elim (llpx_sn_alt_r_inv_bind … H) -H
+  /3 width=1 by llpx_sn_bind/
+| #I #V #T #Hx #L2 #l #H elim (llpx_sn_alt_r_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+(* Alternative definition of llpx_sn (recursive) ****************************)
+
+lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,l. |L1| = |L2| →
+                           (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                              ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                              ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
+                           ) → llpx_sn R l T L1 L2.
+#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
+@llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
+qed.
+
+lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
+                         (∀L1,L2,T,l. |L1| = |L2| → (
+                            ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                            ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                            ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+                         ) → S l T L1 L2) →
+                         ∀L1,L2,T,l. llpx_sn R l T L1 L2 → S l T L1 L2.
+#R #S #IH1 #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
+qed-.
+
+lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
+                         |L1| = |L2| ∧
+                         ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
+                         ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                         ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
+#R #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
+#H elim (llpx_sn_alt_r_inv_alt … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_drop.ma
new file mode 100644 (file)
index 0000000..5a1a75c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_drop.ma
@@ -0,0 +1,431 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/multiple/llpx_sn_lreq.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
+                           ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+                           i < l ∨
+                           ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
+                                    R K1 V1 V2 & l ≤ i.
+#R #L1 #L2 #l #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+  lapply (drop_fwd_length_lt2 … HLK2) -HLK2
+  /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
+  lapply (drop_mono … HLK22 … HLK2) -L2 #H destruct
+  /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
+                           ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
+                           i < l ∨
+                           ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
+                                    R K1 V1 V2 & l ≤ i.
+#R #L1 #L2 #l #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
+[ #H #_ elim (lt_refl_false i)
+  lapply (drop_fwd_length_lt2 … HLK1) -HLK1
+  /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
+  lapply (drop_mono … HLK11 … HLK1) -L1 #H destruct
+  /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
+                              ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+                              ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
+                                       llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #l #i #H #Hli #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
+[ #H elim (ylt_yle_false … H Hli)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
+                              ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
+                              ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
+                                       llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #l #i #H #Hli #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
+[ #H elim (ylt_yle_false … H Hli)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
+                              ∀I1,I2,K1,K2,V1,V2.
+                              ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                              ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #l #i #HL12 #Hli #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -l
+#J #Y #HY lapply (drop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,l0. llpx_sn R l0 T L1 L2 → ∀l. l0 = l + 1 →
+                        ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 →
+                        llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2.
+#R #L1 #L2 #T #l0 #H elim H -L1 -L2 -T -l0
+/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
+[ #L1 #L2 #l0 #i #HL12 #Hil #l #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
+  elim (yle_split_eq i l) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hil
+  #H destruct /2 width=9 by llpx_sn_lref/
+| #I #L1 #L2 #K11 #K22 #V1 #V2 #l0 #i #Hl0i #HLK11 #HLK22 #HK12 #HV12 #_ #l #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
+  /3 width=9 by llpx_sn_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+  /4 width=9 by llpx_sn_bind, drop_drop/
+| #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+  /3 width=9 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_S: ∀R,L1,L2,T,l. llpx_sn R (l + 1) T L1 L2 →
+                     ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 →
+                     llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2.
+/2 width=9 by llpx_sn_inv_S_aux/ qed-.
+
+lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) →
+                          ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+                          llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
+/3 width=9 by drop_pair, conj, llpx_sn_inv_S/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀L. reflexive … (R L)) →
+                             ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
+                             llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
+                           ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
+/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
+
+lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                   ∀T,L1,L2,l. Decidable (llpx_sn R l T L1 L2).
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#x #IH #L1 * *
+[ #k #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
+| #i #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|))
+  [ #HL12 #l elim (ylt_split i l) /3 width=1 by llpx_sn_skip, or_introl/
+    #Hli elim (lt_or_ge i (|L1|)) #HiL1
+    elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
+    elim (drop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
+    elim (drop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
+    elim (eq_bind2_dec I2 I1)
+    [ #H2 elim (HR K1 V1 V2) -HR
+      [ #H3 elim (IH K1 V1 … K2 0) destruct
+        /3 width=9 by llpx_sn_lref, drop_fwd_rfw, or_introl/
+      ]
+    ]
+    -IH #H3 @or_intror
+    #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
+    [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+    |7,8,9: /2 width=4 by ylt_yle_false/
+    ]
+    #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
+    lapply (drop_mono … HLY1 … HLK1) -HLY1 -HLK1
+    lapply (drop_mono … HLY2 … HLK2) -HLY2 -HLK2
+    #H #H0 destruct /2 width=1 by/
+  ]
+| #p #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
+| #a #I #V #T #Hx #L2 #l destruct
+  elim (IH L1 V … L2 l) /2 width=1 by/
+  elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯l)) -IH /3 width=1 by or_introl, llpx_sn_bind/
+  #H1 #H2 @or_intror
+  #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
+| #I #V #T #Hx #L2 #l destruct
+  elim (IH L1 V … L2 l) /2 width=1 by/
+  elim (IH L1 T … L2 l) -IH /3 width=1 by or_introl, llpx_sn_flat/
+  #H1 #H2 @or_intror
+  #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+]
+-x /4 width=4 by llpx_sn_fwd_length, or_intror/
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma llpx_sn_lift_le: ∀R. d_liftable R →
+                       ∀K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
+                       ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                       ∀U. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 U L1 L2.
+#R #HR #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
+[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
+  /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
+  * #Hli #H destruct
+  [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
+    /2 width=1 by llpx_sn_skip/
+  | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -m -Hil0
+    /3 width=3 by yle_trans, yle_inj/
+  ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
+  * #Hli #H destruct [ -HK12 | -IHK12 ]
+  [ elim (drop_trans_lt … HLK1 … HK11) // -K1
+    elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2
+    /3 width=18 by llpx_sn_lref/
+  | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+    lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2
+    /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
+  ]
+| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
+  * #Hil #H destruct
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+  [ /3 width=7 by llpx_sn_free, drop_fwd_be/
+  | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
+    lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
+    @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+  ]
+| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -m
+  /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
+  #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
+| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
+  #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
+                       ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                       ∀U. ⬆[l, m] T ≡ U → l ≤ l0 → llpx_sn R (l0+m) U L1 L2.
+#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
+[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
+  /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
+  * #_ #H destruct
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
+  [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
+  | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
+  ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
+  * #Hil #H destruct
+  [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0
+    /3 width=3 by ylt_yle_trans, ylt_inj/
+  | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
+    lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2
+    /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
+  ]
+| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
+  * #Hil #H destruct
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
+  [ /3 width=7 by llpx_sn_free, drop_fwd_be/
+  | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
+    lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
+    @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+  ]
+| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
+  lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
+  /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
+  #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/
+| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
+  #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R →
+                           ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
+                           ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                           ∀T. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2.
+#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m
+  /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
+  * #_ #H destruct
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+  [ /2 width=1 by llpx_sn_skip/
+  | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
+  ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
+  * #Hil #H destruct [ -HK12 | -IHK12 ]
+  [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
+    elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
+    elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
+    lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
+    /3 width=10 by llpx_sn_lref/
+  | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
+    lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0
+    elim (le_inv_plus_l … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
+  ]
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
+  * #_ #H destruct
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
+  [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
+    lapply (drop_fwd_length_le4 … HLK2) -HLK2
+    #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+  | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+    lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+    /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+  ]
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m
+  /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H
+  #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H
+  #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
+                           ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                           ∀T. ⬆[l, m] T ≡ U → l ≤ l0 → l0 ≤ yinj l + m → llpx_sn R l T K1 K2.
+#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m
+  /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
+  * #Hil #H destruct
+  [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+    -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/
+  | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0
+    /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
+  ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
+  * #Hil #H destruct
+  [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0
+    /3 width=3 by ylt_yle_trans, ylt_inj/
+  | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
+    lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m
+    elim (le_inv_plus_l … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/
+  ]
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
+  * #_ #H destruct
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
+  [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
+    lapply (drop_fwd_length_le4 … HLK2) -HLK2
+    #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+  | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+    lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+    /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+  ]
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m
+  /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_bind2 … H) -H
+  >commutative_plus #V #T #HVW #HTU #H destruct
+  @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+  @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_flat2 … H) -H
+  #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
+                           ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                           ∀T. ⬆[l, m] T ≡ U → yinj l + m ≤ l0 → llpx_sn R (l0-m) T K1 K2.
+#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
+  /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
+  * #Hil #H destruct [ -Hil0 | -Hlml0 ]
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
+  [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/
+  | elim (le_inv_plus_l … Hil) -Hil #_
+    /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
+  ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
+  * #Hil #H destruct
+  [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0
+    /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
+  | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
+    lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil
+    /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
+  ]
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
+  * #_ #H destruct
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
+  [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
+    lapply (drop_fwd_length_le4 … HLK2) -HLK2
+    #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+  | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
+    lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
+    /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+  ]
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
+  lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
+  /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H
+  #V #T #HVW #HTU #H destruct
+  @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+  <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
+  @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_flat2 … H) -H
+  #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+                          ∀K1,K2,m. ⬇[m] L1 ≡ K1 → ⬇[m] L2 ≡ K2 →
+                          ∀T. ⬆[0, m] T ≡ U → llpx_sn R 0 T K1 K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma llpx_sn_drop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+                           ∀K1,m. ⬇[m] L1 ≡ K1 → ∀T. ⬆[0, m] T ≡ U →
+                           ∃∃K2. ⬇[m] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K1 #m #HLK1 #T #HTU elim (llpx_sn_fwd_drop_sn … HU … HLK1)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+lemma llpx_sn_drop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
+                            ∀K2,m. ⬇[m] L2 ≡ K2 → ∀T. ⬆[0, m] T ≡ U →
+                            ∃∃K1. ⬇[m] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K2 #m #HLK2 #T #HTU elim (llpx_sn_fwd_drop_dx … HU … HLK2)
+/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
+qed-.
+
+(* Inversion lemmas on negated lazy pointwise extension *********************)
+
+lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                         ∀a,I,L1,L2,V,T,l. (llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+                         (llpx_sn R l V L1 L2 → ⊥) ∨ (llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
+/4 width=1 by llpx_sn_bind, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                         ∀I,L1,L2,V,T,l. (llpx_sn R l (ⓕ{I}V.T) L1 L2 → ⊥) →
+                         (llpx_sn R l V L1 L2 → ⊥) ∨ (llpx_sn R l T L1 L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
+/4 width=1 by llpx_sn_flat, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                           ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
+                           (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
+/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_frees.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_frees.ma
new file mode 100644 (file)
index 0000000..0f1dff0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_frees.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/frees.ma".
+include "basic_2A/multiple/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on context-sensitive free variables ***************************)
+
+fact llpx_sn_frees_trans_aux: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                              ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                              ∀L1. llpx_sn R l U L1 L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#R #H1R #H2R #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+elim (llpx_sn_inv_alt_r … HL12) -HL12 #HL12 #IH
+lapply (drop_fwd_length_lt2 … HLK2) #Hj
+elim (drop_O1_lt (Ⓕ) L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
+elim (IH … HnU HLK1 HLK2) // -IH -HLK2 /5 width=11 by frees_be/
+qed-.
+
+lemma llpx_sn_frees_trans: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                           ∀L1,L2,U,l. llpx_sn R l U L1 L2 →
+                           ∀i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+/2 width=6 by llpx_sn_frees_trans_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_llor.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_llor.ma
new file mode 100644 (file)
index 0000000..63f54b2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_llor.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn_alt.ma".
+include "basic_2A/multiple/llor.ma".
+include "basic_2A/multiple/lleq_alt.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Inversion lemmas on pointwise union for local environments ****************)
+
+lemma llpx_sn_llor_fwd_sn: ∀R. (∀L. reflexive … (R L)) →
+                           ∀L1,L2,T,l. llpx_sn R l T L1 L2 →
+                           ∀L. L1 ⋓[T, l] L2 ≡ L → lpx_sn R L1 L.
+#R #HR #L1 #L2 #T #l #H1 #L #H2
+elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
+elim H2 -H2 #_ #HL1 #IH2
+@lpx_sn_intro_alt // #I1 #I #K1 #K #V1 #V #i #HLK1 #HLK
+lapply (drop_fwd_length_lt2 … HLK) #HiL
+elim (drop_O1_lt (Ⓕ) L2 i) // -HiL -HL1 -HL12 #I2 #K2 #V2 #HLK2
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK * /2 width=1 by conj/
+#HnT #H1 #H2 elim (IH1 … HnT … HLK1 HLK2) -IH1 -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lpx_sn.ma
new file mode 100644 (file)
index 0000000..870315f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lpx_sn.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn_drop.ma".
+include "basic_2A/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on pointwise extensions ***************************************)
+
+lemma lpx_sn_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
+                      ∀T,L1,L2,l. lpx_sn R L1 L2 → llpx_sn R l T L1 L2.
+#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
+#x #IH #L1 * *
+[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
+| -HR #i elim (lt_or_ge i (|L1|))
+  [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
+  #Hi #Hx #L2 #l elim (ylt_split i l) 
+  [ -x /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
+  #Hli #HL12 elim (drop_O1_lt (Ⓕ) L1 i) //
+  #I #K1 #V1 #HLK1 elim (lpx_sn_drop_conf … HL12 … HLK1) -HL12
+  /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
+| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
+| /4 width=1 by llpx_sn_bind, lpx_sn_pair/
+| -HR /3 width=1 by llpx_sn_flat/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lreq.ma
new file mode 100644 (file)
index 0000000..ab4eb09
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_lreq.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_lreq.ma".
+include "basic_2A/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_llpx_sn_trans: ∀R,L2,L,T,l. llpx_sn R l T L2 L →
+                          ∀L1. L1 ⩬[l, ∞] L2 → llpx_sn R l T L1 L.
+#R #L2 #L #T #l #H elim H -L2 -L -T -l
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/  
+[ #I #L2 #L #K2 #K #V2 #V #l #i #Hli #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
+  elim (lreq_drop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
+  lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, lreq_succ/
+]
+qed-.
+
+lemma llpx_sn_lreq_trans: ∀R,L,L1,T,l. llpx_sn R l T L L1 →
+                          ∀L2. L1 ⩬[l, ∞] L2 → llpx_sn R l T L L2.
+#R #L #L1 #T #l #H elim H -L -L1 -T -l
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/  
+[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
+  elim (lreq_drop_conf_be … HL12 … HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
+  lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, lreq_succ/
+]
+qed-.
+
+lemma llpx_sn_lreq_repl: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → ∀K1. K1 ⩬[l, ∞] L1 → 
+                         ∀K2. L2 ⩬[l, ∞] K2 → llpx_sn R l T K1 K2.
+/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-.
+
+lemma llpx_sn_bind_repl_SO: ∀R,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+                            ∀J1,J2,W1,W2. llpx_sn R 1 T (L1.ⓑ{J1}W1) (L2.ⓑ{J2}W2).
+#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R … 1 … HT) -HT
+/3 width=7 by llpx_sn_lreq_repl, lreq_succ/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_tc.ma
new file mode 100644 (file)
index 0000000..97e0580
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/llpx_sn_tc.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_drop.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties about transitive closure **************************************)
+
+lemma llpx_sn_TC_pair_dx: ∀R. (∀L. reflexive … (R L)) →
+                          ∀I,L,V1,V2,T. LTC … R L V1 V2 →
+                          LTC … (llpx_sn R 0) T (L.ⓑ{I}V1) (L.ⓑ{I}V2).
+#R #HR #I #L #V1 #V2 #T #H @(TC_star_ind … V2 H) -V2
+/4 width=9 by llpx_sn_bind_repl_O, llpx_sn_refl, step, inj/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2.ma
new file mode 100644 (file)
index 0000000..34f1691
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2.ma
@@ -0,0 +1,74 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rat_3.ma".
+include "basic_2A/grammar/term_vector.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+inductive at: list2 nat nat → relation nat ≝
+| at_nil: ∀i. at (◊) i i
+| at_lt : ∀cs,l,m,i1,i2. i1 < l →
+          at cs i1 i2 → at ({l, m} @ cs) i1 i2
+| at_ge : ∀cs,l,m,i1,i2. l ≤ i1 →
+          at cs (i1 + m) i2 → at ({l, m} @ cs) i1 i2
+.
+
+interpretation "application (multiple relocation with pairs)"
+   'RAt i1 cs i2 = (at cs i1 i2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact at_inv_nil_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = i2.
+#cs #i1 #i2 * -cs -i1 -i2
+[ //
+| #cs #l #m #i1 #i2 #_ #_ #H destruct
+| #cs #l #m #i1 #i2 #_ #_ #H destruct
+]
+qed-.
+
+lemma at_inv_nil: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = i2.
+/2 width=3 by at_inv_nil_aux/ qed-.
+
+fact at_inv_cons_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 →
+                      ∀l,m,cs0. cs = {l, m} @ cs0 →
+                      i1 < l ∧ @⦃i1, cs0⦄ ≡ i2 ∨
+                      l ≤ i1 ∧ @⦃i1 + m, cs0⦄ ≡ i2.
+#cs #i1 #i2 * -cs -i1 -i2
+[ #i #l #m #cs #H destruct
+| #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
+| #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
+]
+qed-.
+
+lemma at_inv_cons: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+                   i1 < l ∧ @⦃i1, cs⦄ ≡ i2 ∨
+                   l ≤ i1 ∧ @⦃i1 + m, cs⦄ ≡ i2.
+/2 width=3 by at_inv_cons_aux/ qed-.
+
+lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+                      i1 < l → @⦃i1, cs⦄ ≡ i2.
+#cs #l #m #i1 #m2 #H
+elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
+lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
+elim (lt_refl_false … Hl)
+qed-.
+
+lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+                      l ≤ i1 → @⦃i1 + m, cs⦄ ≡ i2.
+#cs #l #m #i1 #m2 #H
+elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
+lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
+elim (lt_refl_false … Hl)
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_minus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_minus.ma
new file mode 100644 (file)
index 0000000..8eb1a55
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_minus.ma
@@ -0,0 +1,76 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rminus_3.ma".
+include "basic_2A/multiple/mr2.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+inductive minuss: nat → relation (list2 nat nat) ≝
+| minuss_nil: ∀i. minuss i (◊) (◊)
+| minuss_lt : ∀cs1,cs2,l,m,i. i < l → minuss i cs1 cs2 →
+              minuss i ({l, m} @ cs1) ({l - i, m} @ cs2)
+| minuss_ge : ∀cs1,cs2,l,m,i. l ≤ i → minuss (m + i) cs1 cs2 →
+              minuss i ({l, m} @ cs1) cs2
+.
+
+interpretation "minus (multiple relocation with pairs)"
+   'RMinus cs1 i cs2 = (minuss i cs1 cs2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact minuss_inv_nil1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 → cs1 = ◊ → cs2 = ◊.
+#cs1 #cs2 #i * -cs1 -cs2 -i
+[ //
+| #cs1 #cs2 #l #m #i #_ #_ #H destruct
+| #cs1 #cs2 #l #m #i #_ #_ #H destruct
+]
+qed-.
+
+lemma minuss_inv_nil1: ∀cs2,i. ◊ ▭ i ≡ cs2 → cs2 = ◊.
+/2 width=4 by minuss_inv_nil1_aux/ qed-.
+
+fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 →
+                           ∀l,m,cs. cs1 = {l, m} @ cs →
+                           l ≤ i ∧ cs ▭ m + i ≡ cs2 ∨
+                           ∃∃cs0. i < l & cs ▭ i ≡ cs0 &
+                                   cs2 = {l - i, m} @ cs0.
+#cs1 #cs2 #i * -cs1 -cs2 -i
+[ #i #l #m #cs #H destruct
+| #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #cs1 #cs #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
+]
+qed-.
+
+lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                        l ≤ i ∧ cs1 ▭ m + i ≡ cs2 ∨
+                        ∃∃cs. i < l & cs1 ▭ i ≡ cs &
+                               cs2 = {l - i, m} @ cs.
+/2 width=3 by minuss_inv_cons1_aux/ qed-.
+
+lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                           l ≤ i → cs1 ▭ m + i ≡ cs2.
+#cs1 #cs2 #l #m #i #H
+elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
+lapply (lt_to_le_to_lt … Hil Hli) -Hil -Hli #Hi
+elim (lt_refl_false … Hi)
+qed-.
+
+lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                           i < l →
+                           ∃∃cs. cs1 ▭ i ≡ cs & cs2 = {l - i, m} @ cs.
+#cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
+#Hli #_ #Hil lapply (lt_to_le_to_lt … Hil Hli) -Hil -Hli
+#Hi elim (lt_refl_false … Hi)
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_mr2.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_mr2.ma
new file mode 100644 (file)
index 0000000..20702f6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_mr2.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/mr2.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem at_mono: ∀cs,i,i1. @⦃i, cs⦄ ≡ i1 → ∀i2. @⦃i, cs⦄ ≡ i2 → i1 = i2.
+#cs #i #i1 #H elim H -cs -i -i1
+[ #i #x #H <(at_inv_nil … H) -x //
+| #cs #l #m #i #i1 #Hil #_ #IHi1 #x #H
+  lapply (at_inv_cons_lt … H Hil) -H -Hil /2 width=1 by/
+| #cs #l #m #i #i1 #Hli #_ #IHi1 #x #H
+  lapply (at_inv_cons_ge … H Hli) -H -Hli /2 width=1 by/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_plus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_plus.ma
new file mode 100644 (file)
index 0000000..fe66fa1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/multiple/mr2_plus.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/mr2.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+let rec pluss (cs:list2 nat nat) (i:nat) on cs ≝ match cs with
+[ nil2          ⇒ ◊
+| cons2 l m cs ⇒ {l + i, m} @ pluss cs i
+].
+
+interpretation "plus (multiple relocation with pairs)"
+   'plus x y = (pluss x y).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma pluss_inv_nil2: ∀i,cs. cs + i = ◊ → cs = ◊.
+#i * // normalize
+#l #m #cs #H destruct
+qed.
+
+lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l, m} @ cs2 →
+                       ∃∃cs1. cs1 + i = cs2 & cs = {l - i, m} @ cs1.
+#i #l #m #cs2 * normalize
+[ #H destruct
+| #l1 #m1 #cs1 #H destruct /2 width=3 by ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabbr_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabbr_2.ma
new file mode 100644 (file)
index 0000000..afd2f18
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabbr_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 . break ⓓ T2 )"
+ left associative with precedence 48
+ for @{ 'DxAbbr $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabst_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabst_2.ma
new file mode 100644 (file)
index 0000000..43e324e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxabst_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 . break ⓛ T2 )"
+ left associative with precedence 49
+ for @{ 'DxAbst $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxbind2_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxbind2_3.ma
new file mode 100644 (file)
index 0000000..bbded1e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/dxbind2_3.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation > "hvbox( T . break ②{ term 46 I } break term 47 T1 )"
+ non associative with precedence 46
+ for @{ 'DxBind2 $T $I $T1 }.
+
+notation "hvbox( T . break ⓑ { term 46 I } break term 48 T1 )"
+ non associative with precedence 47
+ for @{ 'DxBind2 $T $I $T1 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/gref_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/gref_1.ma
new file mode 100644 (file)
index 0000000..d227488
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/gref_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( § term 90 p )"
+ non associative with precedence 55
+ for @{ 'GRef $p }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_0.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_0.ma
new file mode 100644 (file)
index 0000000..b6e471d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "⓪"
+ non associative with precedence 55
+ for @{ 'Item0 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_1.ma
new file mode 100644 (file)
index 0000000..2c9e41a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/item0_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓪ { term 46 I } )"
+ non associative with precedence 55
+ for @{ 'Item0 $I }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/lref_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/lref_1.ma
new file mode 100644 (file)
index 0000000..ebab03c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/lref_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( # term 90 i )"
+ non associative with precedence 55
+ for @{ 'LRef $i }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbr_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbr_3.ma
new file mode 100644 (file)
index 0000000..4411605
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbr_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓓ { term 46 a } break term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbr $a $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrneg_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrneg_2.ma
new file mode 100644 (file)
index 0000000..13786d9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrneg_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓓ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbrNeg $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrpos_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrpos_2.ma
new file mode 100644 (file)
index 0000000..ae76f19
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabbrpos_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓓ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbrPos $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabst_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabst_3.ma
new file mode 100644 (file)
index 0000000..8ba3449
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabst_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓛ { term 46 a } break term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbst $a $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstneg_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstneg_2.ma
new file mode 100644 (file)
index 0000000..277b58b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstneg_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓛ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbstNeg $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstpos_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstpos_2.ma
new file mode 100644 (file)
index 0000000..9e22bc7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snabstpos_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓛ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbstPos $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snappl_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snappl_2.ma
new file mode 100644 (file)
index 0000000..c1acadb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snappl_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓐ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAppl $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2_4.ma
new file mode 100644 (file)
index 0000000..ceae735
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓑ { term 46 a , break term 46 I } break term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2 $a $I $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2neg_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2neg_3.ma
new file mode 100644 (file)
index 0000000..428fe73
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2neg_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓑ { term 46 I } break term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2Neg $I $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2pos_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2pos_3.ma
new file mode 100644 (file)
index 0000000..b89b958
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snbind2pos_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓑ { term 46 I } break term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2Pos $I $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/sncast_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/sncast_2.ma
new file mode 100644 (file)
index 0000000..55565d0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/sncast_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓝ term 55 T1 . break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnCast $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snflat2_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snflat2_3.ma
new file mode 100644 (file)
index 0000000..ef0bdb8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snflat2_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓕ { term 46 I } break term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnFlat2 $I $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_2.ma
new file mode 100644 (file)
index 0000000..82044ea
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ② term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnItem2 $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_3.ma
new file mode 100644 (file)
index 0000000..9fdf70b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/snitem2_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ② { term 46 I } break term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnItem2 $I $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_0.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_0.ma
new file mode 100644 (file)
index 0000000..8cf6da5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "⋆"
+ non associative with precedence 46
+ for @{ 'Star }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_1.ma
new file mode 100644 (file)
index 0000000..6307ed9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/constructors/star_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⋆ term 90 k )"
+ non associative with precedence 55
+ for @{ 'Star $k }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabbr_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabbr_2.ma
new file mode 100644 (file)
index 0000000..4051c20
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabbr_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓓ term 55 T . break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnAbbr $T $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabst_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabst_2.ma
new file mode 100644 (file)
index 0000000..838bd00
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snabst_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓛ term 55 T . break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnAbst $T $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snapplvector_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snapplvector_2.ma
new file mode 100644 (file)
index 0000000..e595202
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snapplvector_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( Ⓐ term 55 T1 . break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnApplVector $T1 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snbind2_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snbind2_3.ma
new file mode 100644 (file)
index 0000000..fa44f29
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/snbind2_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓑ { term 46 I } break term 55 T . break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnBind2 $I $T $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_1.ma
new file mode 100644 (file)
index 0000000..31b4fe6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 x } )"
+ non associative with precedence 90
+ for @{ 'Weight $x }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_2.ma
new file mode 100644 (file)
index 0000000..a2b9b47
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 L , break term 46 T } )"
+ non associative with precedence 90
+ for @{ 'Weight $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_3.ma
new file mode 100644 (file)
index 0000000..42b3c60
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/functions/weight_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 G , break term 46 L , break term 46 T } )"
+ non associative with precedence 90
+ for @{ 'Weight $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/notation.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/notation.ma
new file mode 100644 (file)
index 0000000..0996c7c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/notation.ma
@@ -0,0 +1,47 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃G, L⦄ ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
+   non associative with precedence 45
+   for @{ 'ICM $L $T $k }.
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )"
+   non associative with precedence 45
+   for @{ 'BinaryArity $h $L $T $A }.
+
+notation "hvbox( h ⊢ break term 46 L1 ÷ ⫃ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqB $h $L1 $L2 }.
+
+notation "hvbox( L1 ⊢ ⬌ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PConvSn $L1 $L2 }.
+
+notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PConvSnStar $L1 $L2 }.
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'NativeType $h $L $T1 $T2 }.
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'NativeTypeAlt $h $L $T1 $T2 }.
+
+notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'NativeTypeStar $h $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/atomicarity_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/atomicarity_4.ma
new file mode 100644 (file)
index 0000000..e6a1b09
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/atomicarity_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃term 46 G, break term 46 L⦄ ⊢ break term 46 T ⁝ break term 46 A )"
+   non associative with precedence 45
+   for @{ 'AtomicArity $G $L $T $A }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpred_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpred_8.ma
new file mode 100644 (file)
index 0000000..80e40a1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpred_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRed $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredalt_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredalt_8.ma
new file mode 100644 (file)
index 0000000..c3f111a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredalt_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ ≽ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRedAlt $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredproper_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredproper_8.ma
new file mode 100644 (file)
index 0000000..bb6d3d1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredproper_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstar_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstar_8.ma
new file mode 100644 (file)
index 0000000..3943a33
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstar_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRedStar $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstaralt_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstaralt_8.ma
new file mode 100644 (file)
index 0000000..227a666
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btpredstaralt_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ ≥ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRedStarAlt $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsn_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsn_5.ma
new file mode 100644 (file)
index 0000000..207ba13
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsn_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦥ [ term 46 h, break term 46 g ] break ⦃ term 46 G, break term 46 L, break term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTSN $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsnalt_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsnalt_5.ma
new file mode 100644 (file)
index 0000000..7c6d691
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/btsnalt_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦥ ⦥ [ term 46 h, break term 46 g ] break ⦃ term 46 G, break term 46 L, break term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTSNAlt $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/cosn_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/cosn_5.ma
new file mode 100644 (file)
index 0000000..be4f9a4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/cosn_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ~ ⬊ * break [ term 46 h , break term 46 g , break term 46 l ] break term 46 L )"
+   non associative with precedence 45
+   for @{ 'CoSN $h $g $l $G $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/degree_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/degree_6.ma
new file mode 100644 (file)
index 0000000..7602a65
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/degree_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ▪ break [ term 46 h , break term 46 g ] break term 46 d )"
+   non associative with precedence 45
+   for @{ 'Degree $h $g $G $L $T $d }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpconvstar_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpconvstar_8.ma
new file mode 100644 (file)
index 0000000..9aa3568
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpconvstar_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • * ⬌ * break [ term 46 h , break term 46 g , break term 46 n1 , break term 46 n2 ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'DPConvStar $h $g $n1 $n2 $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpredstar_7.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpredstar_7.ma
new file mode 100644 (file)
index 0000000..c5f55ae
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/dpredstar_7.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • * ➡ * break [ term 46 h , break term 46 g , break term 46 n ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'DPRedStar $h $g $n $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/freestar_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/freestar_4.ma
new file mode 100644 (file)
index 0000000..e464806
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/freestar_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L ⊢ break term 46 i ϵ 𝐅 * [ break term 46 l ] ⦃ break term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'FreeStar $L $i $l $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/ineint_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/ineint_5.ma
new file mode 100644 (file)
index 0000000..83f23ef
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/ineint_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L, break term 46 T ⦄ ϵ break [ term 46 R ] break 〚term 46  A 〛 )"
+   non associative with precedence 45
+   for @{ 'InEInt $R $G $L $T $A }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazybtpredstarproper_8.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazybtpredstarproper_8.ma
new file mode 100644 (file)
index 0000000..e56e02e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazybtpredstarproper_8.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ >≡ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'LazyBTPRedStarProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_4.ma
new file mode 100644 (file)
index 0000000..a6b02b3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ≡ break [ term 46 T , break term 46 l ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LazyEq $T $l $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_7.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_7.ma
new file mode 100644 (file)
index 0000000..79a79e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyeq_7.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≡ break [ term 46 l ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'LazyEq $l $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyor_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyor_5.ma
new file mode 100644 (file)
index 0000000..6479c1f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lazyor_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⋓ break [ term 46 T , break term 46 l ] break term 46 L2 ≡ break term 46 L )"
+   non associative with precedence 45
+   for @{ 'LazyOr $L1 $T $l $L2 $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeq_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeq_4.ma
new file mode 100644 (file)
index 0000000..994e6b2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeq_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEq $L1 $l $m $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqa_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqa_3.ma
new file mode 100644 (file)
index 0000000..c3b74a9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqa_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⫃ ⁝ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqA $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_2.ma
new file mode 100644 (file)
index 0000000..3a6e5ed
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⫃ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqC $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_4.ma
new file mode 100644 (file)
index 0000000..823d6be
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqc_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⫃ break [ term 46 R ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqC $R $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqd_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqd_5.ma
new file mode 100644 (file)
index 0000000..d90e351
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqd_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⫃ ▪ break [ term 46 h, break term 46 g ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqD $h $g $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqv_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqv_5.ma
new file mode 100644 (file)
index 0000000..f6595e1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/lrsubeqv_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⫃ ¡ break [ term 46 h, break term 46 g ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'LRSubEqV $h $g $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/midiso_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/midiso_4.ma
new file mode 100644 (file)
index 0000000..dc380d8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/midiso_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⩬ break [ term 46 l , break term 46 m ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'MidIso $l $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_5.ma
new file mode 100644 (file)
index 0000000..d6426f6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 h, break term 46 g ] )"
+   non associative with precedence 45
+   for @{ 'NativeValid $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_6.ma
new file mode 100644 (file)
index 0000000..449fe14
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/nativevalid_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 h , break term 46 g , break term 46 d ] )"
+   non associative with precedence 45
+   for @{ 'NativeValid $h $g $d $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconv_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconv_4.ma
new file mode 100644 (file)
index 0000000..a7563d0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconv_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⬌ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PConv $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconvstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconvstar_4.ma
new file mode 100644 (file)
index 0000000..f263e5d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pconvstar_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⬌* break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PConvStar $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_4.ma
new file mode 100644 (file)
index 0000000..397ac93
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 ➡ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRed $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_6.ma
new file mode 100644 (file)
index 0000000..1a423e4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/pred_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ break [ term 46 h , break term 46 g ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRed $h $g $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_4.ma
new file mode 100644 (file)
index 0000000..e26cfe6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedEval $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_6.ma
new file mode 100644 (file)
index 0000000..8360d14
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predeval_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break [ term 46 h , break term 46 g ] break 𝐍 ⦃ term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedEval $h $g $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_3.ma
new file mode 100644 (file)
index 0000000..a8806a1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐍 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedNormal $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_5.ma
new file mode 100644 (file)
index 0000000..d1e001d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednormal_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 h , break term 46 g ] 𝐍 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedNormal $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_3.ma
new file mode 100644 (file)
index 0000000..4ad7cbc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐈 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedNotReducible $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_5.ma
new file mode 100644 (file)
index 0000000..cb33d31
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/prednotreducible_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 h , break term 46 g ] 𝐈 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedNotReducible $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_3.ma
new file mode 100644 (file)
index 0000000..5d84a36
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐑 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedReducible $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_5.ma
new file mode 100644 (file)
index 0000000..82ded9c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predreducible_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 h , break term 46 g ] 𝐑 break ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'PRedReducible $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_3.ma
new file mode 100644 (file)
index 0000000..d06064a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PRedSn $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_5.ma
new file mode 100644 (file)
index 0000000..2065660
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsn_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ break [ term 46 h , break term 46 g ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PRedSn $h $g $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_3.ma
new file mode 100644 (file)
index 0000000..2bcf326
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ➡* break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PRedSnStar $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_5.ma
new file mode 100644 (file)
index 0000000..aef4f27
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predsnstar_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ * break [ term 46 h , break term 46 g ] break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'PRedSnStar $h $g $G $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_4.ma
new file mode 100644 (file)
index 0000000..2498427
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , term 46 L ⦄ ⊢ break term 46 T1 ➡ * break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRedStar $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_6.ma
new file mode 100644 (file)
index 0000000..54ac666
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/predstar_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break [ term 46 h , break term 46 g ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PRedStar $h $g $G $L $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubst_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubst_6.ma
new file mode 100644 (file)
index 0000000..e35f7ca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubst_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PSubst $G $L $T1 $l $m $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststar_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststar_6.ma
new file mode 100644 (file)
index 0000000..2dd1f2d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststar_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PSubstStar $G $L $T1 $l $m $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststaralt_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststaralt_6.ma
new file mode 100644 (file)
index 0000000..97227dc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/psubststaralt_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'PSubstStarAlt $G $L $T1 $l $m $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rat_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rat_3.ma
new file mode 100644 (file)
index 0000000..a63dc95
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rat_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( @ ⦃ term 46 T1 , break term 46 f ⦄ ≡ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'RAt $T1 $f $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_3.ma
new file mode 100644 (file)
index 0000000..a761d47
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'RDrop $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_4.ma
new file mode 100644 (file)
index 0000000..2345f06
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ [ term 46 l , break term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'RDrop $l $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_5.ma
new file mode 100644 (file)
index 0000000..d8c0672
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdrop_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ [ term 46 s , break term 46 l , break term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'RDrop $s $l $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_3.ma
new file mode 100644 (file)
index 0000000..8a6f2f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ * [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'RDropStar $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_4.ma
new file mode 100644 (file)
index 0000000..291de4c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rdropstar_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ * [ term 46 s , break term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'RDropStar $s $m $L1 $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rlift_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rlift_4.ma
new file mode 100644 (file)
index 0000000..e2a0328
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rlift_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬆ [ term 46 l , break term 46 m ] break term 46 T1 ≡ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'RLift $l $m $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rliftstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rliftstar_3.ma
new file mode 100644 (file)
index 0000000..af4de47
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rliftstar_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬆ * [ term 46 m ] break term 46 T1 ≡ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'RLiftStar $m $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rminus_3.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rminus_3.ma
new file mode 100644 (file)
index 0000000..c896a98
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/rminus_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ▭ break term 46 T2 ≡ break term 46 T )"
+   non associative with precedence 45
+   for @{ 'RMinus $T1 $T2 $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/simple_1.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/simple_1.ma
new file mode 100644 (file)
index 0000000..0acfb4e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/simple_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( 𝐒 ⦃ term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'Simple $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_5.ma
new file mode 100644 (file)
index 0000000..bfc7d0c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬊ * break [ term 46 h , break term 46 g ] break term 46 T )"
+   non associative with precedence 45
+   for @{ 'SN $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_6.ma
new file mode 100644 (file)
index 0000000..c402cdc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/sn_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 l ] break term 46 L )"
+   non associative with precedence 45
+   for @{ 'SN $h $g $T $l $G $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_5.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_5.ma
new file mode 100644 (file)
index 0000000..13fed06
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_5.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬊ ⬊ * break [ term 46 h , break term 46 g ] break term 46 T )"
+   non associative with precedence 45
+   for @{ 'SNAlt $h $g $G $L $T }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_6.ma
new file mode 100644 (file)
index 0000000..4beb9fd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/snalt_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ⬊ ⬊ * break [ term 46 h , break term 46 g , break term 46 T , break term 46 l ] break term 46 L )"
+   non associative with precedence 45
+   for @{ 'SNAlt $h $g $T $l $G $L }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/statictypestar_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/statictypestar_6.ma
new file mode 100644 (file)
index 0000000..b7d6a71
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/statictypestar_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 •* break [ term 46 h , break term 46 n ] break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'StaticTypeStar $h $G $L $n $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/supterm_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/supterm_6.ma
new file mode 100644 (file)
index 0000000..1300f0f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/supterm_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTerm $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermopt_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermopt_6.ma
new file mode 100644 (file)
index 0000000..8ed1123
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermopt_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⸮ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTermOpt $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermoptalt_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermoptalt_6.ma
new file mode 100644 (file)
index 0000000..7748c18
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermoptalt_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⊐⸮ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTermOptAlt $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermplus_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermplus_6.ma
new file mode 100644 (file)
index 0000000..533b88c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermplus_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + break ⦃ term 46 G2, term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermstar_6.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermstar_6.ma
new file mode 100644 (file)
index 0000000..f2abfc2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/suptermstar_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * break ⦃ term 46 G2, term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/topiso_2.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/topiso_2.ma
new file mode 100644 (file)
index 0000000..059b437
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/topiso_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ≂ break term 46 T2 )"
+   non associative with precedence 45
+   for @{ 'TopIso $T1 $T2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/unfold_4.ma b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/unfold_4.ma
new file mode 100644 (file)
index 0000000..3381c56
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/notation/relations/unfold_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ⧫ * break term 46 T ≡ break term 46 L2 )"
+   non associative with precedence 45
+   for @{ 'Unfold $G $L1 $T $L2 }.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir.ma
new file mode 100644 (file)
index 0000000..6868ce3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir.ma
@@ -0,0 +1,79 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/prednotreducible_3.ma".
+include "basic_2A/reduction/crr.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************)
+
+definition cir: relation3 genv lenv term ≝ λG,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⊥.
+
+interpretation "irreducibility for context-sensitive reduction (term)"
+   'PRedNotReducible G L T = (cir G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cir_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐈⦃#i⦄ → ⊥.
+/3 width=3 by crr_delta/ qed-.
+
+lemma cir_inv_ri2: ∀I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃②{I}V.T⦄ → ⊥.
+/3 width=1 by crr_ri2/ qed-.
+
+lemma cir_inv_ib2: ∀a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄.
+/4 width=1 by crr_ib2_sn, crr_ib2_dx, conj/ qed-.
+
+lemma cir_inv_bind: ∀a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ & ib2 a I.
+#a * [ elim a -a ]
+#G #L #V #T #H [ elim H -H /3 width=1 by crr_ri2, or_introl/ ]
+elim (cir_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
+qed-.
+
+lemma cir_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#G #L #V #T #HVT @and3_intro /3 width=1 by crr_appl_sn, crr_appl_dx/
+generalize in match HVT; -HVT elim T -T //
+* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crr_beta, crr_theta/
+qed-.
+
+lemma cir_inv_flat: ∀I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+* #G #L #V #T #H
+[ elim (cir_inv_appl … H) -H /2 width=1 by and4_intro/
+| elim (cir_inv_ri2 … H) -H //
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cir_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆k⦄.
+/2 width=4 by crr_inv_sort/ qed.
+
+lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐈⦃§p⦄.
+/2 width=4 by crr_inv_gref/ qed.
+
+lemma tir_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃⓪{I}⦄.
+/2 width=3 by trr_inv_atom/ qed.
+
+lemma cir_ib2: ∀a,I,G,L,V,T.
+               ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄.
+#a #I #G #L #V #T #HI #HV #HT #H
+elim (crr_inv_ib2 … HI H) -HI -H /2 width=1 by/
+qed.
+
+lemma cir_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄.
+#G #L #V #T #HV #HT #H1 #H2
+elim (crr_inv_appl … H2) -H2 /2 width=1 by/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir_lift.ma
new file mode 100644 (file)
index 0000000..672cf11
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cir_lift.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/crr_lift.ma".
+include "basic_2A/reduction/cir.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************)
+
+(* Properties on relocation *************************************************)
+
+lemma cir_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄.
+/3 width=8 by crr_inv_lift/ qed.
+
+lemma cir_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                    ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=8 by crr_lift/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix.ma
new file mode 100644 (file)
index 0000000..bcce8ae
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/prednotreducible_5.ma".
+include "basic_2A/reduction/cir.ma".
+include "basic_2A/reduction/crx.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
+
+definition cix: ∀h. sd h → relation3 genv lenv term ≝
+                λh,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⊥.
+
+interpretation "irreducibility for context-sensitive extended reduction (term)"
+   'PRedNotReducible h g G L T = (cix h g G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cix_inv_sort: ∀h,g,G,L,k,d. deg h g k (d+1) → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄ → ⊥.
+/3 width=2 by crx_sort/ qed-.
+
+lemma cix_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄ → ⊥.
+/3 width=4 by crx_delta/ qed-.
+
+lemma cix_inv_ri2: ∀h,g,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃②{I}V.T⦄ → ⊥.
+/3 width=1 by crx_ri2/ qed-.
+
+lemma cix_inv_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+/4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-.
+
+lemma cix_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & ib2 a I.
+#h #g #a * [ elim a -a ]
+#G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ]
+elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
+qed-.
+
+lemma cix_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#h #g #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/
+generalize in match HVT; -HVT elim T -T //
+* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/
+qed-.
+
+lemma cix_inv_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+#h #g * #G #L #V #T #H
+[ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/
+| elim (cix_inv_ri2 … H) -H //
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cix_inv_cir: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
+/3 width=1 by crr_crx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cix_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄.
+#h #g #G #L #k #Hk #H elim (crx_inv_sort … H) -L #d #Hkd
+lapply (deg_mono … Hk Hkd) -h -k <plus_n_Sm #H destruct
+qed.
+
+lemma tix_lref: ∀h,g,G,i. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄.
+#h #g #G #i #H elim (trx_inv_atom … H) -H #k #d #_ #H destruct
+qed.
+
+lemma cix_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃§p⦄.
+#h #g #G #L #p #H elim (crx_inv_gref … H)
+qed.
+
+lemma cix_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ →
+                               ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄.
+#h #g #a #I #G #L #V #T #HI #HV #HT #H
+elim (crx_inv_ib2 … HI H) -HI -H /2 width=1 by/
+qed.
+
+lemma cix_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄.
+#h #g #G #L #V #T #HV #HT #H1 #H2
+elim (crx_inv_appl … H2) -H2 /2 width=1 by/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix_lift.ma
new file mode 100644 (file)
index 0000000..43cb45c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cix_lift.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/crx_lift.ma".
+include "basic_2A/reduction/cix.ma".
+
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma cix_lref: ∀h,g,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄.
+#h #g #G #L #i #HL #H elim (crx_inv_lref … H) -h #I #K #V #HLK
+lapply (drop_mono … HLK … HL) -L -i #H destruct
+qed.
+
+(* Properties on relocation *************************************************)
+
+lemma cix_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃U⦄.
+/3 width=8 by crx_inv_lift/ qed.
+
+lemma cix_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                    ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+/3 width=8 by crx_lift/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr.ma
new file mode 100644 (file)
index 0000000..f1cdcf6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr.ma
@@ -0,0 +1,114 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/prednormal_3.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
+
+definition cnr: relation3 genv lenv term ≝ λG,L. NF … (cpr G L) (eq …).
+
+interpretation
+   "normality for context-sensitive reduction (term)"
+   'PRedNormal G L T = (cnr G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cnr_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄ → ⊥.
+#G #L #K #V #i #HLK #H
+elim (lift_total V 0 (i+1)) #W #HVW
+lapply (H W ?) -H [ /3 width=6 by cpr_delta/ ] -HLK #H destruct
+elim (lift_inv_lref2_be … HVW) -HVW //
+qed-.
+
+lemma cnr_inv_abst: ∀a,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}V.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡ 𝐍⦃T⦄.
+#a #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnr_inv_abbr: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃-ⓓV.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡ 𝐍⦃T⦄.
+#G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnr_inv_zeta: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃+ⓓV.T⦄ → ⊥.
+#G #L #V #T #H elim (is_lift_dec T 0 1)
+[ * #U #HTU
+  lapply (H U ?) -H /2 width=3 by cpr_zeta/ #H destruct
+  elim (lift_inv_pair_xy_y … HTU)
+| #HT
+  elim (cpr_delift G (⋆) V T (⋆. ⓓV) 0) //
+  #T2 #T1 #HT2 #HT12 lapply (H (+ⓓV.T2) ?) -H /4 width=1 by tpr_cpr, cpr_bind/ -HT2
+  #H destruct /3 width=2 by ex_intro/
+]
+qed-.
+
+lemma cnr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#G #L #V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
+| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
+  [ elim (lift_total V1 0 1) #V2 #HV12
+    lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by tpr_cpr, cpr_theta/ -HV12 #H destruct
+  | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by tpr_cpr, cpr_beta/ #H destruct
+]
+qed-.
+
+lemma cnr_inv_eps: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓝV.T⦄ → ⊥.
+#G #L #V #T #H lapply (H T ?) -H
+/2 width=4 by cpr_eps, discr_tpair_xy_y/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_sort *)
+lemma cnr_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆k⦄.
+#G #L #k #X #H
+>(cpr_inv_sort1 … H) //
+qed.
+
+lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
+#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // *
+#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+(* Basic_1: was only: nf2_csort_lref *)
+lemma cnr_lref_atom: ∀G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
+#G #L #i #HL @cnr_lref_free >(drop_fwd_length … HL) -HL //
+qed.
+
+(* Basic_1: was: nf2_abst *)
+lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄.
+#a #G #L #W #T #HW #HT #X #H
+elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+>(HW … HW0) -W0 >(HT … HT0) -T0 //
+qed.
+
+(* Basic_1: was only: nf2_appl_lref *)
+lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄.
+#G #L #V #T #HV #HT #HS #X #H
+elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
+>(HV … HV0) -V0 >(HT … HT0) -T0 //
+qed.
+
+(* Basic_1: was: nf2_dec *)
+axiom cnr_dec: ∀G,L,T1. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T1⦄ ∨
+               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
+
+(* Basic_1: removed theorems 1: nf2_abst_shift *)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma
new file mode 100644 (file)
index 0000000..786611f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_cir.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpr_cir.ma".
+include "basic_2A/reduction/cnr_crr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
+
+(* Main properties on irreducibility ****************************************)
+
+theorem cir_cnr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+/2 width=4 by cpr_fwd_cir/ qed.
+
+(* Main inversion lemmas on irreducibility **********************************)
+
+theorem cnr_inv_cir: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
+/2 width=5 by cnr_inv_crr/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_crr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_crr.ma
new file mode 100644 (file)
index 0000000..e8b5dfb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_crr.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/crr.ma".
+include "basic_2A/reduction/cnr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
+
+(* Advanced inversion lemmas on reducibility ********************************)
+
+(* Note: this property is unusual *)
+lemma cnr_inv_crr: ∀G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → ⊥.
+#G #L #T #H elim H -L -T
+[ #L #K #V #i #HLK #H
+  elim (cnr_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+  elim (cnr_inv_appl … H) -H /2 width=1 by/
+| #L #V #T #_ #IHT #H
+  elim (cnr_inv_appl … H) -H /2 width=1 by/
+| #I #L #V #T * #H1 #H2 destruct
+  [ elim (cnr_inv_zeta … H2)
+  | elim (cnr_inv_eps … H2)
+  ]
+|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+  [1,3: elim (cnr_inv_abbr … H2) -H2 /2 width=1 by/
+  |*: elim (cnr_inv_abst … H2) -H2 /2 width=1 by/
+  ]
+| #a #L #V #W #T #H
+  elim (cnr_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+  elim (cnr_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma
new file mode 100644 (file)
index 0000000..4ff641f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnr_lift.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpr_lift.ma".
+include "basic_2A/reduction/cnr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was: nf2_lref_abst *)
+lemma cnr_lref_abst: ∀G,L,K,V,i. ⬇[i] L ≡ K. ⓛV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
+#G #L #K #V #i #HLK #X #H
+elim (cpr_inv_lref1 … H) -H // *
+#K0 #V1 #V2 #HLK0 #_ #_
+lapply (drop_mono … HLK … HLK0) -L #H destruct
+qed.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: nf2_lift *)
+lemma cnr_lift: ∀G,L0,L,T,T0,s,l,m. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ →
+                ⬇[s, l, m] L0 ≡ L → ⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄.
+#G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
+elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
+<(HLT … HT1) in HT0; -L #HT0
+>(lift_mono … HT10 … HT0) -T1 -X //
+qed.
+
+(* Note: this was missing in basic_1 *)
+lemma cnr_inv_lift: ∀G,L0,L,T,T0,s,l,m. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ →
+                    ⬇[s, l, m] L0 ≡ L → ⬆[l, m] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+#G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l m) #X0 #HX0
+lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0
+>(HLT0 … HTX0) in HX0; -L0 -X0 #H
+>(lift_inj … H … HT0) -T0 -X -s -l -m //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma
new file mode 100644 (file)
index 0000000..e754c0a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx.ma
@@ -0,0 +1,136 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/prednormal_5.ma".
+include "basic_2A/reduction/cnr.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
+
+definition cnx: ∀h. sd h → relation3 genv lenv term ≝
+                λh,g,G,L. NF … (cpx h g G L) (eq …).
+
+interpretation
+   "normality for context-sensitive extended reduction (term)"
+   'PRedNormal h g L T = (cnx h g L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cnx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄ → deg h g k 0.
+#h #g #G #L #k #H elim (deg_total h g k)
+#d @(nat_ind_plus … d) -d // #d #_ #Hkd
+lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H
+lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *)
+lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *)
+lapply (next_lt h k) >H -H #H elim (lt_refl_false … H)
+qed-.
+
+lemma cnx_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄ → ⊥.
+#h #g #I #G #L #K #V #i #HLK #H
+elim (lift_total V 0 (i+1)) #W #HVW
+lapply (H W ?) -H [ /3 width=7 by cpx_delta/ ] -HLK #H destruct
+elim (lift_inv_lref2_be … HVW) -HVW //
+qed-.
+
+lemma cnx_inv_abst: ∀h,g,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}V.T⦄ →
+                    ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
+#h #g #a #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnx_inv_abbr: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃-ⓓV.T⦄ →
+                    ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
+#h #g #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
+]
+qed-.
+
+lemma cnx_inv_zeta: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃+ⓓV.T⦄ → ⊥.
+#h #g #G #L #V #T #H elim (is_lift_dec T 0 1)
+[ * #U #HTU
+  lapply (H U ?) -H /2 width=3 by cpx_zeta/ #H destruct
+  elim (lift_inv_pair_xy_y … HTU)
+| #HT
+  elim (cpr_delift G(⋆) V T (⋆.ⓓV) 0) // #T2 #T1 #HT2 #HT12
+  lapply (H (+ⓓV.T2) ?) -H /5 width=1 by cpr_cpx, tpr_cpr, cpr_bind/ -HT2
+  #H destruct /3 width=2 by ex_intro/
+]
+qed-.
+
+lemma cnx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#h #g #G #L #V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2 #H destruct //
+| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
+  [ elim (lift_total V1 0 1) #V2 #HV12
+    lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by cpr_cpx, cpr_theta/ -HV12 #H destruct
+  | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by cpr_cpx, cpr_beta/ #H destruct
+  ]
+]
+qed-.
+
+lemma cnx_inv_eps: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #g #G #L #V #T #H lapply (H T ?) -H
+/2 width=4 by cpx_eps, discr_tpair_xy_y/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cnx_fwd_cnr: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+#h #g #G #L #T #H #U #HTU
+@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *)
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cnx_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄.
+#h #g #G #L #k #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_
+lapply (deg_mono … Hkd Hk) -h -L <plus_n_Sm #H destruct
+qed.
+
+lemma cnx_sort_iter: ∀h,g,G,L,k,d. deg h g k d → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆((next h)^d k)⦄.
+#h #g #G #L #k #d #Hkd
+lapply (deg_iter … d Hkd) -Hkd <minus_n_n /2 width=6 by cnx_sort/
+qed.
+
+lemma cnx_lref_free: ∀h,g,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
+#h #g #G #L #i #Hi #X #H elim (cpx_inv_lref1 … H) -H // *
+#I #K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+lemma cnx_lref_atom: ∀h,g,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
+#h #g #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
+qed.
+
+lemma cnx_abst: ∀h,g,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
+                ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}W.T⦄.
+#h #g #a #G #L #W #T #HW #HT #X #H
+elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+>(HW … HW0) -W0 >(HT … HT0) -T0 //
+qed.
+
+lemma cnx_appl_simple: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
+                       ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄.
+#h #g #G #L #V #T #HV #HT #HS #X #H
+elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
+>(HV … HV0) -V0 >(HT … HT0) -T0 //
+qed.
+
+axiom cnx_dec: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T1⦄ ∨
+               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & (T1 = T2 → ⊥).

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma
new file mode 100644 (file)
index 0000000..40f69e0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_cix.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_cix.ma".
+include "basic_2A/reduction/cnx_crx.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
+
+(* Main properties on irreducibility ****************************************)
+
+theorem cix_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
+/2 width=6 by cpx_fwd_cix/ qed.
+
+(* Main inversion lemmas on irreducibility **********************************)
+
+theorem cnx_inv_cix: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+/2 width=7 by cnx_inv_crx/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_crx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_crx.ma
new file mode 100644 (file)
index 0000000..2be5d0d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_crx.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/crx.ma".
+include "basic_2A/reduction/cnx.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
+
+(* Advanced inversion lemmas on reducibility ********************************)
+
+(* Note: this property is unusual *)
+lemma cnx_inv_crx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⊥.
+#h #g #G #L #T #H elim H -L -T
+[ #L #k #d #Hkd #H
+  lapply (cnx_inv_sort … H) -H #H
+  lapply (deg_mono … H Hkd) -h -L -k <plus_n_Sm #H destruct
+| #I #L #K #V #i #HLK #H
+  elim (cnx_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+  elim (cnx_inv_appl … H) -H /2 width=1 by/
+| #L #V #T #_ #IHT #H
+  elim (cnx_inv_appl … H) -H /2 width=1 by/
+| #I #L #V #T * #H1 #H2 destruct
+  [ elim (cnx_inv_zeta … H2)
+  | elim (cnx_inv_eps … H2)
+  ]
+|6,7: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+  [1,3: elim (cnx_inv_abbr … H2) -H2 /2 width=1 by/
+  |*: elim (cnx_inv_abst … H2) -H2 /2 width=1 by/
+  ]
+| #a #L #V #W #T #H
+  elim (cnx_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+  elim (cnx_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma
new file mode 100644 (file)
index 0000000..0839511
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cnx_lift.ma
@@ -0,0 +1,37 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_lift.ma".
+include "basic_2A/reduction/cnx.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
+
+(* Relocation properties ****************************************************)
+
+lemma cnx_lift: ∀h,g,G,L0,L,T,T0,s,l,m. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⬇[s, l, m] L0 ≡ L →
+                ⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄.
+#h #g #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
+elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
+<(HLT … HT1) in HT0; -L #HT0
+>(lift_mono … HT10 … HT0) -T1 -X //
+qed.
+
+lemma cnx_inv_lift: ∀h,g,G,L0,L,T,T0,s,l,m. ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄ → ⬇[s, l, m] L0 ≡ L →
+                    ⬆[l, m] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
+#h #g #G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l m) #X0 #HX0
+lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0
+>(HLT0 … HTX0) in HX0; -L0 -X0 #H
+>(lift_inj … H … HT0) -T0 -X -l -m //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma
new file mode 100644 (file)
index 0000000..d79fba9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr.ma
@@ -0,0 +1,309 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/pred_4.ma".
+include "basic_2A/static/lsubr.ma".
+include "basic_2A/unfold/lstas.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* activate genv *)
+(* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
+(* Note: cpr_flat: does not hold in basic_1 *)
+inductive cpr: relation4 genv lenv term term ≝
+| cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
+| cpr_delta: ∀G,L,K,V,V2,W2,i.
+             ⬇[i] L ≡ K. ⓓV → cpr G K V V2 →
+             ⬆[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
+| cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
+             cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
+             cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpr_flat : ∀I,G,L,V1,V2,T1,T2.
+             cpr G L V1 V2 → cpr G L T1 T2 →
+             cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
+             ⬆[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
+| cpr_eps  : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
+| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
+             cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
+             cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
+| cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+             cpr G L V1 V → ⬆[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
+             cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
+.
+
+interpretation "context-sensitive parallel reduction (term)"
+   'PRed G L T1 T2 = (cpr G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
+#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
+  /3 width=6 by cpr_delta/
+|3,7: /4 width=1 by lsubr_pair, cpr_bind, cpr_beta/
+|4,6: /3 width=1 by cpr_flat, cpr_eps/
+|5,8: /4 width=3 by lsubr_pair, cpr_zeta, cpr_theta/
+]
+qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
+#G #T1 #T2 #HT12 #L
+lapply (lsubr_cpr_trans … HT12 L ?) //
+qed.
+
+(* Basic_1: includes by definition: pr0_refl *)
+lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
+#G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
+qed.
+
+(* Basic_1: was: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
+* /2 width=1 by cpr_bind, cpr_flat/ qed.
+
+lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) →
+                  ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2.
+#G #K #V #T1 elim T1 -T1
+[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
+  #i #L #l #HLK elim (lt_or_eq_or_gt i l)
+  #Hil [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+  destruct
+  elim (lift_total V 0 (i+1)) #W #HVW
+  elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+  [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
+  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
+  ]
+]
+qed-.
+
+fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 →
+                    d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
+/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
+[ #G #L #d #k #H0 destruct normalize //
+| #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
+  /3 width=6 by cpr_delta/
+| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+/2 width=4 by lstas_cpr_aux/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
+                        T2 = ⓪{I} ∨
+                        ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                                    ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
+#G #L #T1 #T2 * -G -L -T1 -T2
+[ #I #G #L #J #H destruct /2 width=1 by or_introl/
+| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T2 #_ #J #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
+                     T2 = ⓪{I} ∨
+                     ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                                 ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
+/2 width=3 by cpr_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
+lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+#G #L #T2 #k #H
+elim (cpr_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
+                     T2 = #i ∨
+                     ∃∃K,V,V2. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                               ⬆[O, i + 1] V2 ≡ T2.
+#G #L #T2 #i #H
+elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
+qed-.
+
+lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
+#G #L #T2 #p #H
+elim (cpr_inv_atom1 … H) -H //
+* #K #V #V2 #i #_ #_ #_ #H destruct
+qed-.
+
+fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
+                        ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
+                                 U2 = ⓑ{a,I}V2.T2
+                        ) ∨
+                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
+                             a = true & I = Abbr.
+#G #L #U1 #U2 * -L -U1 -U2
+[ #I #G #L #b #J #W1 #U1 #H destruct
+| #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
+| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
+]
+qed-.
+
+lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓑ{a,I}V2.T2
+                     ) ∨
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
+                          a = true & I = Abbr.
+/2 width=3 by cpr_inv_bind1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓓ{a}V2.T2
+                     ) ∨
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T & a = true.
+#a #G #L #V1 #T1 #U2 #H
+elim (cpr_inv_bind1 … H) -H *
+/3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
+qed-.
+
+(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
+lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
+                              U2 = ⓛ{a}V2.T2.
+#a #G #L #V1 #T1 #U2 #H
+elim (cpr_inv_bind1 … H) -H *
+[ /3 width=5 by ex3_2_intro/
+| #T #_ #_ #_ #H destruct
+]
+qed-.
+
+fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
+                        ∀I,V1,U1. U = ⓕ{I}V1.U1 →
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                                    U2 = ⓕ{I} V2. T2
+                         | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+                                               U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                                 U1 = ⓓ{a}W1.T1 &
+                                                 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
+#G #L #U #U2 * -L -U -U2
+[ #I #G #L #J #W1 #U1 #H destruct
+| #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
+| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or4_intro2, ex6_6_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or4_intro3, ex7_7_intro/
+]
+qed-.
+
+lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                                 U2 = ⓕ{I}V2.T2
+                      | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
+                                            U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                              U1 = ⓓ{a}W1.T1 &
+                                              U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
+/2 width=3 by cpr_inv_flat1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                                 U2 = ⓐV2.T2
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
+                                            U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
+                                              U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
+#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+| #_ #H destruct
+| /3 width=11 by or3_intro1, ex5_6_intro/
+| /3 width=13 by or3_intro2, ex6_7_intro/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
+                                     U = ⓐV2. T2.
+#G #L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5 by ex3_2_intro/
+| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
+lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+                              U2 = ⓝ V2. T2
+                     ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
+#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
+[ /3 width=5 by ex3_2_intro, or_introl/
+| /2 width=1 by or_intror/
+| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
+| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+                                    T = -ⓑ{I}V2.T2.
+#I #G #L #V1 #T1 #T #H #b
+elim (cpr_inv_bind1 … H) -H *
+[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
+| #T2 #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: removed theorems 11:
+            pr0_subst0_back pr0_subst0_fwd pr0_subst0
+            pr2_head_2 pr2_cflat clear_pr2_trans
+            pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
+            pr2_gen_ctail pr2_ctail
+*)
+(* Basic_1: removed local theorems 4:
+            pr0_delta_eps pr0_cong_delta
+            pr2_free_free pr2_free_delta
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_cir.ma
new file mode 100644 (file)
index 0000000..3bbd1be
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_cir.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cir.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Advanced forward lemmas on irreducibility ********************************)
+
+lemma cpr_fwd_cir: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T1⦄ → T2 = T1.
+#G #L #T1 #T2 #H elim H -G -L -T1 -T2
+[ //
+| #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
+  elim (cir_inv_delta … HLK) //
+| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+  [ elim (cir_inv_bind … H) -H #HV1 #HT1 * #H destruct
+    lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+    lapply (IHT1 … HT1) -IHT1 #H destruct //
+  | elim (cir_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/
+  ]
+| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+  [ elim (cir_inv_appl … H) -H #HV1 #HT1 #_
+    >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+  | elim (cir_inv_ri2 … H) /2 width=1 by/
+  ]
+| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H
+  elim (cir_inv_ri2 … H) /2 width=1 by or_introl/
+| #G #L #V1 #T1 #T2 #_ #_ #H
+  elim (cir_inv_ri2 … H) /2 width=1 by/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H
+  elim (cir_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+  elim (cir_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_lift.ma
new file mode 100644 (file)
index 0000000..4a89afa
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_lift.ma
@@ -0,0 +1,110 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: includes: pr0_lift pr2_lift *)
+lemma cpr_lift: ∀G. d_liftable (cpr G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
+  >(lift_mono … H1 … H2) -H1 -H2 //
+| #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+    elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil
+    #K #Y #HKV #HVY #H destruct /3 width=9 by cpr_delta/
+  | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+    lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil /3 width=6 by cpr_delta, drop_inv_gen/
+  ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpr_bind, drop_skip/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpr_flat/
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+  elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+  elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpr_zeta, drop_skip/
+| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpr_eps/
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+  elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+  elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+  elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
+  elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpr_beta, drop_skip/
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+  elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+  elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+  elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+  elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+  elim (lift_trans_ge … HV2 … HV3) -V2 /4 width=6 by cpr_theta, drop_skip/
+]
+qed.
+
+(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
+lemma cpr_inv_lift1: ∀G. d_deliftable_sn (cpr G).
+#G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ * #i #G #L #K #s #l #m #_ #T1 #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  ]
+| #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ elim (drop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+    elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+    elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=8 by cpr_delta, ex2_intro/
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    lapply (drop_conf_ge … HLK … HLV ?) -L // #HKLV
+    elim (lift_split … HVW2 l (i - m + 1)) -HVW2 [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
+    #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O
+    /3 width=8 by cpr_delta, ex2_intro/
+  ]
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpr_bind, drop_skip, lift_bind, ex2_intro/
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1) -V1
+  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=6 by cpr_flat, lift_flat, ex2_intro/
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHU1 (K.ⓓW1) s  … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
+  elim (lift_div_le … HU2 … HTU) -U /3 width=6 by cpr_zeta, ex2_intro/
+| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpr_eps, ex2_intro/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+  elim (IHV12 … HLK … HV01) -V1
+  elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by drop_skip/
+  elim (IHW12 … HLK … HW01) -W1 /4 width=7 by cpr_beta, lift_flat, lift_bind, ex2_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+  elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
+  elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+  elim (IHW12 … HLK … HW01) -W1
+  elim (lift_trans_le … HV3 … HV2) -V
+  /4 width=9 by cpr_theta, lift_flat, lift_bind, ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_llpx_sn.ma
new file mode 100644 (file)
index 0000000..c633003
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpr_llpx_sn.ma
@@ -0,0 +1,47 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_drop.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma cpr_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R →
+                        ∀G. s_r_confluent1 … (cpr G) (llpx_sn R 0).
+#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ //
+| #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+  #Kd #V1l #HLKd #HV1s #HV1sd
+  lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+  lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+  @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+  /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+  /3 width=10 by llpx_sn_inv_lift_le, drop_drop/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
+  /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
+| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
+  #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
+  [ /3 width=10 by llpx_sn_lift_le, drop_drop/
+  | /3 width=4 by llpx_sn_bind_repl_O/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx.ma
new file mode 100644 (file)
index 0000000..b732758
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx.ma
@@ -0,0 +1,304 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/pred_6.ma".
+include "basic_2A/static/sd.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* avtivate genv *)
+inductive cpx (h) (g): relation4 genv lenv term term ≝
+| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
+| cpx_st   : ∀G,L,k,d. deg h g k (d+1) → cpx h g G L (⋆k) (⋆(next h k))
+| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
+             ⬇[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
+             ⬆[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
+| cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
+             cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
+             cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpx_flat : ∀I,G,L,V1,V2,T1,T2.
+             cpx h g G L V1 V2 → cpx h g G L T1 T2 →
+             cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
+             ⬆[0, 1] T2 ≡ T → cpx h g G L (+ⓓV.T1) T2
+| cpx_eps  : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
+| cpx_ct   : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
+| cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
+             cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
+             cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
+| cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+             cpx h g G L V1 V → ⬆[0, 1] V ≡ V2 → cpx h g G L W1 W2 →
+             cpx h g G (L.ⓓW1) T1 T2 →
+             cpx h g G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
+.
+
+interpretation
+   "context-sensitive extended parallel reduction (term)"
+   'PRed h g G L T1 T2 = (cpx h g G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_st/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lsubr_fwd_drop2_pair … HL12 … HLK1) -HL12 -HLK1 *
+  /4 width=7 by cpx_delta, cpx_ct/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsubr_pair/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_pair/
+]
+qed-.
+
+(* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
+lemma cpx_refl: ∀h,g,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
+#h #g #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
+qed.
+
+lemma cpr_cpx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
+qed.
+
+lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
+#h #g * /2 width=1 by cpx_bind, cpx_flat/
+qed.
+
+lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
+                  ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⬆[l, 1] T ≡ T2.
+#h #g #I #G #K #V #T1 elim T1 -T1
+[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
+  elim (lt_or_eq_or_gt i l) #Hil [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+  destruct
+  elim (lift_total V 0 (i+1)) #W #HVW
+  elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+  [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, drop_drop, lift_bind, ex2_2_intro/
+  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
+  ]
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
+                        ∨∨ T2 = ⓪{J}
+                         | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
+                         | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                         ⬆[O, i+1] V2 ≡ T2 & J = LRef i.
+#G #h #g #L #T1 #T2 * -L -T1 -T2
+[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
+| #G #L #k #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T2 #_ #J #H destruct
+| #G #L #V1 #V2 #T #_ #J #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
+                     ∨∨ T2 = ⓪{J}
+                      | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
+                      | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                      ⬆[O, i+1] V2 ≡ T2 & J = LRef i.
+/2 width=3 by cpx_inv_atom1_aux/ qed-.
+
+lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
+                     ∃∃d. deg h g k (d+1) & T2 = ⋆(next h k).
+#h #g #G #L #T2 #k #H
+elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+[ #k0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
+| #I #K #V #V2 #i #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
+                     T2 = #i ∨
+                     ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                 ⬆[O, i+1] V2 ≡ T2.
+#h #g #G #L #T2 #i #H
+elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+[ #k #d #_ #_ #H destruct
+| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
+]
+qed-.
+
+lemma cpx_inv_lref1_ge: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 → |L| ≤ i → T2 = #i.
+#h #g #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
+#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma cpx_inv_gref1: ∀h,g,G,L,T2,p.  ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
+#h #g #G #L #T2 #p #H
+elim (cpx_inv_atom1 … H) -H // *
+[ #k #d #_ #_ #H destruct
+| #I #K #V #V2 #i #_ #_ #_ #H destruct
+]
+qed-.
+
+fact cpx_inv_bind1_aux: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
+                        ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
+                                 U2 = ⓑ{a,J}V2.T2
+                        ) ∨
+                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+                             a = true & J = Abbr.
+#h #g #G #L #U1 #U2 * -L -U1 -U2
+[ #I #G #L #b #J #W #U1 #H destruct
+| #G #L #k #d #_ #b #J #W #U1 #H destruct
+| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
+| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
+| #G #L #V1 #V2 #T #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
+]
+qed-.
+
+lemma cpx_inv_bind1: ∀h,g,a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, g] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
+                              U2 = ⓑ{a,I} V2. T2
+                     ) ∨
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+                          a = true & I = Abbr.
+/2 width=3 by cpx_inv_bind1_aux/ qed-.
+
+lemma cpx_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
+                              U2 = ⓓ{a} V2. T2
+                     ) ∨
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T & a = true.
+#h #g #a #G #L #V1 #T1 #U2 #H
+elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
+qed-.
+
+lemma cpx_inv_abst1: ∀h,g,a,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 &  ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, g] T2 &
+                              U2 = ⓛ{a} V2. T2.
+#h #g #a #G #L #V1 #T1 #U2 #H
+elim (cpx_inv_bind1 … H) -H *
+[ /3 width=5 by ex3_2_intro/
+| #T #_ #_ #_ #H destruct
+]
+qed-.
+
+fact cpx_inv_flat1_aux: ∀h,g,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
+                        ∀J,V1,U1. U = ⓕ{J}V1.U1 →
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+                                    U2 = ⓕ{J}V2.T2
+                         | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ J = Cast)
+                         | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ J = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                               U1 = ⓛ{a}W1.T1 &
+                                               U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                                 U1 = ⓓ{a}W1.T1 &
+                                                 U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
+#h #g #G #L #U #U2 * -L -U -U2
+[ #I #G #L #J #W #U1 #H destruct
+| #G #L #k #d #_ #J #W #U1 #H destruct
+| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
+| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or5_intro1, conj/
+| #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=1 by or5_intro2, conj/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or5_intro3, ex6_6_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or5_intro4, ex7_7_intro/
+]
+qed-.
+
+lemma cpx_inv_flat1: ∀h,g,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+                                 U2 = ⓕ{I} V2. T2
+                      | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ I = Cast)
+                      | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                            U1 = ⓛ{a}W1.T1 &
+                                            U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                              U1 = ⓓ{a}W1.T1 &
+                                              U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
+/2 width=3 by cpx_inv_flat1_aux/ qed-.
+
+lemma cpx_inv_appl1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+                                 U2 = ⓐ V2. T2
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                            U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                                              U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
+#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+|2,3: #_ #H destruct
+| /3 width=11 by or3_intro1, ex5_6_intro/
+| /3 width=13 by or3_intro2, ex6_7_intro/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpx_inv_appl1_simple: ∀h,g,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g] U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 &
+                                     U = ⓐV2.T2.
+#h #g #G #L #V1 #T1 #U #H #HT1
+elim (cpx_inv_appl1 … H) -H *
+[ /2 width=5 by ex3_2_intro/
+| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+]
+qed-.
+
+lemma cpx_inv_cast1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+                                 U2 = ⓝ V2. T2
+                      | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
+                      | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
+#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+|2,3: /2 width=1 by or3_intro1, or3_intro2/
+| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
+| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpx_fwd_bind1_minus: ∀h,g,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, g] T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, g] ⓑ{b,I}V2.T2 &
+                                    T = -ⓑ{I}V2.T2.
+#h #g #I #G #L #V1 #T1 #T #H #b
+elim (cpx_inv_bind1 … H) -H *
+[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpx_bind, ex2_2_intro/
+| #T2 #_ #_ #H destruct
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_cix.ma
new file mode 100644 (file)
index 0000000..1091156
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_cix.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cix.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Advanced forward lemmas on irreducibility ********************************)
+
+lemma cpx_fwd_cix: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T1⦄ → T2 = T1.
+#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
+[ //
+| #G #L #k #d #Hkd #H elim (cix_inv_sort … Hkd H)
+| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
+  elim (cix_inv_delta … HLK) //
+| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+  [ elim (cix_inv_bind … H) -H #HV1 #HT1 * #H destruct
+    lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+    lapply (IHT1 … HT1) -IHT1 #H destruct //
+  | elim (cix_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/
+  ]
+| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+  [ elim (cix_inv_appl … H) -H #HV1 #HT1 #_
+    >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+  | elim (cix_inv_ri2 … H) /2 width=1 by/
+  ]
+| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H
+  elim (cix_inv_ri2 … H) /2 width=1 by or_introl/
+| #G #L #V1 #T1 #T2 #_ #_ #H
+  elim (cix_inv_ri2 … H) /2 width=1 by/
+| #G #L #V1 #V2 #T #_ #_ #H
+  elim (cix_inv_ri2 … H) /2 width=1 by/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H
+  elim (cix_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+  elim (cix_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lift.ma
new file mode 100644 (file)
index 0000000..6cc63f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lift.ma
@@ -0,0 +1,264 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/multiple/fqus_alt.ma".
+include "basic_2A/static/da.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Advanced properties ******************************************************)
+
+fact sta_cpx_aux: ∀h,g,G,L,T1,T2,d2,d1. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → d2 = 1 →
+                  ⦃G, L⦄ ⊢ T1 ▪[h, g] d1+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2
+[ #G #L #d2 #k #H0 destruct normalize
+  /3 width=4 by cpx_st, da_inv_sort/
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct
+  elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
+| #G #L #K #V1 #V2 #i #_ #_ #_ #H destruct
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #HV12 #HVW2 #_ #H0 #H
+  lapply (discr_plus_xy_y … H0) -H0 #H0 destruct
+  elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+  /4 width=7 by cpx_delta, cpr_cpx, lstas_cpr/
+| /4 width=2 by cpx_bind, da_inv_bind/
+| /4 width=3 by cpx_flat, da_inv_flat/
+| /4 width=3 by cpx_eps, da_inv_flat/
+]
+qed-.
+
+lemma sta_cpx: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
+               ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+/2 width=3 by sta_cpx_aux/ qed.
+
+(* Relocation properties ****************************************************)
+
+lemma cpx_lift: ∀h,g,G. d_liftable (cpx h g G).
+#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
+  >(lift_mono … H1 … H2) -H1 -H2 //
+| #G #K #k #d #Hkd #L #s #l #m #_ #U1 #H1 #U2 #H2
+  >(lift_inv_sort1 … H1) -U1
+  >(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_st/
+| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
+    elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil
+    #K #Y #HKV #HVY #H destruct /3 width=10 by cpx_delta/
+  | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+    lapply (drop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpx_delta, drop_inv_gen/
+  ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpx_bind, drop_skip/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpx_flat/
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+  elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+  elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpx_zeta, drop_skip/
+| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_eps/
+| #G #K #V1 #V2 #T #_ #IHV12 #L #s #l #m #HLK #U1 #H #U2 #HVU2
+  elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ct/
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+  elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+  elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+  elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
+  elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpx_beta, drop_skip/
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+  elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
+  elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+  elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+  elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+  elim (lift_trans_ge … HV2 … HV3) -V2 /4 width=6 by cpx_theta, drop_skip/
+]
+qed.
+
+lemma cpx_inv_lift1: ∀h,g,G. d_deliftable_sn (cpx h g G).
+#h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ * #i #G #L #K #s #l #m #_ #T1 #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_sort, ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
+  ]
+| #G #L #k #d #Hkd #K #s #l #m #_ #T1 #H
+  lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_st, lift_sort, ex2_intro/
+| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ elim (drop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
+    elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
+    elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpx_delta, ex2_intro/
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    lapply (drop_conf_ge … HLK … HLV ?) -L // #HKLV
+    elim (lift_split … HVW2 l (i - m + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hil -Hlim
+    #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
+  ]
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpx_bind, drop_skip, lift_bind, ex2_intro/
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1) -V1
+  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHU1 (K.ⓓW1) s … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
+  elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
+| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_eps, ex2_intro/
+| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ct, ex2_intro/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+  elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
+  elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+  elim (IHW12 … HLK … HW01) -W1
+  /4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
+  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+  elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
+  elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+  elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
+  elim (lift_trans_le … HV3 … HV2) -V
+  /4 width=9 by cpx_theta, lift_bind, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 
+/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
+[ #I #G #L #V2 #U2 #HVU2
+  elim (lift_total U2 0 1)
+  /4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
+| #G #L #K #T1 #U1 #m #HLK1 #HTU1 #T2 #HTU2
+  elim (lift_total T2 0 (m+1))
+  /3 width=11 by cpx_lift, fqu_drop, ex2_intro/
+]
+qed-.
+
+lemma fqu_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 →
+                     ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d+1 →
+                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-.
+
+lemma fquq_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fquq_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 →
+                      ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d+1 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-.
+
+lemma fqup_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
+  /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
+  elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
+  elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma fqus_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fqu_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+                         ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+  #U2 #HVU2 @(ex3_intro … U2)
+  [1,3: /3 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop/
+  | #H destruct /2 width=7 by lift_inv_lref2_be/
+  ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+  [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+  [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+  [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
+  | #H0 destruct /2 width=1 by/
+  ]
+| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
+  #U2 #HTU2 @(ex3_intro … U2)
+  [1,3: /2 width=10 by cpx_lift, fqu_drop/
+  | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+  #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_neq … H1 … HTU1 H) -T1
+  /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
+  /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lleq.ma
new file mode 100644 (file)
index 0000000..93c49c0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lleq.ma
@@ -0,0 +1,55 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_drop.ma".
+include "basic_2A/reduction/cpx_llpx_sn.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+                      ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
+[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
+  [ #H elim (ylt_yle_false … H) //
+  | * /3 width=7 by cpx_delta/
+  ]
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=1 by cpx_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=1 by cpx_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=3 by cpx_zeta/
+| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=1 by cpx_eps/
+| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=1 by cpx_ct/
+| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
+  #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=1 by cpx_beta/
+| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
+  #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=3 by cpx_theta/
+]
+qed-.
+
+lemma cpx_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+                     ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
+
+lemma cpx_lleq_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
+/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
+
+lemma cpx_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+                        ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
+/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_llpx_sn.ma
new file mode 100644 (file)
index 0000000..41137f2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_llpx_sn.ma
@@ -0,0 +1,50 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_drop.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+(* Note: lemma 1000 *)
+lemma cpx_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R →
+                        ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx_sn R 0).
+#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+[ //
+| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #I #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+  #Kd #V1l #HLKd #HV1s #HV1sd
+  lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+  lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+  @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+  /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+  /3 width=10 by llpx_sn_inv_lift_le, drop_drop/
+| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
+| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  #HV1 #H elim (llpx_sn_inv_bind_O … H) -H
+  /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/
+| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  #HV1 #H elim (llpx_sn_inv_bind_O … H) -H //
+  #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *)
+  [ /3 width=10 by llpx_sn_lift_le, drop_drop/
+  | /3 width=4 by llpx_sn_bind_repl_O/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lreq.ma
new file mode 100644 (file)
index 0000000..84e58fe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/cpx_lreq.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_lreq.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) (lreq 0 (∞)).
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_st/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lreq_drop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lreq_pair_O_Y/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lreq_pair_O_Y/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr.ma
new file mode 100644 (file)
index 0000000..b18e71c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr.ma
@@ -0,0 +1,145 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predreducible_3.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
+
+(* reducible binary items *)
+definition ri2: predicate item2 ≝
+                λI. I = Bind2 true Abbr ∨ I = Flat2 Cast.
+
+(* irreducible binary binders *)
+definition ib2: relation2 bool bind2 ≝
+                λa,I. I = Abst ∨ Bind2 a I = Bind2 false Abbr.
+
+(* activate genv *)
+(* reducible terms *)
+inductive crr (G:genv): relation2 lenv term ≝
+| crr_delta  : ∀L,K,V,i. ⬇[i] L ≡ K.ⓓV → crr G L (#i)
+| crr_appl_sn: ∀L,V,T. crr G L V → crr G L (ⓐV.T)
+| crr_appl_dx: ∀L,V,T. crr G L T → crr G L (ⓐV.T)
+| crr_ri2    : ∀I,L,V,T. ri2 I → crr G L (②{I}V.T)
+| crr_ib2_sn : ∀a,I,L,V,T. ib2 a I → crr G L V → crr G L (ⓑ{a,I}V.T)
+| crr_ib2_dx : ∀a,I,L,V,T. ib2 a I → crr G (L.ⓑ{I}V) T → crr G L (ⓑ{a,I}V.T)
+| crr_beta   : ∀a,L,V,W,T. crr G L (ⓐV.ⓛ{a}W.T)
+| crr_theta  : ∀a,L,V,W,T. crr G L (ⓐV.ⓓ{a}W.T)
+.
+
+interpretation
+   "reducibility for context-sensitive reduction (term)"
+   'PRedReducible G L T = (crr G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact crr_inv_sort_aux: ∀G,L,T,k. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⋆k → ⊥.
+#G #L #T #k0 * -L -T
+[ #L #K #V #i #HLK #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crr_inv_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐑⦃⋆k⦄ → ⊥.
+/2 width=6 by crr_inv_sort_aux/ qed-.
+
+fact crr_inv_lref_aux: ∀G,L,T,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = #i →
+                       ∃∃K,V. ⬇[i] L ≡ K.ⓓV.
+#G #L #T #j * -L -T
+[ #L #K #V #i #HLK #H destruct /2 width=3 by ex1_2_intro/
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crr_inv_lref: ∀G,L,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃#i⦄ → ∃∃K,V. ⬇[i] L ≡ K.ⓓV.
+/2 width=4 by crr_inv_lref_aux/ qed-.
+
+fact crr_inv_gref_aux: ∀G,L,T,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = §p → ⊥.
+#G #L #T #q * -L -T
+[ #L #K #V #i #HLK #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crr_inv_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐑⦃§p⦄ → ⊥.
+/2 width=6 by crr_inv_gref_aux/ qed-.
+
+lemma trr_inv_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃⓪{I}⦄ → ⊥.
+#G * #i #H
+[ elim (crr_inv_sort … H)
+| elim (crr_inv_lref … H) -H #L #V #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| elim (crr_inv_gref … H)
+]
+qed-.
+
+fact crr_inv_ib2_aux: ∀a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓑ{a,I}W.U →
+                      ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃U⦄.
+#G #b #J #L #W0 #U #T #HI * -L -T
+[ #L #K #V #i #_ #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #H1 #H2 destruct
+  elim H1 -H1 #H destruct
+  elim HI -HI #H destruct
+| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/
+| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crr_inv_ib2: ∀a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓑ{a,I}W.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡ 𝐑⦃T⦄.
+/2 width=5 by crr_inv_ib2_aux/ qed-.
+
+fact crr_inv_appl_aux: ∀G,L,W,U,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⓐW.U →
+                       ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+#G #L #W0 #U #T * -L -T
+[ #L #K #V #i #_ #H destruct
+| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/
+| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/
+| #I #L #V #T #H1 #H2 destruct
+  elim H1 -H1 #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+  @or3_intro2 #H elim (simple_inv_bind … H)
+| #a #L #V #W #T #H destruct
+  @or3_intro2 #H elim (simple_inv_bind … H)
+]
+qed-.
+
+lemma crr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃ⓐV.T⦄ →
+                              ∨∨ ⦃G, L⦄ ⊢ ➡ 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+/2 width=3 by crr_inv_appl_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr_lift.ma
new file mode 100644 (file)
index 0000000..12dafe5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crr_lift.ma
@@ -0,0 +1,74 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/reduction/crr.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
+
+(* Properties on relocation *************************************************)
+
+lemma crr_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄.
+#G #K #T #H elim H -K -T
+[ #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+  elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+  [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crr_delta/
+  | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=4 by crr_delta, drop_inv_gen/
+  ]
+| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
+| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
+| #I #K #V #T #HI #L #s #l #m #_ #X #H
+  elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crr_ri2/
+| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
+  elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
+| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+  elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_beta/
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+  elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_theta/
+]
+qed.
+
+lemma crr_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                    ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L #U #H elim H -L -U
+[ #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crr_delta/
+  | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crr_delta/
+  ]
+| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
+| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
+| #I #L #W #U #HI #K #s #l #m #_ #X #H
+  elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crr_ri2/
+| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
+| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+  elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_beta/
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+  elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_theta/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx.ma
new file mode 100644 (file)
index 0000000..9e3e031
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx.ma
@@ -0,0 +1,152 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predreducible_5.ma".
+include "basic_2A/static/sd.ma".
+include "basic_2A/reduction/crr.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
+
+(* activate genv *)
+(* extended reducible terms *)
+inductive crx (h) (g) (G:genv): relation2 lenv term ≝
+| crx_sort   : ∀L,k,d. deg h g k (d+1) → crx h g G L (⋆k)
+| crx_delta  : ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → crx h g G L (#i)
+| crx_appl_sn: ∀L,V,T. crx h g G L V → crx h g G L (ⓐV.T)
+| crx_appl_dx: ∀L,V,T. crx h g G L T → crx h g G L (ⓐV.T)
+| crx_ri2    : ∀I,L,V,T. ri2 I → crx h g G L (②{I}V.T)
+| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h g G L V → crx h g G L (ⓑ{a,I}V.T)
+| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h g G (L.ⓑ{I}V) T → crx h g G L (ⓑ{a,I}V.T)
+| crx_beta   : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓛ{a}W.T)
+| crx_theta  : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓓ{a}W.T)
+.
+
+interpretation
+   "reducibility for context-sensitive extended reduction (term)"
+   'PRedReducible h g G L T = (crx h g G L T).
+
+(* Basic properties *********************************************************)
+
+lemma crr_crx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+#h #g #G #L #T #H elim H -L -T
+/2 width=4 by crx_delta, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact crx_inv_sort_aux: ∀h,g,G,L,T,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⋆k →
+                       ∃d. deg h g k (d+1).
+#h #g #G #L #T #k0 * -L -T
+[ #L #k #d #Hkd #H destruct /2 width=2 by ex_intro/
+| #I #L #K #V #i #HLK #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃⋆k⦄ → ∃d. deg h g k (d+1).
+/2 width=5 by crx_inv_sort_aux/ qed-.
+
+fact crx_inv_lref_aux: ∀h,g,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = #i →
+                       ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
+#h #g #G #L #T #j * -L -T
+[ #L #k #d #_ #H destruct
+| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crx_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
+/2 width=6 by crx_inv_lref_aux/ qed-.
+
+fact crx_inv_gref_aux: ∀h,g,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = §p → ⊥.
+#h #g #G #L #T #q * -L -T
+[ #L #k #d #_ #H destruct
+| #I #L #K #V #i #HLK #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crx_inv_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃§p⦄ → ⊥.
+/2 width=8 by crx_inv_gref_aux/ qed-.
+
+lemma trx_inv_atom: ∀h,g,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐑⦃⓪{I}⦄ →
+                    ∃∃k,d. deg h g k (d+1) & I = Sort k.
+#h #g * #i #G #H
+[ elim (crx_inv_sort … H) -H /2 width=4 by ex2_2_intro/
+| elim (crx_inv_lref … H) -H #I #L #V #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| elim (crx_inv_gref … H)
+]
+qed-.
+
+fact crx_inv_ib2_aux: ∀h,g,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ →
+                      T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃U⦄.
+#h #g #b #J #G #L #W0 #U #T #HI * -L -T
+[ #L #k #d #_ #H destruct
+| #I #L #K #V #i #_ #H destruct
+| #L #V #T #_ #H destruct
+| #L #V #T #_ #H destruct
+| #I #L #V #T #H1 #H2 destruct
+  elim H1 -H1 #H destruct
+  elim HI -HI #H destruct
+| #a #I #L #V #T #_ #HV #H destruct /2 width=1 by or_introl/
+| #a #I #L #V #T #_ #HT #H destruct /2 width=1 by or_intror/
+| #a #L #V #W #T #H destruct
+| #a #L #V #W #T #H destruct
+]
+qed-.
+
+lemma crx_inv_ib2: ∀h,g,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓑ{a,I}W.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+/2 width=5 by crx_inv_ib2_aux/ qed-.
+
+fact crx_inv_appl_aux: ∀h,g,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⓐW.U →
+                       ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+#h #g #G #L #W0 #U #T * -L -T
+[ #L #k #d #_ #H destruct
+| #I #L #K #V #i #_ #H destruct
+| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/
+| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/
+| #I #L #V #T #H1 #H2 destruct
+  elim H1 -H1 #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #I #L #V #T #_ #_ #H destruct
+| #a #L #V #W #T #H destruct
+  @or3_intro2 #H elim (simple_inv_bind … H)
+| #a #L #V #W #T #H destruct
+  @or3_intro2 #H elim (simple_inv_bind … H)
+]
+qed-.
+
+lemma crx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓐV.T⦄ →
+                    ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+/2 width=3 by crx_inv_appl_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx_lift.ma
new file mode 100644 (file)
index 0000000..b77db17
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/crx_lift.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/reduction/crx.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION *****************)
+
+(* Properties on relocation *************************************************)
+
+lemma crx_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄.
+#h #g #G #K #T #H elim H -K -T
+[ #K #k #d #Hkd #L #s #l #m #_ #X #H
+  >(lift_inv_sort1 … H) -X /2 width=2 by crx_sort/
+| #I #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+  elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+  [ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crx_delta/
+  | lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=5 by crx_delta, drop_inv_gen/
+  ]
+| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
+| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
+| #I #K #V #T #HI #L #s #l #m #_ #X #H
+  elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crx_ri2/
+| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
+  elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
+| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
+  elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+  elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_beta/
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+  elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_theta/
+]
+qed.
+
+lemma crx_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                    ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+#h #g #G #L #U #H elim H -L -U
+[ #L #k #d #Hkd #K #s #l #m #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=2 by crx_sort/
+| #I #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crx_delta/
+  | lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crx_delta/
+  ]
+| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
+| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
+| #I #L #W #U #HI #K #s #l #m #_ #X #H
+  elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crx_ri2/
+| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
+| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
+  elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+  elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_beta/
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+  elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
+  elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_theta/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb.ma
new file mode 100644 (file)
index 0000000..d95512b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btpredproper_8.ma".
+include "basic_2A/substitution/fqu.ma".
+include "basic_2A/multiple/lleq.ma".
+include "basic_2A/reduction/lpx.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+inductive fpb (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpb h g G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpb h g G1 L1 T1 G1 L2 T1
+.
+
+interpretation
+   "'rst' proper parallel reduction (closure)"
+   'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpb h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
+               ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+/3 width=1 by fpb_cpx, cpr_cpx/ qed.
+
+lemma lpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) →
+               ⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpb_lpx, lpr_lpx/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_fleq.ma
new file mode 100644 (file)
index 0000000..ec8c001
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_fleq.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fleq.ma".
+include "basic_2A/reduction/fpb_lleq.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fleq_fpb_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
+                      ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
+                      ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
+#h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+#K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpb_trans … HK12 … H12) -K2
+/3 width=5 by fleq_intro, ex2_3_intro/
+qed-.
+
+(* Inversion lemmas on lazy equivalence for closures ************************)
+
+lemma fpb_inv_fleq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #H elim (fleq_inv_gen … H) -H
+  /3 width=8 by lleq_fwd_length, fqu_inv_eq/
+| #T2 #_ #HnT #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+| #L2 #_ #HnL #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+]
+qed-.
+

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lift.ma
new file mode 100644 (file)
index 0000000..fa4af73
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lift.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/unfold/lstas_da.ma".
+include "basic_2A/reduction/cpx_lift.ma".
+include "basic_2A/reduction/fpb.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpb: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 →
+               ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2)
+/3 width=2 by fpb_cpx, sta_cpx/ #H destruct
+elim (lstas_inv_refl_pos h G L T2 0) //
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lleq.ma
new file mode 100644 (file)
index 0000000..f2756f3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpb_lleq.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_fqus.ma".
+include "basic_2A/multiple/lleq_lleq.ma".
+include "basic_2A/reduction/lpx_lleq.ma".
+include "basic_2A/reduction/fpb.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fpb_trans: ∀h,g,F,K1,K2,T. K1 ≡[T, 0] K2 →
+                      ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, g] ⦃G, L2, U⦄ →
+                      ∃∃L1. ⦃F, K1, T⦄ ≻[h, g] ⦃G, L1, U⦄ & L1 ≡[U, 0] L2.
+#h #g #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+[ #G #L2 #U #H2 elim (lleq_fqu_trans … H2 … HT) -K2
+  /3 width=3 by fpb_fqu, ex2_intro/
+| /4 width=10 by fpb_cpx, cpx_lleq_conf_sn, lleq_cpx_trans, ex2_intro/
+| #L2 #HKL2 #HnKL2 elim (lleq_lpx_trans … HKL2 … HT) -HKL2
+  /6 width=3 by fpb_lpx, lleq_canc_sn, ex2_intro/ (* 2 lleq_canc_sn *)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq.ma
new file mode 100644 (file)
index 0000000..e1d0d39
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btpred_8.ma".
+include "basic_2A/substitution/fquq.ma".
+include "basic_2A/multiple/lleq.ma".
+include "basic_2A/reduction/lpx.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+inductive fpbq (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h g G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → fpbq h g G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → fpbq h g G1 L1 T1 G1 L2 T1
+| fpbq_lleq: ∀L2. L1 ≡[T1, 0] L2 → fpbq h g G1 L1 T1 G1 L2 T1
+.
+
+interpretation
+   "'qrst' parallel reduction (closure)"
+   'BTPRed h g G1 L1 T1 G2 L2 T2 = (fpbq h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpbq_refl: ∀h,g. tri_reflexive … (fpbq h g).
+/2 width=1 by fpbq_cpx/ qed.
+
+lemma cpr_fpbq: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄. 
+/3 width=1 by fpbq_cpx, cpr_cpx/ qed.
+
+lemma lpr_fpbq: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpbq_lpx, lpr_lpx/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_aaa.ma
new file mode 100644 (file)
index 0000000..c4619f5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_aaa.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_fqus.ma".
+include "basic_2A/static/aaa_lleq.ma".
+include "basic_2A/reduction/lpx_aaa.ma".
+include "basic_2A/reduction/fpbq.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma fpbq_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                     ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=6 by aaa_lleq_conf, lpx_aaa_conf, cpx_aaa_conf, aaa_fquq_conf, ex_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_alt.ma
new file mode 100644 (file)
index 0000000..e2e1ebb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_alt.ma
@@ -0,0 +1,86 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/btpredalt_8.ma".
+include "basic_2A/reduction/fpb_fleq.ma".
+include "basic_2A/reduction/fpbq.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+(* alternative definition of fpbq *)
+definition fpbqa: ∀h. sd h → tri_relation genv lenv term ≝
+                  λh,g,G1,L1,T1,G2,L2,T2.
+                  ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
+
+interpretation
+   "'qrst' parallel reduction (closure) alternative"
+   'BTPRedAlt h g G1 L1 T1 G2 L2 T2 = (fpbqa h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fleq_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2.
+                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fpbq_lleq/
+qed.
+
+lemma fpb_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem fpbq_fpbqa: ∀h,g,G1,G2,L1,L2,T1,T2.
+                    ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1, T1⦄ ≽≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
+  [ /3 width=1 by fpb_fqu, or_intror/
+  | * #H1 #H2 #H3 destruct /2 width=1 by or_introl/
+  ]
+| #T2 #HT12 elim (eq_term_dec T1 T2)
+  #HnT12 destruct /4 width=1 by fpb_cpx, or_intror, or_introl/
+| #L2 #HL12 elim (lleq_dec … T1 L1 L2 0)
+  /4 width=1 by fpb_lpx, fleq_intro, or_intror, or_introl/
+| /3 width=1 by fleq_intro, or_introl/
+]
+qed.
+
+theorem fpbqa_inv_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2.
+                        ⦃G1, L1, T1⦄ ≽≽[h, g] ⦃G2, L2, T2⦄ →
+                        ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fleq_fpbq, fpb_fpbq/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fpbq_ind_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ∀R:Prop.
+                    (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → R) →
+                    (⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R) →
+                    ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #R #HI1 #HI2 #H elim (fpbq_fpbqa … H) /2 width=1 by/
+qed-.
+
+(* aternative definition of fpb *********************************************)
+
+lemma fpb_fpbq_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ ∧ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥).
+/3 width=10 by fpb_fpbq, fpb_inv_fleq, conj/ qed.
+
+lemma fpbq_inv_fpb_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                        (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 @(fpbq_ind_alt … H) -H //
+#H elim H0 -H0 //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_lift.ma
new file mode 100644 (file)
index 0000000..bb93ef2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/fpbq_lift.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/reduction/cpx_lift.ma".
+include "basic_2A/reduction/fpbq.ma".
+
+(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma sta_fpbq: ∀h,g,G,L,T1,T2,d.
+                 ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
+                 ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+/3 width=4 by fpbq_cpx, sta_cpx/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr.ma
new file mode 100644 (file)
index 0000000..5901ee5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr.ma
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predsn_3.ma".
+include "basic_2A/substitution/lpx_sn.ma".
+include "basic_2A/reduction/cpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+definition lpr: relation3 genv lenv lenv ≝ λG. lpx_sn (cpr G).
+
+interpretation "parallel reduction (local environment, sn variant)"
+   'PRedSn G L1 L2 = (lpr G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: includes: wcpr0_gen_sort *)
+lemma lpr_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡ L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: wcpr0_gen_head *)
+lemma lpr_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ L2 →
+                     ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L2 = K2.ⓑ{I}V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpr_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡ ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpr_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡ K2.ⓑ{I}V2 →
+                     ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡ K2 & ⦃G, K1⦄ ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Note: lemma 250 *)
+lemma lpr_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡ L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpr_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡ K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 →
+                ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡ K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpr_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.
+
+(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
+                                pr0_subst1_back
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_drop.ma
new file mode 100644 (file)
index 0000000..4f515f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_drop.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn_drop.ma".
+include "basic_2A/substitution/fquq_alt.ma".
+include "basic_2A/reduction/cpr_lift.ma".
+include "basic_2A/reduction/lpr.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Properties on local environment slicing ***********************************)
+
+(* Basic_1: includes: wcpr0_drop *)
+lemma lpr_drop_conf: ∀G. dropable_sn (lpr G).
+/3 width=6 by lpx_sn_deliftable_dropable, cpr_inv_lift1/ qed-.
+
+(* Basic_1: includes: wcpr0_drop_back *)
+lemma drop_lpr_trans: ∀G. dedropable_sn (lpr G).
+/3 width=10 by lpx_sn_liftable_dedropable, cpr_lift/ qed-.
+
+lemma lpr_drop_trans_O1: ∀G. dropable_dx (lpr G).
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                        ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+                        ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
+#G #L #K #U #T #m #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (m+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+                         ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                        ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+                        ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
+#G #L #K #U #T #m #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (m+1)) #T2 #HUT2
+lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
+qed-.
+
+lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
+                         ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma fqu_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
+                     ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpr_pair, ex3_2_intro/
+[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 … H) -H
+  #K2 #W2 #HLK2 #HVW2 #H destruct
+  /3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
+  elim (drop_lpr_trans … HLK1 … HK12) -HK12
+  /3 width=7 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
+                      ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_lpr_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_lpr.ma
new file mode 100644 (file)
index 0000000..9fbe5ee
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpr_lpr.ma
@@ -0,0 +1,357 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn_lpx_sn.ma".
+include "basic_2A/multiple/fqup.ma".
+include "basic_2A/reduction/lpr_drop.ma".
+
+(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties on context-sensitive parallel reduction for terms ********)
+
+fact cpr_conf_lpr_atom_atom:
+   ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
+/2 width=3 by cpr_atom, ex2_intro/ qed-.
+
+fact cpr_conf_lpr_atom_delta:
+   ∀G,L0,i. (
+      ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀K0,V0. ⬇[i] L0 ≡ K0.ⓓV0 →
+   ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⬆[O, i + 1] V2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_drop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpr_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1))
+/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
+qed-.
+
+(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
+fact cpr_conf_lpr_delta_delta:
+   ∀G,L0,i. (
+      ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀K0,V0. ⬇[i] L0 ≡ K0.ⓓV0 →
+   ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⬆[O, i + 1] V1 ≡ T1 →
+   ∀KX,VX. ⬇[i] L0 ≡ KX.ⓓVX →
+   ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⬆[O, i + 1] V2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (drop_mono … H … HLK0) -H #H destruct
+elim (lpr_drop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (drop_fwd_drop2 … HLK1) -W1 #HLK1
+elim (lpr_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_bind:
+   ∀a,I,G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by lpr_pair, cpr_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_zeta:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
+   ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⬆[O, 1] X2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, drop_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_zeta_zeta:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⬆[O, 1] X1 ≡ T1 →
+   ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⬆[O, 1] X2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by drop_drop/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by drop_drop/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_flat:
+   ∀I,G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_eps:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_eps, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_eps_eps:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01
+#T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_beta:
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
+/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
+qed-.
+
+(* Basic-1: includes:
+            pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
+            pr0_cong_upsilon_cong pr0_cong_upsilon_delta
+*)
+fact cpr_conf_lpr_flat_theta:
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⬆[O, 1] V2 ≡ U2 →
+   ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/ #HU2
+elim (cpr_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+  elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+  elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+  /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
+| #T1 #HT01 #HXT1 #H destruct
+  elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+  elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
+  /4 width=9 by cpr_flat, cpr_zeta, drop_drop, lift_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_lpr_beta_beta:
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
+/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+(* Basic_1: was: pr0_upsilon_upsilon *)
+fact cpr_conf_lpr_theta_theta:
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
+   ) →
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⬆[O, 1] V1 ≡ U1 →
+   ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⬆[O, 1] V2 ≡ U2 →
+   ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+elim (lift_total V 0 1) #U #HVU
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by drop_drop/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/
+/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+qed-.
+
+theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G).
+#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+  elim (cpr_inv_atom1 … H1) -H1
+  elim (cpr_inv_atom1 … H2) -H2
+  [ #H2 #H1 destruct
+    /2 width=1 by cpr_conf_lpr_atom_atom/
+  | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+    /3 width=10 by cpr_conf_lpr_atom_delta/
+  | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+    /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+  | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+    * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+    /3 width=17 by cpr_conf_lpr_delta_delta/
+  ]
+| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+  elim (cpr_inv_bind1 … H1) -H1 *
+  [ #V1 #T1 #HV01 #HT01 #H1
+  | #T1 #HT01 #HXT1 #H11 #H12
+  ]
+  elim (cpr_inv_bind1 … H2) -H2 *
+  [1,3: #V2 #T2 #HV02 #HT02 #H2
+  |2,4: #T2 #HT02 #HXT2 #H21 #H22
+  ] destruct
+  [ /3 width=10 by cpr_conf_lpr_bind_bind/
+  | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
+  | /3 width=11 by cpr_conf_lpr_bind_zeta/
+  | /3 width=12 by cpr_conf_lpr_zeta_zeta/
+  ]
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+  elim (cpr_inv_flat1 … H1) -H1 *
+  [ #V1 #T1 #HV01 #HT01 #H1
+  | #HX1 #H1
+  | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+  | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+  ]
+  elim (cpr_inv_flat1 … H2) -H2 *
+  [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
+  |2,6,10,14: #HX2 #H2
+  |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+  |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
+  ] destruct
+  [ /3 width=10 by cpr_conf_lpr_flat_flat/
+  | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_eps/
+  | /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
+  | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
+  | /3 width=8 by cpr_conf_lpr_flat_eps/
+  | /3 width=7 by cpr_conf_lpr_eps_eps/
+  | /3 width=12 by cpr_conf_lpr_flat_beta/
+  | /3 width=13 by cpr_conf_lpr_beta_beta/
+  | /3 width=14 by cpr_conf_lpr_flat_theta/
+  | /3 width=17 by cpr_conf_lpr_theta_theta/
+  ]
+]
+qed-.
+
+(* Basic_1: includes: pr0_confluence pr2_confluence *)
+theorem cpr_conf: ∀G,L. confluent … (cpr G L).
+/2 width=6 by cpr_conf_lpr/ qed-.
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+lemma lpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                       ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+lemma lpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                       ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpr_conf: ∀G. confluent … (lpr G).
+/3 width=6 by lpx_sn_conf, cpr_conf_lpr/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx.ma
new file mode 100644 (file)
index 0000000..5b93217
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx.ma
@@ -0,0 +1,65 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/predsn_5.ma".
+include "basic_2A/reduction/lpr.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
+                λh,g,G. lpx_sn (cpx h g G).
+
+interpretation "extended parallel reduction (local environment, sn variant)"
+   'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
+                     ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+                              L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+lemma lpx_inv_atom2: ∀h,g,G,L1.  ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2.  ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
+                     ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+                             L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2.  ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
+                    ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
+/2 width=1 by lpx_sn_inv_pair/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀h,g,G,L.  ⦃G, L⦄ ⊢ ➡[h, g] L.
+/2 width=1 by lpx_sn_refl/ qed.
+
+lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
+                ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
+/2 width=1 by lpx_sn_pair/ qed.
+
+lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_aaa.ma
new file mode 100644 (file)
index 0000000..58387c3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_aaa.ma
@@ -0,0 +1,83 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_lift.ma".
+include "basic_2A/static/lsuba_aaa.ma".
+include "basic_2A/reduction/lpx_drop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+(* Note: lemma 500 *)
+lemma cpx_lpx_aaa_conf: ∀h,g,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+                        ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+                        ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #g #L1 #k #X #H
+  elim (cpx_inv_sort1 … H) -H // * //
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+  elim (cpx_inv_lref1 … H) -H
+  [ #H destruct
+    elim (lpx_drop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+    elim (lpx_inv_pair1 … H) -H
+    #K2 #V2 #HK12 #HV12 #H destruct /3 width=6 by aaa_lref/
+  | * #J #Y #Z #V2 #H #HV12 #HV2
+    lapply (drop_mono … H … HLK1) -H #H destruct
+    elim (lpx_drop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+    elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+    /3 width=8 by aaa_lift, drop_fwd_drop2/
+  ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_abbr1 … H) -H *
+  [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2 by lpx_pair, aaa_abbr/
+  | #T2 #HT12 #HT2 #H destruct -IHV1
+    /4 width=8 by lpx_pair, aaa_inv_lift, drop_drop/
+  ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /4 width=1 by lpx_pair, aaa_abst/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_appl1 … H) -H *
+  [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
+  | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+    lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
+    lapply (IHT1 (ⓛ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+    elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+    /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
+  | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+    lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by drop_drop/ #HV2
+    lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+    elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
+  ]
+| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_cast1 … H) -H
+  [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
+  | -IHV1 /2 width=1 by/
+  | -IHT1 /2 width=1 by/
+  ]
+]
+qed-.
+
+lemma cpx_aaa_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/2 width=7 by cpx_lpx_aaa_conf/ qed-.
+
+lemma lpx_aaa_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/2 width=7 by cpx_lpx_aaa_conf/ qed-.
+
+lemma cpr_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/3 width=5 by cpx_aaa_conf, cpr_cpx/ qed-.
+
+lemma lpr_aaa_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by lpx_aaa_conf, lpr_lpx/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_drop.ma
new file mode 100644 (file)
index 0000000..6509da5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_drop.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn_drop.ma".
+include "basic_2A/reduction/cpx_lift.ma".
+include "basic_2A/reduction/lpx.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on local environment slicing ***********************************)
+
+lemma lpx_drop_conf: ∀h,g,G. dropable_sn (lpx h g G).
+/3 width=6 by lpx_sn_deliftable_dropable, cpx_inv_lift1/ qed-.
+
+lemma drop_lpx_trans: ∀h,g,G. dedropable_sn (lpx h g G).
+/3 width=10 by lpx_sn_liftable_dedropable, cpx_lift/ qed-.
+
+lemma lpx_drop_trans_O1: ∀h,g,G. dropable_dx (lpx h g G).
+/2 width=3 by lpx_sn_dropable/ qed-.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
+                     ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
+[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
+  #K2 #W2 #HLK2 #HVW2 #H destruct
+  /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
+  elim (drop_lpx_trans … HLK1 … HK12) -HK12
+  /3 width=7 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma fquq_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
+                      ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                     ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/
+[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H
+  #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
+  /4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #L0 #HL01
+  elim (lpx_drop_trans_O1 … HL01 … HLK1) -L1
+  /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
+[ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_frees.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_frees.ma
new file mode 100644 (file)
index 0000000..6a187d4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_frees.ma
@@ -0,0 +1,88 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/frees_lreq.ma".
+include "basic_2A/multiple/frees_lift.ma".
+include "basic_2A/reduction/lpx_drop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on context-sensitive free variables ***************************)
+
+lemma lpx_cpx_frees_trans: ∀h,g,G,L1,U1,U2. ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2 →
+                           ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+                           ∀i. L2 ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U1⦄.
+#h #g #G #L1 #U1 @(fqup_wf_ind_eq … G L1 U1) -G -L1 -U1
+#G0 #L0 #U0 #IH #G #L1 * *
+[ -IH #k #HG #HL #HU #U2 #H1 #L2 #_ #i #H2 elim (cpx_inv_sort1 … H1) -H1
+  [| * #d #_ ] #H destruct elim (frees_inv_sort … H2)
+| #j #HG #HL #HU #U2 #H1 #L2 #HL12 #i #H2 elim (cpx_inv_lref1 … H1) -H1
+  [ #H destruct elim (frees_inv_lref … H2) -H2 //
+    * #I #K2 #W2 #Hj #Hji #HLK2 #HW2
+    elim (lpx_drop_trans_O1 … HL12 … HLK2) -HL12 #Y #HLK1 #H
+    elim (lpx_inv_pair2 … H) -H #K1 #W1 #HK12 #HW12 #H destruct
+    /4 width=11 by frees_lref_be, fqup_lref/
+  | * #I #K1 #W1 #W0 #HLK1 #HW10 #HW0U2
+    lapply (drop_fwd_drop2 … HLK1) #H0
+    elim (lpx_drop_conf … H0 … HL12) -H0 -HL12 #K2 #HK12 #HLK2
+    elim (lt_or_ge i (j+1)) #Hji
+    [ -IH elim (frees_inv_lift_be … H2 … HLK2 … HW0U2) /2 width=1 by monotonic_pred/
+    | lapply (frees_inv_lift_ge … H2 … HLK2 … HW0U2 ?) -L2 -U2 // <minus_plus destruct
+      /4 width=11 by frees_lref_be, fqup_lref/
+    ]
+  ]
+| -IH #p #HG #HL #HU #U2 #H1 >(cpx_inv_gref1 … H1) -H1 destruct
+   #L2 #_ #i #H2 elim (frees_inv_gref … H2)
+| #a #I #W1 #U1 #HG #HL #HU #X #HX #L2 #HL12 #i #Hi destruct
+  elim (cpx_inv_bind1 … HX) -HX *
+  [ #W2 #U2 #HW12 #HU12 #H destruct
+    elim (frees_inv_bind_O … Hi) -Hi
+    /4 width=7 by frees_bind_dx_O, frees_bind_sn, lpx_pair/
+  | #U2 #HU12 #HXU2 #H1 #H2 destruct
+    lapply (frees_lift_ge … Hi (L2.ⓓW1) (Ⓕ) … HXU2 ?)
+    /4 width=7 by frees_bind_dx_O, lpx_pair, drop_drop/
+  ]
+| #I #W1 #X1 #HG #HL #HU #X2 #HX2 #L2 #HL12 #i #Hi destruct
+  elim (cpx_inv_flat1 … HX2) -HX2 *
+  [ #W2 #U2 #HW12 #HU12 #H destruct
+    elim (frees_inv_flat … Hi) -Hi /3 width=7 by frees_flat_dx, frees_flat_sn/
+  | #HU12 #H destruct /3 width=7 by frees_flat_dx/
+  | #HW12 #H destruct /3 width=7 by frees_flat_sn/
+  | #b #W2 #V1 #V2 #U1 #U2 #HW12 #HV12 #HU12 #H1 #H2 #H3 destruct
+    elim (frees_inv_bind … Hi) -Hi #Hi
+    [ elim (frees_inv_flat … Hi) -Hi
+      /4 width=7 by frees_flat_dx, frees_flat_sn, frees_bind_sn/
+    | lapply (lreq_frees_trans … Hi (L2.ⓛV2) ?) /2 width=1 by lreq_succ/ -Hi #HU2
+      lapply (frees_weak … HU2 0 ?) -HU2
+      /5 width=7 by frees_bind_dx_O, frees_flat_dx, lpx_pair/
+    ]
+  | #b #W2 #W0 #V1 #V2 #U1 #U2 #HW12 #HW20 #HV12 #HU12 #H1 #H2 #H3 destruct
+    elim (frees_inv_bind_O … Hi) -Hi #Hi
+    [ /4 width=7 by frees_flat_dx, frees_bind_sn/
+    | elim (frees_inv_flat … Hi) -Hi
+      [ #HW0 lapply (frees_inv_lift_ge … HW0 L2 (Ⓕ) … HW20 ?) -W0
+        /3 width=7 by frees_flat_sn, drop_drop/
+      | /5 width=7 by frees_bind_dx_O, frees_flat_dx, lpx_pair/
+      ]
+    ]
+  ]
+]
+qed-.
+
+lemma cpx_frees_trans: ∀h,g,G. frees_trans (cpx h g G).
+/2 width=8 by lpx_cpx_frees_trans/ qed-.
+
+lemma lpx_frees_trans: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+                       ∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄.
+/2 width=8 by lpx_cpx_frees_trans/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_lleq.ma
new file mode 100644 (file)
index 0000000..0563bae
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/reduction/lpx_lleq.ma
@@ -0,0 +1,136 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llor_drop.ma".
+include "basic_2A/multiple/llpx_sn_llor.ma".
+include "basic_2A/multiple/llpx_sn_lpx_sn.ma".
+include "basic_2A/multiple/lleq_lreq.ma".
+include "basic_2A/multiple/lleq_llor.ma".
+include "basic_2A/reduction/cpx_lreq.ma".
+include "basic_2A/reduction/cpx_lleq.ma".
+include "basic_2A/reduction/lpx_frees.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+(* Note: contains a proof of llpx_cpx_conf *)
+lemma lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
+                      ∀L1,T,l. L1 ≡[T, l] L2 →
+                      ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & K1 ≡[T, l] K2.
+#h #g #G #L2 #K2 #HLK2 #L1 #T #l #HL12
+lapply (lpx_fwd_length … HLK2) #H1
+lapply (lleq_fwd_length … HL12) #H2
+lapply (lpx_sn_llpx_sn … T … l HLK2) // -HLK2 #H
+lapply (lleq_llpx_sn_trans … HL12 … H) /2 width=3 by lleq_cpx_trans/ -HL12 -H #H
+elim (llor_total L1 K2 T l) // -H1 -H2 #K1 #HLK1
+lapply (llpx_sn_llor_dx_sym … H … HLK1)
+[ /2 width=6 by cpx_frees_trans/
+| /3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/
+| /3 width=5 by llpx_sn_llor_fwd_sn, ex2_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                          ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
+  #K0 #V0 #H1KL1 #_ #H destruct
+  elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
+  #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
+  /2 width=4 by fqu_lref_O, ex3_intro/
+| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
+  [ elim (lleq_inv_bind … H)
+  | elim (lleq_inv_flat … H)
+  ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
+| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
+| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
+  /2 width=4 by fqu_flat_dx, ex3_intro/
+| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+  elim (drop_O1_le (Ⓕ) (m+1) K1)
+  [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
+    #H2KL elim (lpx_drop_trans_O1 … H1KL1 … HL1) -L1
+    #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
+    /3 width=4 by fqu_drop, ex3_intro/
+  | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
+    lapply (lleq_fwd_length … H2KL1) //
+  ]
+]
+qed-.
+
+lemma lpx_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fquq_inv_gen … H) -H
+[ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
+  #K #HT1 #H1KL #H2KL elim (lpx_lleq_fqu_trans … HT2 … H1KL H2KL) -L
+  /3 width=5 by fqup_strap1, ex3_intro/
+]
+qed-.
+
+lemma lpx_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fqus_inv_gen … H) -H
+[ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
+  /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+fact lreq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
+                              ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                              ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                                   (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
+[ #l #m #_ #L2 #H >(lpx_inv_atom1 … H) -H
+  /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpx_pair, lreq_cpx_trans, lreq_pair/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
+  elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+  @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpx_pair, lreq_succ/
+  #T elim (IH T) #HL0dx #HL0sn
+  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
+]
+qed-.
+
+lemma lreq_lpx_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+                           ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
+                           ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+                                (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+/2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa.ma
new file mode 100644 (file)
index 0000000..2870142
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa.ma
@@ -0,0 +1,145 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/atomicarity_4.ma".
+include "basic_2A/grammar/aarity.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* activate genv *)
+inductive aaa: relation4 genv lenv term aarity ≝
+| aaa_sort: ∀G,L,k. aaa G L (⋆k) (⓪)
+| aaa_lref: ∀I,G,L,K,V,B,i. ⬇[i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
+| aaa_abbr: ∀a,G,L,V,T,B,A.
+            aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{a}V.T) A
+| aaa_abst: ∀a,G,L,V,T,B,A.
+            aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{a}V.T) (②B.A)
+| aaa_appl: ∀G,L,V,T,B,A. aaa G L V B → aaa G L T (②B.A) → aaa G L (ⓐV.T) A
+| aaa_cast: ∀G,L,V,T,A. aaa G L V A → aaa G L T A → aaa G L (ⓝV.T) A
+.
+
+interpretation "atomic arity assignment (term)"
+   'AtomicArity G L T A = (aaa G L T A).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
+#G #L #T #A * -G -L -T -A
+[ //
+| #I #G #L #K #V #B #i #_ #_ #k #H destruct
+| #a #G #L #V #T #B #A #_ #_ #k #H destruct
+| #a #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #A #_ #_ #k #H destruct
+]
+qed-.
+
+lemma aaa_inv_sort: ∀G,L,A,k. ⦃G, L⦄ ⊢ ⋆k ⁝ A → A = ⓪.
+/2 width=6 by aaa_inv_sort_aux/ qed-.
+
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
+                       ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #i #H destruct
+| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5 by ex2_3_intro/
+| #a #G #L #V #T #B #A #_ #_ #i #H destruct
+| #a #G #L #V #T #B #A #_ #_ #i #H destruct
+| #G #L #V #T #B #A #_ #_ #i #H destruct
+| #G #L #V #T #A #_ #_ #i #H destruct
+]
+qed-.
+
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #i ⁝ A →
+                    ∃∃I,K,V. ⬇[i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+/2 width=3 by aaa_inv_lref_aux/ qed-.
+
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p. T = §p → ⊥.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #q #H destruct
+| #I #G #L #K #V #B #i #HLK #HB #q #H destruct
+| #a #G #L #V #T #B #A #_ #_ #q #H destruct
+| #a #G #L #V #T #B #A #_ #_ #q #H destruct
+| #G #L #V #T #B #A #_ #_ #q #H destruct
+| #G #L #V #T #A #_ #_ #q #H destruct
+]
+qed-.
+
+lemma aaa_inv_gref: ∀G,L,A,p. ⦃G, L⦄ ⊢ §p ⁝ A → ⊥.
+/2 width=7 by aaa_inv_gref_aux/ qed-.
+
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
+                       ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #a #W #U #H destruct
+| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2 by ex2_intro/
+| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
+| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_abbr: ∀a,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{a}V. T ⁝ A →
+                    ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
+/2 width=4 by aaa_inv_abbr_aux/ qed-.
+
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
+                       ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #a #W #U #H destruct
+| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
+| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
+| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_abst: ∀a,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{a}W. T ⁝ A →
+                    ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
+/2 width=4 by aaa_inv_abst_aux/ qed-.
+
+fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
+                       ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #W #U #H destruct
+| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
+| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #A #_ #_ #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
+                    ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
+/2 width=3 by aaa_inv_appl_aux/ qed-.
+
+fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
+                       ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #k #W #U #H destruct
+| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
+| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW. T ⁝ A →
+                    ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
+/2 width=3 by aaa_inv_cast_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_aaa.ma
new file mode 100644 (file)
index 0000000..a47130c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_aaa.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
+#G #L #T #A1 #H elim H -G -L -T -A1
+[ #G #L #k #A2 #H
+  >(aaa_inv_sort … H) -H //
+| #I1 #G #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H
+  elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2
+  lapply (drop_mono … HLK1 … HLK2) -L #H destruct /2 width=1 by/
+| #a #G #L #V #T #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
+  elim (aaa_inv_abbr … H) -H /2 width=1 by/
+| #a #G #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H
+  elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1 by eq_f2/
+| #G #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
+  elim (aaa_inv_appl … H) -H #B2 #_ #HA2
+  lapply (IHA1 … HA2) -L #H destruct //
+| #G #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H
+  elim (aaa_inv_cast … H) -H /2 width=1 by/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_fqus.ma
new file mode 100644 (file)
index 0000000..922c971
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_fqus.ma
@@ -0,0 +1,63 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/fqus_alt.ma".
+include "basic_2A/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                    ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #A #H elim (aaa_inv_lref … H) -H
+  #J #K #V #H #HA lapply (drop_inv_O2 … H) -H
+  #H destruct /2 width=2 by ex_intro/
+| * [ #a ] * #G #L #V #T #X #H
+  [ elim (aaa_inv_abbr … H)
+  | elim (aaa_inv_abst … H)
+  | elim (aaa_inv_appl … H)
+  | elim (aaa_inv_cast … H)
+  ] -H /2 width=2 by ex_intro/
+| #a * #G #L #V #T #X #H
+  [ elim (aaa_inv_abbr … H)
+  | elim (aaa_inv_abst … H)
+  ] -H /2 width=2 by ex_intro/
+| * #G #L #V #T #X #H
+  [ elim (aaa_inv_appl … H)
+  | elim (aaa_inv_cast … H)
+  ] -H /2 width=2 by ex_intro/
+| /3 width=9 by aaa_inv_lift, ex_intro/
+]
+qed-.
+
+lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                     ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fquq_inv_gen … H) -H /2 width=6 by aaa_fqu_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.
+
+lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                     ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
+/2 width=6 by aaa_fqu_conf/
+qed-.
+
+lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                     ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_gen … H) -H /2 width=6 by aaa_fqup_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lift.ma
new file mode 100644 (file)
index 0000000..7a05536
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lift.ma
@@ -0,0 +1,73 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on basic relocation *******************************************)
+
+lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,s,l,m. ⬇[s, l, m] L2 ≡ L1 →
+                ∀T2. ⬆[l, m] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #k #L2 #s #l #m #_ #T2 #H
+  >(lift_inv_sort1 … H) -H //
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #l #m #HL21 #T2 #H
+  elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+  [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by aaa_lref/
+  | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=9 by aaa_lref, drop_inv_gen/
+  ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /4 width=5 by aaa_abbr, drop_skip/
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /4 width=5 by aaa_abst, drop_skip/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /3 width=5 by aaa_appl/
+| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /3 width=5 by aaa_cast/
+]
+qed.
+
+lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,s,l,m. ⬇[s, l, m] L2 ≡ L1 →
+                    ∀T1. ⬆[l, m] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
+[ #G #L2 #k #L1 #s #l #m #_ #T1 #H
+  >(lift_inv_sort2 … H) -H //
+| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #l #m #HL21 #T1 #H
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
+  ]
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /4 width=5 by aaa_abbr, drop_skip/
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /4 width=5 by aaa_abst, drop_skip/
+| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /3 width=5 by aaa_appl/
+| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /3 width=5 by aaa_cast/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lifts.ma
new file mode 100644 (file)
index 0000000..0b08733
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lifts.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/drops.ma".
+include "basic_2A/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties concerning generic relocation *********************************)
+
+lemma aaa_lifts: ∀G,L1,L2,T2,A,s,cs. ⬇*[s, cs] L2 ≡ L1 → ∀T1. ⬆*[cs] T1 ≡ T2 →
+                 ⦃G, L1⦄ ⊢ T1 ⁝ A → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#G #L1 #L2 #T2 #A #s #cs #H elim H -L1 -L2 -cs
+[ #L #T1 #H #HT1
+  <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL1 #T1 #H #HT1
+  elim (lifts_inv_cons … H) -H /3 width=10 by aaa_lift/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lleq.ma
new file mode 100644 (file)
index 0000000..5907427
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/aaa_lleq.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_drop.ma".
+include "basic_2A/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_aaa_trans: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+                      ∀L1. L1 ≡[T, 0] L2 → ⦃G, L1⦄ ⊢ T ⁝ A.
+#G #L2 #T #A #H elim H -G -L2 -T -A /2 width=1 by aaa_sort/
+[ #I #G #L2 #K2 #V2 #A #i #HLK2 #_ #IHV2 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
+  [ #H elim (ylt_yle_false … H) //
+  | * /3 width=5 by aaa_lref/
+  ]
+| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=2 by aaa_abbr/
+| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
+  /3 width=1 by aaa_abst/
+| #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=3 by aaa_appl/
+| #G #L2 #V #T #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
+  /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma aaa_lleq_conf: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+                     ∀L1. L2 ≡[T, 0] L1 → ⦃G, L1⦄ ⊢ T ⁝ A.
+/3 width=3 by lleq_aaa_trans, lleq_sym/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/da.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/da.ma
new file mode 100644 (file)
index 0000000..d083238
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/da.ma
@@ -0,0 +1,108 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/degree_6.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/drop.ma".
+include "basic_2A/static/sd.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* activate genv *)
+inductive da (h:sh) (g:sd h): relation4 genv lenv term nat ≝
+| da_sort: ∀G,L,k,d. deg h g k d → da h g G L (⋆k) d
+| da_ldef: ∀G,L,K,V,i,d. ⬇[i] L ≡ K.ⓓV → da h g G K V d → da h g G L (#i) d
+| da_ldec: ∀G,L,K,W,i,d. ⬇[i] L ≡ K.ⓛW → da h g G K W d → da h g G L (#i) (d+1)
+| da_bind: ∀a,I,G,L,V,T,d. da h g G (L.ⓑ{I}V) T d → da h g G L (ⓑ{a,I}V.T) d
+| da_flat: ∀I,G,L,V,T,d. da h g G L T d → da h g G L (ⓕ{I}V.T) d
+.
+
+interpretation "degree assignment (term)"
+   'Degree h g G L T d = (da h g G L T d).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact da_inv_sort_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
+                      ∀k0. T = ⋆k0 → deg h g k0 d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #Hkd #k0 #H destruct //
+| #G #L #K #V #i #d #_ #_ #k0 #H destruct
+| #G #L #K #W #i #d #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #d #_ #k0 #H destruct
+| #I #G #L #V #T #d #_ #k0 #H destruct
+]
+qed-.
+
+lemma da_inv_sort: ∀h,g,G,L,k,d. ⦃G, L⦄ ⊢ ⋆k ▪[h, g] d → deg h g k d.
+/2 width=5 by da_inv_sort_aux/ qed-.
+
+fact da_inv_lref_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀j. T = #j →
+                      (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] d) ∨
+                      (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] d0 &
+                                 d = d0 + 1
+                       ).
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #j #H destruct
+| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
+| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
+| #a #I #G #L #V #T #d #_ #j #H destruct
+| #I #G #L #V #T #d #_ #j #H destruct
+]
+qed-.
+
+lemma da_inv_lref: ∀h,g,G,L,j,d. ⦃G, L⦄ ⊢ #j ▪[h, g] d →
+                   (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] d) ∨
+                   (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] d0 & d = d0+1).
+/2 width=3 by da_inv_lref_aux/ qed-.
+
+fact da_inv_gref_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀p0. T = §p0 → ⊥.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #p0 #H destruct
+| #G #L #K #V #i #d #_ #_ #p0 #H destruct
+| #G #L #K #W #i #d #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #d #_ #p0 #H destruct
+| #I #G #L #V #T #d #_ #p0 #H destruct
+]
+qed-.
+
+lemma da_inv_gref: ∀h,g,G,L,p,d. ⦃G, L⦄ ⊢ §p ▪[h, g] d → ⊥.
+/2 width=9 by da_inv_gref_aux/ qed-.
+
+fact da_inv_bind_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
+                      ∀b,J,X,Y. T = ⓑ{b,J}Y.X → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, g] d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #b #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct //
+| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct
+]
+qed-.
+
+lemma da_inv_bind: ∀h,g,b,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X ▪[h, g] d → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, g] d.
+/2 width=4 by da_inv_bind_aux/ qed-.
+
+fact da_inv_flat_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
+                      ∀J,X,Y. T = ⓕ{J}Y.X → ⦃G, L⦄ ⊢ X ▪[h, g] d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct
+| #I #G #L #V #T #d #HT #J #X #Y #H destruct //
+]
+qed-.
+
+lemma da_inv_flat: ∀h,g,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓕ{J}Y.X ▪[h, g] d → ⦃G, L⦄ ⊢ X ▪[h, g] d.
+/2 width=5 by da_inv_flat_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/da_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/da_aaa.ma
new file mode 100644 (file)
index 0000000..433a577
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/da_aaa.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_lift.ma".
+include "basic_2A/static/da.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃d. ⦃G, L⦄ ⊢ T ▪[h, g] d.
+#h #g #G #L #T #A #H elim H -G -L -T -A
+[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
+| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/da_da.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/da_da.ma
new file mode 100644 (file)
index 0000000..ecef4ca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/da_da.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/da_lift.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Main properties **********************************************************)
+
+theorem da_mono: ∀h,g,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 →
+                 ∀d2. ⦃G, L⦄ ⊢ T ▪[h, g] d2 → d1 = d2.
+#h #g #G #L #T #d1 #H elim H -G -L -T -d1
+[ #G #L #k #d1 #Hkd1 #d2 #H
+  lapply (da_inv_sort … H) -G -L #Hkd2
+  >(deg_mono … Hkd2 … Hkd1) -h -k -d2 //
+| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H
+  elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ]
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /2 width=1 by/
+| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H
+  elim (da_inv_lref … H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ]
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /3 width=1 by eq_f/
+| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H
+  lapply (da_inv_bind … H) -H /2 width=1 by/
+| #I #G #L #V #T #d1 #_ #IHT #d2 #H
+  lapply (da_inv_flat … H) -H /2 width=1 by/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/da_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/da_lift.ma
new file mode 100644 (file)
index 0000000..8721ebd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/da_lift.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/static/da.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on relocation *************************************************)
+
+lemma da_lift: ∀h,g,G,L1,T1,d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d →
+               ∀L2,s,l,m. ⬇[s, l, m] L2 ≡ L1 → ∀T2. ⬆[l, m] T1 ≡ T2 →
+               ⦃G, L2⦄ ⊢ T2 ▪[h, g] d.
+#h #g #G #L1 #T1 #d #H elim H -G -L1 -T1 -d
+[ #G #L1 #k #d #Hkd #L2 #s #l #m #_ #X #H
+  >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
+| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by da_ldef/
+  | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=8 by da_ldef, drop_inv_gen/
+  ]
+| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
+    /3 width=8 by da_ldec/
+  | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=8 by da_ldec, drop_inv_gen/
+  ]
+| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+  /4 width=5 by da_bind, drop_skip/
+| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+  /3 width=5 by da_flat/
+]
+qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma da_inv_lift: ∀h,g,G,L2,T2,d. ⦃G, L2⦄ ⊢ T2 ▪[h, g] d →
+                   ∀L1,s,l,m. ⬇[s, l, m] L2 ≡ L1 → ∀T1. ⬆[l, m] T1 ≡ T2 →
+                   ⦃G, L1⦄ ⊢ T1 ▪[h, g] d.
+#h #g #G #L2 #T2 #d #H elim H -G -L2 -T2 -d
+[ #G #L2 #k #d #Hkd #L1 #s #l #m #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
+| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HV2 | -IHV2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
+  ]
+| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HW2 | -IHW2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
+  ]
+| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /4 width=5 by da_bind, drop_skip/
+| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /3 width=5 by da_flat/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba.ma
new file mode 100644 (file)
index 0000000..c22a725
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba.ma
@@ -0,0 +1,144 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lrsubeqa_3.ma".
+include "basic_2A/static/lsubr.ma".
+include "basic_2A/static/aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+inductive lsuba (G:genv): relation lenv ≝
+| lsuba_atom: lsuba G (⋆) (⋆)
+| lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+              lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+  "local environment refinement (atomic arity assignment)"
+  'LRSubEqA G L1 L2 = (lsuba G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
+/2 width=4 by lsuba_inv_atom1_aux/ qed-.
+
+fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
+                          ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                      G ⊢ K1 ⫃⁝ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by or_intror, ex6_4_intro/
+]
+qed-.
+
+lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃⁝ L2 →
+                       (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
+                       ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+                                   I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsuba_inv_pair1_aux/ qed-.
+
+fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
+/2 width=4 by lsuba_inv_atom2_aux/ qed-.
+
+fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
+                          (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
+                          ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+                                    G ⊢ K1 ⫃⁝ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by or_intror, ex5_3_intro/
+]
+qed-.
+
+lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⫃⁝ K2.ⓑ{I}W →
+                       (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
+                       ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+                                 I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 ⫃ L2.
+#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
+#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
+                          ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L2 ≡ K2.
+#G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+                           ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L1 ≡ K1.
+#G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_aaa.ma
new file mode 100644 (file)
index 0000000..3a5cd5d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_aaa.ma
@@ -0,0 +1,56 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_aaa.ma".
+include "basic_2A/static/lsuba.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Properties concerning atomic arity assignment ****************************)
+
+lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
+                      ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
+#G #L1 #V #A #H elim H -G -L1 -V -A
+[ //
+| #I #G #L1 #K1 #V #A #i #HLK1 #HV #IHV #L2 #HL12
+  elim (lsuba_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+  elim (lsuba_inv_pair1 … H) -H * #K2
+  [ #HK12 #H destruct /3 width=5 by aaa_lref/
+  | #W0 #V0 #A0 #HV0 #HW0 #_ #H1 #H2 #H3 destruct
+    lapply (aaa_mono … HV0 … HV) #H destruct -V0 /2 width=5 by aaa_lref/
+  ]
+| /4 width=2 by lsuba_pair, aaa_abbr/
+| /4 width=1 by lsuba_pair, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
+                       ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
+#G #L2 #V #A #H elim H -G -L2 -V -A
+[ //
+| #I #G #L2 #K2 #V #A #i #HLK2 #H1V #IHV #L1 #HL12
+  elim (lsuba_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+  elim (lsuba_inv_pair2 … H) -H * #K1
+  [ #HK12 #H destruct /3 width=5 by aaa_lref/
+  | #V0 #A0 #HV0 #H2V #_ #H1 #H2 destruct
+    lapply (aaa_mono … H2V … H1V) #H destruct -K2 /2 width=5 by aaa_lref/
+  ]
+| /4 width=2 by lsuba_pair, aaa_abbr/
+| /4 width=1 by lsuba_pair, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /3 width=1 by aaa_cast/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_lsuba.ma
new file mode 100644 (file)
index 0000000..40384ad
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsuba_lsuba.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsuba_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Main properties **********************************************************)
+
+theorem lsuba_trans: ∀G,L1,L. G ⊢ L1 ⫃⁝ L → ∀L2. G ⊢ L ⫃⁝ L2 → G ⊢ L1 ⫃⁝ L2.
+#G #L1 #L #H elim H -L1 -L
+[ #X #H >(lsuba_inv_atom1 … H) -H //
+| #I #L1 #L #Y #HL1 #IHL1 #X #H
+  elim (lsuba_inv_pair1 … H) -H * #L2
+  [ #HL2 #H destruct /3 width=1 by lsuba_pair/
+  | #W #V #A #HV #HW #HL2 #H1 #H2 #H3 destruct
+    /3 width=3 by lsuba_beta, lsuba_aaa_trans/
+  ]
+| #L1 #L #W #V #A #HV #HW #HL1 #IHL1 #X #H
+  elim (lsuba_inv_pair1 … H) -H * #L2
+  [ #HL2 #H destruct /3 width=5 by lsuba_beta, lsuba_aaa_conf/
+  | #W0 #V0 #A0 #_ #_ #_ #H destruct
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd.ma
new file mode 100644 (file)
index 0000000..e3d01c2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd.ma
@@ -0,0 +1,151 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lrsubeqd_5.ma".
+include "basic_2A/static/lsubr.ma".
+include "basic_2A/static/da.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+inductive lsubd (h) (g) (G): relation lenv ≝
+| lsubd_atom: lsubd h g G (⋆) (⋆)
+| lsubd_pair: ∀I,L1,L2,V. lsubd h g G L1 L2 →
+              lsubd h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubd_beta: ∀L1,L2,W,V,d. ⦃G, L1⦄ ⊢ V ▪[h, g] d+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d →
+              lsubd h g G L1 L2 → lsubd h g G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+  "local environment refinement (degree assignment)"
+  'LRSubEqD h g G L1 L2 = (lsubd h g G L1 L2).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubd_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L1 ⫃ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubd_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubd_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃▪[h, g] L2 → L2 = ⋆.
+/2 width=6 by lsubd_inv_atom1_aux/ qed-.
+
+fact lsubd_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
+                          ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          (∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                          ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
+                                      G ⊢ K1 ⫃▪[h, g] K2 &
+                                      I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#h #g #G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
+]
+qed-.
+
+lemma lsubd_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃▪[h, g] L2 →
+                       (∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                       ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
+                                   G ⊢ K1 ⫃▪[h, g] K2 &
+                                   I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsubd_inv_pair1_aux/ qed-.
+
+fact lsubd_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L2 = ⋆ → L1 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubd_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃▪[h, g] ⋆ → L1 = ⋆.
+/2 width=6 by lsubd_inv_atom2_aux/ qed-.
+
+fact lsubd_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
+                          ∀I,K2,W. L2 = K2.ⓑ{I}W →
+                          (∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                          ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
+                                    G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+#h #g #G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
+]
+qed-.
+
+lemma lsubd_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃▪[h, g] K2.ⓑ{I}W →
+                       (∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                       ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
+                                 G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+/2 width=3 by lsubd_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubd_refl: ∀h,g,G,L. G ⊢ L ⫃▪[h, g] L.
+#h #g #G #L elim L -L /2 width=1 by lsubd_pair/
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubd_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
+                          ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
+                          ∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+  [ destruct
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubd_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
+                           ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+                           ∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+  [ destruct
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_da.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_da.ma
new file mode 100644 (file)
index 0000000..45d3551
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_da.ma
@@ -0,0 +1,65 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/da_da.ma".
+include "basic_2A/static/lsubd.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+(* Properties on degree assignment ******************************************)
+
+lemma lsubd_da_trans: ∀h,g,G,L2,T,d. ⦃G, L2⦄ ⊢ T ▪[h, g] d →
+                      ∀L1. G ⊢ L1 ⫃▪[h, g] L2 → ⦃G, L1⦄ ⊢ T ▪[h, g] d.
+#h #g #G #L2 #T #d #H elim H -G -L2 -T -d
+[ /2 width=1 by da_sort/
+| #G #L2 #K2 #V #i #d #HLK2 #_ #IHV #L1 #HL12
+  elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+  elim (lsubd_inv_pair2 … H) -H * #K1 [ | -IHV -HLK1 ]
+  [ #HK12 #H destruct /3 width=4 by da_ldef/
+  | #W #d0 #_ #_ #_ #H destruct
+  ]
+| #G #L2 #K2 #W #i #d #HLK2 #HW #IHW #L1 #HL12
+  elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+  elim (lsubd_inv_pair2 … H) -H * #K1 [ -HW | -IHW ]
+  [ #HK12 #H destruct /3 width=4 by da_ldec/
+  | #V #d0 #HV #H0W #_ #_ #H destruct
+    lapply (da_mono … H0W … HW) -H0W -HW #H destruct /3 width=7 by da_ldef, da_flat/
+  ]
+| /4 width=1 by lsubd_pair, da_bind/
+| /3 width=1 by da_flat/
+]
+qed-.
+
+lemma lsubd_da_conf: ∀h,g,G,L1,T,d. ⦃G, L1⦄ ⊢ T ▪[h, g] d →
+                     ∀L2. G ⊢ L1 ⫃▪[h, g] L2 → ⦃G, L2⦄ ⊢ T ▪[h, g] d.
+#h #g #G #L1 #T #d #H elim H -G -L1 -T -d
+[ /2 width=1 by da_sort/
+| #G #L1 #K1 #V #i #d #HLK1 #HV #IHV #L2 #HL12
+  elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+  elim (lsubd_inv_pair1 … H) -H * #K2 [ -HV | -IHV ]
+  [ #HK12 #H destruct /3 width=4 by da_ldef/
+  | #W0 #V0 #d0 #HV0 #HW0 #_ #_ #H1 #H2 destruct
+    lapply (da_inv_flat … HV) -HV #H0V0
+    lapply (da_mono … H0V0 … HV0) -H0V0 -HV0 #H destruct /2 width=4 by da_ldec/
+  ]
+| #G #L1 #K1 #W #i #d #HLK1 #HW #IHW #L2 #HL12
+  elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+  elim (lsubd_inv_pair1 … H) -H * #K2 [ -HW | -IHW ]
+  [ #HK12 #H destruct /3 width=4 by da_ldec/
+  | #W0 #V0 #d0 #HV0 #HW0 #_ #H destruct
+  ]
+| /4 width=1 by lsubd_pair, da_bind/
+| /3 width=1 by da_flat/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_lsubd.ma
new file mode 100644 (file)
index 0000000..6ff3a9f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubd_lsubd.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsubd_da.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+(* Main properties **********************************************************)
+
+theorem lsubd_trans: ∀h,g,G. Transitive … (lsubd h g G).
+#h #g #G #L1 #L #H elim H -L1 -L
+[ #X #H >(lsubd_inv_atom1 … H) -H //
+| #I #L1 #L #Y #HL1 #IHL1 #X #H
+  elim (lsubd_inv_pair1 … H) -H * #L2
+  [ #HL2 #H destruct /3 width=1 by lsubd_pair/
+  | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct
+    /3 width=3 by lsubd_beta, lsubd_da_trans/
+  ]
+| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H
+  elim (lsubd_inv_pair1 … H) -H * #L2
+  [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/
+  | #W0 #V0 #d0 #_ #_ #_ #H destruct
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr.ma
new file mode 100644 (file)
index 0000000..e176dd9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr.ma
@@ -0,0 +1,107 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/lrsubeqc_2.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+inductive lsubr: relation lenv ≝
+| lsubr_atom: ∀L. lsubr L (⋆)
+| lsubr_pair: ∀I,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+  "local environment refinement (restricted)"
+  'LRSubEqC L1 L2 = (lsubr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_refl: ∀L. L ⫃ L.
+#L elim L -L /2 width=1 by lsubr_atom, lsubr_pair/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⫃ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
+/2 width=3 by lsubr_inv_atom1_aux/ qed-.
+
+fact lsubr_inv_abst1_aux: ∀L1,L2. L1 ⫃ L2 → ∀K1,W. L1 = K1.ⓛW →
+                          L2 = ⋆ ∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW.
+#L1 #L2 * -L1 -L2
+[ #L #K1 #W #H destruct /2 width=1 by or_introl/
+| #I #L1 #L2 #V #HL12 #K1 #W #H destruct /3 width=3 by ex2_intro, or_intror/
+| #L1 #L2 #V1 #V2 #_ #K1 #W #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+                       L2 = ⋆ ∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW.
+/2 width=3 by lsubr_inv_abst1_aux/ qed-.
+
+fact lsubr_inv_abbr2_aux: ∀L1,L2. L1 ⫃ L2 → ∀K2,W. L2 = K2.ⓓW →
+                          ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓW.
+#L1 #L2 * -L1 -L2
+[ #L #K2 #W #H destruct
+| #I #L1 #L2 #V #HL12 #K2 #W #H destruct /2 width=3 by ex2_intro/
+| #L1 #L2 #V1 #V2 #_ #K2 #W #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⫃ K2.ⓓW →
+                       ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓW.
+/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⫃ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
+qed-.
+
+lemma lsubr_fwd_drop2_pair: ∀L1,L2. L1 ⫃ L2 →
+                            ∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
+                            (∃∃K1. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W) ∨
+                            ∃∃K1,V. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #I #K2 #W #s #i #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #s #i #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+  [ /3 width=3 by drop_pair, ex2_intro, or_introl/
+  | elim (IHL12 … HLK2) -IHL12 -HLK2 *
+    /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
+  ]
+| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #s #i #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
+  [ /3 width=4 by drop_pair, ex3_2_intro, or_intror/
+  | elim (IHL12 … HLK2) -IHL12 -HLK2 *
+    /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
+  ]
+]
+qed-.
+
+lemma lsubr_fwd_drop2_abbr: ∀L1,L2. L1 ⫃ L2 →
+                            ∀K2,V,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓓV →
+                            ∃∃K1. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓓV.
+#L1 #L2 #HL12 #K2 #V #s #i #HLK2 elim (lsubr_fwd_drop2_pair … HL12 … HLK2) -L2 // *
+#K1 #W #_ #_ #H destruct
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr_lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr_lsubr.ma
new file mode 100644 (file)
index 0000000..fbc688a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/lsubr_lsubr.ma
@@ -0,0 +1,53 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/lsubr.ma".
+
+(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+
+(* Auxiliary inversion lemmas ***********************************************)
+
+fact lsubr_inv_pair1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                          ∨∨ L2 = ⋆
+                           | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
+                           | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+                                       I = Abbr & X = ⓝW.V.
+#L1 #L2 * -L1 -L2
+[ #L #J #K1 #X #H destruct /2 width=1 by or3_intro0/
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by or3_intro1, ex2_intro/
+| #L1 #L2 #V #W #HL12 #J #K1 #X #H destruct /3 width=6 by or3_intro2, ex4_3_intro/
+]
+qed-.
+
+lemma lsubr_inv_pair1: ∀I,K1,L2,X. K1.ⓑ{I}X ⫃ L2 →
+                       ∨∨ L2 = ⋆
+                        | ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓑ{I}X
+                        | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+                                    I = Abbr & X = ⓝW.V.
+/2 width=3 by lsubr_inv_pair1_aux/ qed-.
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H
+  lapply (lsubr_inv_atom1 … H) -H //
+| #I #L1 #L #V #_ #IHL1 #X #H
+  elim (lsubr_inv_pair1 … H) -H // *
+  #L2 [2: #V2 #W2 ] #HL2 #H1 [ #H2 #H3 ] destruct /3 width=1 by lsubr_pair, lsubr_beta/
+| #L1 #L #V1 #W #_ #IHL1 #X #H
+  elim (lsubr_inv_abst1 … H) -H // *
+  #L2 #HL2 #H destruct /3 width=1 by lsubr_beta/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/sd.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/sd.ma
new file mode 100644 (file)
index 0000000..7b0947d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/sd.ma
@@ -0,0 +1,131 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/sh.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* sort degree specification *)
+record sd (h:sh): Type[0] ≝ {
+   deg      : relation nat;                            (* degree of the sort *)
+   deg_total: ∀k. ∃d. deg k d;                         (* functional relation axioms *)
+   deg_mono : ∀k,d1,d2. deg k d1 → deg k d2 → d1 = d2;
+   deg_next : ∀k,d. deg k d → deg (next h k) (d - 1)   (* compatibility condition *)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition deg_O: relation nat ≝ λk,d. d = 0.
+
+definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
+
+inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat ≝
+| deg_SO_pos : ∀d0. (next h)^d0 k0 = k → deg_SO h k k0 (d0 + 1)
+| deg_SO_zero: ((∃d0. (next h)^d0 k0 = k) → ⊥) → deg_SO h k k0 0
+.
+
+fact deg_SO_inv_pos_aux: ∀h,k,k0,d0. deg_SO h k k0 d0 → ∀d. d0 = d + 1 →
+                         (next h)^d k0 = k.
+#h #k #k0 #d0 * -d0
+[ #d0 #Hd0 #d #H
+  lapply (injective_plus_l … H) -H #H destruct //
+| #_ #d0 <plus_n_Sm #H destruct
+]
+qed.
+
+lemma deg_SO_inv_pos: ∀h,k,k0,d0. deg_SO h k k0 (d0 + 1) → (next h)^d0 k0 = k.
+/2 width=3 by deg_SO_inv_pos_aux/ qed-.
+
+lemma deg_SO_refl: ∀h,k. deg_SO h k k 1.
+#h #k @(deg_SO_pos … 0 ?) //
+qed.
+
+lemma deg_SO_gt: ∀h,k1,k2. k1 < k2 → deg_SO h k1 k2 0.
+#h #k1 #k2 #HK12 @deg_SO_zero * #d elim d -d normalize
+[ #H destruct
+  elim (lt_refl_false … HK12)
+| #d #_ #H
+  lapply (next_lt h ((next h)^d k2)) >H -H #H
+  lapply (transitive_lt … H HK12) -k1 #H1
+  lapply (nexts_le h k2 d) #H2
+  lapply (le_to_lt_to_lt … H2 H1) -h -d #H
+  elim (lt_refl_false … H)
+]
+qed.
+
+definition sd_SO: ∀h. nat → sd h ≝ λh,k. mk_sd h (deg_SO h k) ….
+[ #k0
+  lapply (nexts_dec h k0 k) *
+  [ * /3 width=2 by deg_SO_pos, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
+| #K0 #d1 #d2 * [ #d01 ] #H1 * [1,3: #d02 ] #H2 //
+  [ < H2 in H1; -H2 #H
+    lapply (nexts_inj … H) -H #H destruct //
+  | elim H1 /2 width=2 by ex_intro/
+  | elim H2 /2 width=2 by ex_intro/
+  ]
+| #k0 #d0 *
+  [ #d #H destruct elim d -d normalize
+    /2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
+  | #H1 @deg_SO_zero * #d #H2 destruct
+    @H1 -H1 @(ex_intro … (S d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+  ]
+]
+defined.
+
+let rec sd_d (h:sh) (k:nat) (d:nat) on d : sd h ≝
+   match d with
+   [ O   ⇒ sd_O h
+   | S d ⇒ match d with
+           [ O ⇒ sd_SO h k
+           | _ ⇒ sd_d h (next h k) d
+           ]
+   ].
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma deg_inv_pred: ∀h,g,k,d. deg h g (next h k) (d+1) → deg h g k (d+2).
+#h #g #k #d #H1
+elim (deg_total h g k) #d0 #H0
+lapply (deg_next … H0) #H2
+lapply (deg_mono … H1 H2) -H1 -H2 #H
+<(associative_plus d 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+qed-.
+
+lemma deg_inv_prec: ∀h,g,k,d,d0. deg h g ((next h)^d k) (d0+1) → deg h g k (d+d0+1).
+#h #g #k #d @(nat_ind_plus … d) -d //
+#d #IHd #d0 >iter_SO #H
+lapply (deg_inv_pred … H) -H <(associative_plus d0 1 1) #H
+lapply (IHd … H) -IHd -H //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter: ∀h,g,k,d1,d2. deg h g k d1 → deg h g ((next h)^d2 k) (d1-d2).
+#h #g #k #d1 #d2 @(nat_ind_plus … d2) -d2  [ <minus_n_O // ]
+#d2 #IHd2 #Hkd1 >iter_SO <minus_plus /3 width=1 by deg_next/
+qed.
+
+lemma deg_next_SO: ∀h,g,k,d. deg h g k (d+1) → deg h g (next h k) d.
+#h #g #k #d #Hkd
+lapply (deg_next … Hkd) -Hkd <minus_plus_m_m //
+qed-.
+
+lemma sd_d_SS: ∀h,k,d. sd_d h k (d + 2) = sd_d h (next h k) (d + 1).
+#h #k #d <plus_n_Sm <plus_n_Sm //
+qed.
+
+lemma sd_d_correct: ∀h,d,k. deg h (sd_d h k d) k d.
+#h #d @(nat_ind_plus … d) -d // #d @(nat_ind_plus … d) -d /3 width=1 by deg_inv_pred/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/static/sh.ma b/matita/matita/contribs/lambdadelta/basic_2A/static/sh.ma
new file mode 100644 (file)
index 0000000..490c9f9
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/static/sh.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/lib/arith.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specification *)
+record sh: Type[0] ≝ {
+   next   : nat → nat;     (* next sort in the hierarchy *)
+   next_lt: ∀k. k < next k (* strict monotonicity condition *)
+}.
+
+definition sh_N: sh ≝ mk_sh S ….
+// defined.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le: ∀h,k,d. k ≤ (next h)^d k.
+#h #k #d elim d -d // normalize #d #IHd
+lapply (next_lt h ((next h)^d k)) #H
+lapply (le_to_lt_to_lt … IHd H) -IHd -H /2 width=2 by lt_to_le/
+qed.
+
+lemma nexts_lt: ∀h,k,d. k < (next h)^(d+1) k.
+#h #k #d >iter_SO
+lapply (nexts_le h k d) #H
+@(le_to_lt_to_lt … H) //
+qed.
+
+axiom nexts_dec: ∀h,k1,k2. Decidable (∃d. (next h)^d k1 = k2).
+
+axiom nexts_inj: ∀h,k,d1,d2. (next h)^d1 k = (next h)^d2 k → d1 = d2.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma
new file mode 100644 (file)
index 0000000..4961f85
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy.ma
@@ -0,0 +1,296 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_max.ma".
+include "basic_2A/notation/relations/psubst_6.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
+| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
+| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
+             ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
+| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+             cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
+             cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+             cpy l m G L V1 V2 → cpy l m G L T1 T2 →
+             cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation "context-sensitive extended ordinary substritution (term)"
+   'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
+[ //
+| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
+  elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
+#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
+                ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
+#I #G #K #V #T1 elim T1 -T1
+[ * #i #L #l #HLK
+  /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
+  elim (lt_or_eq_or_gt i l) #Hil
+  /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
+  destruct
+  elim (lift_total V 0 (i+1)) #W #HVW
+  elim (lift_split … HVW i i)
+  /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+  [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
+    /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
+  | elim (IHU1 … HLK) -IHU1 -HLK
+    /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
+  ]
+]
+qed-.
+
+lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
+                ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+                ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
+                    ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
+[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
+  lapply (drop_fwd_length_lt2 … HLK)
+  /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
+| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
+  /2 width=1 by cpy_bind/
+| /2 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
+                     ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
+#G #L #T1 #T2 #l #m #HT12
+lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12
+/2 width=2 by cpy_weak_top/
+qed-.
+
+lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, i-l] T & ⦃G, L⦄ ⊢ T ▶[i, l+m-i] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+  elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
+  /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // #V
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
+  >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+  /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
+  /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
+                      ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+  elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
+  /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // #V
+  elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
+  >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+  /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+  elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
+  /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                  ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+                  l ≤ lt → l + m ≤ lt + mt →
+                  ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #T1 #l #m #H #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  ]
+| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
+  [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
+    #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
+    @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+  ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HVW1) -V1 -IHW12 //
+  elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+  <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+  /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
+  /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
+                        T2 = ⓪{J} ∨
+                        ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
+                                   ⬇[i] L ≡ K.ⓑ{I}V &
+                                   ⬆[O, i+1] V ≡ T2 &
+                                   J = LRef i.
+#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
+[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
+                     T2 = ⓪{I} ∨
+                     ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
+                                ⬇[i] L ≡ K.ⓑ{J}V &
+                                ⬆[O, i+1] V ≡ T2 &
+                                I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
+                     T2 = #i ∨
+                     ∃∃I,K,V. l ≤ i & i < l + m &
+                              ⬇[i] L ≡ K.ⓑ{I}V &
+                              ⬆[O, i+1] V ≡ T2.
+#G #L #T2 #i #l #m #H
+elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
+qed-.
+
+lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+                        ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                                 ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
+                                 U2 = ⓑ{a,I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                              ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
+                              U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
+
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+                        ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                                 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+                                 U2 = ⓕ{I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+                              ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+                              U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
+
+
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ //
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
+  elim (ylt_yle_false … Hli) -Hli //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
+]
+qed-.
+
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
+                        ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
+#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+qed-.
+
+(* Basic_1: removed theorems 25:
+            subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+            subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+            subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+            subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+            subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+            subst0_confluence_lift subst0_tlt
+            subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma
new file mode 100644 (file)
index 0000000..3a88571
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_cpy.ma
@@ -0,0 +1,122 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/cpy_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶[l1, m1] T1 →
+                     ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
+  elim (cpy_inv_lref1 … H) -H
+  [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
+  | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+    lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+    >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
+  ]
+| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+  elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
+  elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
+  /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+  elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  elim (IHV01 … HV02) -V0
+  elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶[l1, m1] T1 →
+                      ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+                      (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+                      ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶[l1, m1] T.
+#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
+  elim (cpy_inv_lref1 … H1) -H1
+  [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
+  | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
+    [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
+    | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
+    ]
+  ]
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+  elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
+  elim (IHT01 … HT02) -T0
+  [ -H #T #HT1 #HT2
+    lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+    lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+  | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
+  ]
+| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+  elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  elim (IHV01 … HV02 H) -V0
+  elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Note: the constant 1 comes from cpy_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
+                      ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
+#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
+[ #I #G #L #l #m #T2 #H #Hm
+  elim (cpy_inv_atom1 … H) -H
+  [ #H destruct //
+  | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
+    lapply (ylt_yle_trans … (l+m) … Hilm2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+  ]
+| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
+  lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+  >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+  elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+  lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+  lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+  elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
+]
+qed-.
+
+theorem cpy_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+                        ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+                        ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
+  lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
+  lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
+  >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+  elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+  elim (IHV10 … HV02) -IHV10 -HV02 // #V
+  elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+  lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+  elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+  elim (IHV10 … HV02) -V0 //
+  elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma
new file mode 100644 (file)
index 0000000..a188129
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_lift.ma
@@ -0,0 +1,249 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+  >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
+  lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+  lapply (ylt_inv_inj … Hil) -Hil #Hil
+  lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
+  elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+  elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+  elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
+  >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=7 by cpy_flat/
+]
+qed-.
+
+lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
+  >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
+  elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+  [ -Hltl
+    lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
+    elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
+    elim (drop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
+    >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
+  | -Hlti
+    elim (yle_inv_inj2 … Hltl) -Hltl #ltt #Hltl #H destruct
+    lapply (transitive_le … Hltl Hil) -Hltl #Hlti
+    lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+    lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
+    /4 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
+  ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=7 by cpy_flat/
+]
+qed-.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+                   ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+                   ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+                   l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+  >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
+  lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
+  elim (yle_inv_inj2 … Hil) -Hil #ll #Hlli #H0 destruct
+  lapply (lift_inv_lref1_ge … H … Hlli) -H #H destruct
+  lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+  lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hlli
+  /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /4 width=6 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  /3 width=6 by cpy_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma cpy_inv_lift1_le: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        lt + mt ≤ l →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
+  lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+  lapply (ylt_inv_inj … Hil) -Hil #Hil
+  lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
+  elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+  elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+  elim (IHU12 … HTU1) -IHU12 -HTU1
+  /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -W1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK
+  /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_inv_lift1_be: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        lt ≤ l → yinj l + m ≤ lt + mt →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
+  lapply (yle_fwd_plus_ge_inj … Hltl Hlmlmt) #Hmmt
+  elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
+  [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) /2 width=1 by ylt_inj/
+    [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
+    elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+    elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hil
+    /3 width=5 by cpy_subst, ex2_intro/
+  | elim (le_inv_plus_l … Hil) #Hlim #Hmi
+    lapply (yle_trans … Hltl (i-m) ?) /2 width=1 by yle_inj/ -Hltl #Hltim
+    lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
+    elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
+    #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
+    @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+    >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+  ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+  elim (IHU12 … HTU1) -U1
+  /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -W1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK //
+  /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                        ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                        yinj l + m ≤ lt →
+                        ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+  ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
+  lapply (yle_trans … Hlmlt … Hlti) #Hlmi
+  elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
+  elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
+  lapply (lift_inv_lref2_ge  … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+  lapply (drop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+  elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hlmi -Hlim
+  #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+  @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+  [ /2 width=1 by monotonic_yle_minus_dx/
+  | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+  ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
+  elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+  elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
+  >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+  elim (IHW12 … HLK … HVW1) -W1 //
+  elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation ***********************************)
+
+lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           l ≤ lt → lt ≤ yinj l + m → yinj l + m ≤ lt + mt →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[l, lt + mt - (yinj l + m)] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           lt ≤ l → lt + mt ≤ yinj l + m →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
+lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+                           ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+                           lt ≤ l → l ≤ lt + mt → lt + mt ≤ yinj l + m →
+                           ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
+elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
+lapply (cpy_weak … HU2 l m ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma
new file mode 100644 (file)
index 0000000..ded7af4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/cpy_nlift.ma
@@ -0,0 +1,66 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_neg.ma".
+include "basic_2A/substitution/lift_lift.ma".
+include "basic_2A/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on negated relocation ***********************************)
+
+lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+                         ∀i. l ≤ yinj i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
+                         (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
+                         ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
+                                    (∀V. ⬆[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
+[ /3 width=2 by or_introl/
+| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
+  elim (lt_or_ge j i) #Hij
+  [ @or_intror @(ex5_4_intro … HLK) // -HLK
+    [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
+      #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
+    | -HnW /2 width=7 by lift_inv_lref2_be/
+    ]
+  | elim (lift_split … HVW i j) -HVW //
+    #X #_ #H elim HnW -HnW //
+  ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
+  [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+    [ /4 width=9 by nlift_bind_sn, or_introl/
+    | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
+    ]
+  | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
+    [ /4 width=9 by nlift_bind_dx, or_introl/
+    | * #J #K #W #j #Hlj #Hji #HLK #HnW
+      elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
+      lapply (ylt_O … Hj) -Hj #Hj
+      lapply (drop_inv_drop1_lt … HLK ?) // -HLK #HLK
+      >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
+      #HnU1 <minus_le_minus_minus_comm in HnW;
+      /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
+    ]
+  ]
+| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
+  [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+    [ /4 width=9 by nlift_flat_sn, or_introl/
+    | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
+    ]
+  | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
+    [ /4 width=9 by nlift_flat_dx, or_introl/
+    | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop.ma
new file mode 100644 (file)
index 0000000..73e6829
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop.ma
@@ -0,0 +1,493 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/lib/lstar.ma".
+include "basic_2A/notation/relations/rdrop_5.ma".
+include "basic_2A/notation/relations/rdrop_4.ma".
+include "basic_2A/notation/relations/rdrop_3.ma".
+include "basic_2A/grammar/lenv_length.ma".
+include "basic_2A/grammar/cl_restricted_weight.ma".
+include "basic_2A/substitution/lift.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Basic_1: includes: drop_skip_bind *)
+inductive drop (s:bool): relation4 nat nat lenv lenv ≝
+| drop_atom: ∀l,m. (s = Ⓕ → m = 0) → drop s l m (⋆) (⋆)
+| drop_pair: ∀I,L,V. drop s 0 0 (L.ⓑ{I}V) (L.ⓑ{I}V)
+| drop_drop: ∀I,L1,L2,V,m. drop s 0 m L1 L2 → drop s 0 (m+1) (L1.ⓑ{I}V) L2
+| drop_skip: ∀I,L1,L2,V1,V2,l,m.
+             drop s l m L1 L2 → ⬆[l, m] V2 ≡ V1 →
+             drop s (l+1) m (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+.
+
+interpretation
+   "basic slicing (local environment) abstract"
+   'RDrop s l m L1 L2 = (drop s l m L1 L2).
+(*
+interpretation
+   "basic slicing (local environment) general"
+   'RDrop d e L1 L2 = (drop true d e L1 L2).
+*)
+interpretation
+   "basic slicing (local environment) lget"
+   'RDrop m L1 L2 = (drop false O m L1 L2).
+
+definition d_liftable: predicate (lenv → relation term) ≝
+                       λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
+                       ∀U1. ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → R L U1 U2.
+
+definition d_deliftable_sn: predicate (lenv → relation term) ≝
+                            λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
+                            ∀T1. ⬆[l, m] T1 ≡ U1 →
+                            ∃∃T2. ⬆[l, m] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: predicate (relation lenv) ≝
+                        λR. ∀L1,K1,s,l,m. ⬇[s, l, m] L1 ≡ K1 → ∀L2. R L1 L2 →
+                        ∃∃K2. R K1 K2 & ⬇[s, l, m] L2 ≡ K2.
+
+definition dropable_dx: predicate (relation lenv) ≝
+                        λR. ∀L1,L2. R L1 L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+                        ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & R K1 K2.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact drop_inv_atom1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → L1 = ⋆ →
+                         L2 = ⋆ ∧ (s = Ⓕ → m = 0).
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ /3 width=1 by conj/
+| #I #L #V #H destruct
+| #I #L1 #L2 #V #m #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was: drop_gen_sort *)
+lemma drop_inv_atom1: ∀L2,s,l,m. ⬇[s, l, m] ⋆ ≡ L2 → L2 = ⋆ ∧ (s = Ⓕ → m = 0).
+/2 width=4 by drop_inv_atom1_aux/ qed-.
+
+fact drop_inv_O1_pair1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → l = 0 →
+                            ∀K,I,V. L1 = K.ⓑ{I}V →
+                            (m = 0 ∧ L2 = K.ⓑ{I}V) ∨
+                            (0 < m ∧ ⬇[s, l, m-1] K ≡ L2).
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #K #J #W #H destruct
+| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
+| #I #L1 #L2 #V #m #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ >commutative_plus normalize #H destruct
+]
+qed-.
+
+lemma drop_inv_O1_pair1: ∀I,K,L2,V,s,m. ⬇[s, 0, m] K. ⓑ{I} V ≡ L2 →
+                         (m = 0 ∧ L2 = K.ⓑ{I}V) ∨
+                         (0 < m ∧ ⬇[s, 0, m-1] K ≡ L2).
+/2 width=3 by drop_inv_O1_pair1_aux/ qed-.
+
+lemma drop_inv_pair1: ∀I,K,L2,V,s. ⬇[s, 0, 0] K.ⓑ{I}V ≡ L2 → L2 = K.ⓑ{I}V.
+#I #K #L2 #V #s #H
+elim (drop_inv_O1_pair1 … H) -H * // #H destruct
+elim (lt_refl_false … H)
+qed-.
+
+(* Basic_1: was: drop_gen_drop *)
+lemma drop_inv_drop1_lt: ∀I,K,L2,V,s,m.
+                         ⬇[s, 0, m] K.ⓑ{I}V ≡ L2 → 0 < m → ⬇[s, 0, m-1] K ≡ L2.
+#I #K #L2 #V #s #m #H #Hm
+elim (drop_inv_O1_pair1 … H) -H * // #H destruct
+elim (lt_refl_false … Hm)
+qed-.
+
+lemma drop_inv_drop1: ∀I,K,L2,V,s,m.
+                      ⬇[s, 0, m+1] K.ⓑ{I}V ≡ L2 → ⬇[s, 0, m] K ≡ L2.
+#I #K #L2 #V #s #m #H lapply (drop_inv_drop1_lt … H ?) -H //
+qed-.
+
+fact drop_inv_skip1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → 0 < l →
+                         ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+                         ∃∃K2,V2. ⬇[s, l-1, m] K1 ≡ K2 &
+                                  ⬆[l-1, m] V2 ≡ V1 &
+                                   L2 = K2.ⓑ{I}V2.
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #J #K1 #W1 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #m #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop_gen_skip_l *)
+lemma drop_inv_skip1: ∀I,K1,V1,L2,s,l,m. ⬇[s, l, m] K1.ⓑ{I}V1 ≡ L2 → 0 < l →
+                      ∃∃K2,V2. ⬇[s, l-1, m] K1 ≡ K2 &
+                               ⬆[l-1, m] V2 ≡ V1 &
+                               L2 = K2.ⓑ{I}V2.
+/2 width=3 by drop_inv_skip1_aux/ qed-.
+
+lemma drop_inv_O1_pair2: ∀I,K,V,s,m,L1. ⬇[s, 0, m] L1 ≡ K.ⓑ{I}V →
+                         (m = 0 ∧ L1 = K.ⓑ{I}V) ∨
+                         ∃∃I1,K1,V1. ⬇[s, 0, m-1] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & 0 < m.
+#I #K #V #s #m *
+[ #H elim (drop_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #H
+  elim (drop_inv_O1_pair1 … H) -H *
+  [ #H1 #H2 destruct /3 width=1 by or_introl, conj/
+  | /3 width=5 by ex3_3_intro, or_intror/
+  ]
+]
+qed-.
+
+fact drop_inv_skip2_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → 0 < l →
+                         ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
+                         ∃∃K1,V1. ⬇[s, l-1, m] K1 ≡ K2 &
+                                  ⬆[l-1, m] V2 ≡ V1 &
+                                  L1 = K1.ⓑ{I}V1.
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #J #K2 #W2 #H destruct
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #m #_ #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: drop_gen_skip_r *)
+lemma drop_inv_skip2: ∀I,L1,K2,V2,s,l,m. ⬇[s, l, m] L1 ≡ K2.ⓑ{I}V2 → 0 < l →
+                      ∃∃K1,V1. ⬇[s, l-1, m] K1 ≡ K2 & ⬆[l-1, m] V2 ≡ V1 &
+                               L1 = K1.ⓑ{I}V1.
+/2 width=3 by drop_inv_skip2_aux/ qed-.
+
+lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → |L| < m →
+                      s = Ⓣ ∧ K = ⋆.
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m
+[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
+  #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1m)
+| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct
+  [ elim (lt_zero_false … H1m)
+  | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
+  ]
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma drop_refl_atom_O2: ∀s,l. ⬇[s, l, O] ⋆ ≡ ⋆.
+/2 width=1 by drop_atom/ qed.
+
+(* Basic_1: was by definition: drop_refl *)
+lemma drop_refl: ∀L,l,s. ⬇[s, l, 0] L ≡ L.
+#L elim L -L //
+#L #I #V #IHL #l #s @(nat_ind_plus … l) -l /2 width=1 by drop_pair, drop_skip/
+qed.
+
+lemma drop_drop_lt: ∀I,L1,L2,V,s,m.
+                    ⬇[s, 0, m-1] L1 ≡ L2 → 0 < m → ⬇[s, 0, m] L1.ⓑ{I}V ≡ L2.
+#I #L1 #L2 #V #s #m #HL12 #Hm >(plus_minus_m_m m 1) /2 width=1 by drop_drop/
+qed.
+
+lemma drop_skip_lt: ∀I,L1,L2,V1,V2,s,l,m.
+                    ⬇[s, l-1, m] L1 ≡ L2 → ⬆[l-1, m] V2 ≡ V1 → 0 < l →
+                    ⬇[s, l, m] L1. ⓑ{I} V1 ≡ L2.ⓑ{I}V2.
+#I #L1 #L2 #V1 #V2 #s #l #m #HL12 #HV21 #Hl >(plus_minus_m_m l 1) /2 width=1 by drop_skip/
+qed.
+
+lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K.
+#s #m @(nat_ind_plus … m) -m /2 width=2 by ex_intro/
+#m #IHm *
+[ #H elim (le_plus_xSy_O_false … H)
+| #L #I #V normalize #H elim (IHm L) -IHm /3 width=2 by drop_drop, monotonic_pred, ex_intro/
+]
+qed-.
+
+lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
+#s #L elim L -L
+[ #m #H elim (lt_zero_false … H)
+| #L #I #V #IHL #m @(nat_ind_plus … m) -m /2 width=4 by drop_pair, ex1_3_intro/
+  #m #_ normalize #H elim (IHL m) -IHL /3 width=4 by drop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
+]
+qed-.
+
+lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → m ≤ |L| → ∀I,V.
+                    ∃∃J,W. ⬇[s, 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize #Hm #I #V
+[ elim (drop_inv_atom1 … H) -H #H <(le_n_O_to_eq … Hm) -m
+  #Hs destruct /2 width=3 by ex1_2_intro/
+| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/
+  elim (IHL … HLK … Z X) -IHL -HLK
+  /3 width=3 by drop_drop_lt, le_plus_to_minus, ex1_2_intro/
+]
+qed-.
+
+lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆.
+#L elim L -L [ #m #_ @drop_atom #H destruct ]
+#L #I #V #IHL #m @(nat_ind_plus … m) -m [ #H elim (le_plus_xSy_O_false … H) ]
+normalize /4 width=1 by drop_drop, monotonic_pred/
+qed.
+
+lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆.
+#L elim L -L /2 width=1 by drop_drop, drop_atom/
+qed.
+
+lemma drop_split: ∀L1,L2,l,m2,s. ⬇[s, l, m2] L1 ≡ L2 → ∀m1. m1 ≤ m2 →
+                  ∃∃L. ⬇[s, l, m2 - m1] L1 ≡ L & ⬇[s, l, m1] L ≡ L2.
+#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
+[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro … (⋆))
+  @drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
+| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq … Hm1) -Hm1
+  #H destruct /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus … m1) -m1
+  [ /3 width=3 by drop_drop, ex2_intro/
+  | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r … Hm12) -Hm12
+    #Hm12 elim (IHL12 … Hm12) -IHL12 >minus_plus_plus_l
+    #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
+    [ elim (drop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
+      elim (drop_inv_atom1 … HL2) -HL2 #H #_ destruct
+      @(ex2_intro … (⋆)) [ @drop_O1_ge normalize // ]
+      @drop_atom #H destruct
+    | elim (drop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
+    ]
+  ]
+| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 … Hm12) -IHL12
+  #L #HL1 #HL2 elim (lift_split … HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
+]
+qed-.
+
+lemma drop_FT: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → ⬇[Ⓣ, l, m] L1 ≡ L2.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+/3 width=1 by drop_atom, drop_drop, drop_skip/
+qed.
+
+lemma drop_gen: ∀L1,L2,s,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → ⬇[s, l, m] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drop_FT/
+qed-.
+
+lemma drop_T: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → ⬇[Ⓣ, l, m] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drop_FT/
+qed-.
+
+lemma d_liftable_LTC: ∀R. d_liftable R → d_liftable (LTC … R).
+#R #HR #K #T1 #T2 #H elim H -T2
+[ /3 width=10 by inj/
+| #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
+  elim (lift_total T l m) /4 width=12 by step/
+]
+qed-.
+
+lemma d_deliftable_sn_LTC: ∀R. d_deliftable_sn R → d_deliftable_sn (LTC … R).
+#R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #K #s #l #m #HLK #T1 #HTU1
+  elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
+| #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
+  elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+  elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #s #l #m #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
+  /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
+  /3 width=3 by step, ex2_intro/
+]
+qed-.
+
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #s #m #HLK2 elim (HR … HL12 … HLK2) -HR -L2
+  /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #K2 #s #m #HLK2 elim (HR … HL2 … HLK2) -HR -L2
+  #K #HLK #HK2 elim (IHL1 … HLK) -L
+  /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma d_deliftable_sn_llstar: ∀R. d_deliftable_sn R →
+                              ∀d. d_deliftable_sn (llstar … R d).
+#R #HR #d #L #U1 #U2 #H @(lstar_ind_r … d U2 H) -d -U2
+[ /2 width=3 by lstar_O, ex2_intro/
+| #d #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
+  elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+  elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_1: was: drop_S *)
+lemma drop_fwd_drop2: ∀L1,I2,K2,V2,s,m. ⬇[s, O, m] L1 ≡ K2. ⓑ{I2} V2 →
+                      ⬇[s, O, m + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #s #m #H lapply (drop_inv_atom1 … H) -H * #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #m #H
+  elim (drop_inv_O1_pair1 … H) -H * #Hm #H
+  [ -IHL1 destruct /2 width=1 by drop_drop/
+  | @drop_drop >(plus_minus_m_m m 1) /2 width=3 by/
+  ]
+]
+qed-.
+
+lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // normalize
+[ #I #L1 #L2 #V #m #_ #_ #H elim (le_plus_xSy_O_false … H)
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → m ≤ |L1| - l → |L2| = |L1| - m.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // normalize
+[ /3 width=2 by le_plus_to_le_r/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 >minus_plus_plus_l
+  #Hl #Hm lapply (le_plus_to_le_r … Hl) -Hl
+  #Hl >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
+]
+qed-.
+
+lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| - l ≤ m → |L2| = l.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m normalize
+[ /2 width=1 by le_n_O_to_eq/
+| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false … H)
+| /3 width=2 by le_plus_to_le_r/
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma drop_fwd_length: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| + m.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m // normalize /2 width=1 by/
+qed-.
+
+lemma drop_fwd_length_minus2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| = |L1| - m.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H /2 width=1 by plus_minus, le_n/
+qed-.
+
+lemma drop_fwd_length_minus4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m = |L1| - |L2|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
+qed-.
+
+lemma drop_fwd_length_le2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m ≤ |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
+qed-.
+
+lemma drop_fwd_length_le4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
+qed-.
+
+lemma drop_fwd_length_lt2: ∀L1,I2,K2,V2,l,m.
+                           ⬇[Ⓕ, l, m] L1 ≡ K2. ⓑ{I2} V2 → m < |L1|.
+#L1 #I2 #K2 #V2 #l #m #H
+lapply (drop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
+qed-.
+
+lemma drop_fwd_length_lt4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → |L2| < |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H /2 width=1 by lt_minus_to_plus_r/
+qed-.
+
+lemma drop_fwd_length_eq1: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                           |L1| = |L2| → |K1| = |K2|.
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
+lapply (drop_fwd_length … HLK1) -HLK1
+lapply (drop_fwd_length … HLK2) -HLK2
+/2 width=2 by injective_plus_r/
+qed-.
+
+lemma drop_fwd_length_eq2: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+                           |K1| = |K2| → |L1| = |L2|.
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
+lapply (drop_fwd_length … HLK1) -HLK1
+lapply (drop_fwd_length … HLK2) -HLK2 //
+qed-.
+
+lemma drop_fwd_lw: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m // normalize
+[ /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12
+  >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+lemma drop_fwd_lw_lt: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → ♯{L2} < ♯{L1}.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #H >H -H //
+| #I #L #V #H elim (lt_refl_false … H)
+| #I #L1 #L2 #V #m #HL12 #_ #_
+  lapply (drop_fwd_lw … HL12) -HL12 #HL12
+  @(le_to_lt_to_lt … HL12) -HL12 //
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+  >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
+]
+qed-.
+
+lemma drop_fwd_rfw: ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#I #L #K #V #i #HLK lapply (drop_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact drop_inv_O2_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → m = 0 → L1 = L2.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
+[ //
+| //
+| #I #L1 #L2 #V #m #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H
+  >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -l -m //
+]
+qed-.
+
+(* Basic_1: was: drop_gen_refl *)
+lemma drop_inv_O2: ∀L1,L2,s,l. ⬇[s, l, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5 by drop_inv_O2_aux/ qed-.
+
+lemma drop_inv_length_eq: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| → m = 0.
+#L1 #L2 #l #m #H #HL12 lapply (drop_fwd_length_minus4 … H) //
+qed-.
+
+lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0.
+/2 width=5 by drop_inv_length_eq/ qed-.
+
+fact drop_inv_FT_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
+                      ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → l = 0 →
+                      ⬇[Ⓕ, l, m] L1 ≡ K.ⓑ{I}V.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #_ #J #K #W #H destruct
+| #I #L #V #J #K #W #H destruct //
+| #I #L1 #L2 #V #m #_ #IHL12 #J #K #W #H1 #H2 destruct
+  /3 width=1 by drop_drop/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #J #K #W #_ #_
+  <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma drop_inv_FT: ∀I,L,K,V,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡ K.ⓑ{I}V.
+/2 width=5 by drop_inv_FT_aux/ qed.
+
+lemma drop_inv_gen: ∀I,L,K,V,s,m. ⬇[s, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drop_inv_FT/
+qed-.
+
+lemma drop_inv_T: ∀I,L,K,V,s,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drop_inv_FT/
+qed-.
+
+(* Basic_1: removed theorems 50:
+            drop_ctail drop_skip_flat
+            cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+            drop_clear drop_clear_O drop_clear_S
+            clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+            clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+            getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+            getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+            getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+            drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+            getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+            getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+            getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_append.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_append.ma
new file mode 100644 (file)
index 0000000..49d734c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_append.ma
@@ -0,0 +1,60 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/lenv_append.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on append for local environments ******************************)
+
+fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
+                               l = 0 → m ≤ |L1| →
+                               ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m normalize
+[2,3,4: /4 width=1 by drop_skip_lt, drop_drop, arith_b1, lt_minus_to_plus_r, monotonic_pred/ ]
+#l #m #_ #_ #H <(le_n_O_to_eq … H) -H //
+qed-.
+
+lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, 0, m] L1 ≡ L2 → m ≤ |L1| →
+                            ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
+/2 width=3 by drop_O1_append_sn_le_aux/ qed.
+
+(* Inversion lemmas on append for local environments ************************)
+
+lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K →
+                              |L2| ≤ m → ⬇[s, 0, m - |L2|] L1 ≡ K.
+#K #L1 #L2 elim L2 -L2 normalize //
+#L2 #I #V #IHL2 #s #m #H #H1m
+elim (drop_inv_O1_pair1 … H) -H * #H2m #HL12 destruct
+[ lapply (le_n_O_to_eq … H1m) -H1m -IHL2
+  >commutative_plus normalize #H destruct
+| <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
+]
+qed-.
+
+lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → m ≤ |L2| →
+                              ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 @@ K2.
+#K #L1 #L2 elim L2 -L2 normalize
+[ #s #m #H1 #H2 #K2 #H3 lapply (le_n_O_to_eq … H2) -H2
+  #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct
+  >(drop_inv_O2 … H1) -H1 //
+| #L2 #I #V #IHL2 #s #m @(nat_ind_plus … m) -m [ -IHL2 ]
+  [ #H1 #_ #K2 #H2
+    lapply (drop_inv_O2 … H1) -H1 #H1
+    lapply (drop_inv_O2 … H2) -H2 #H2 destruct //
+  | /4 width=7 by drop_inv_drop1, le_plus_to_le_r/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_drop.ma
new file mode 100644 (file)
index 0000000..0ee0f74
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_drop.ma
@@ -0,0 +1,208 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_lift.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop_mono *)
+theorem drop_mono: ∀L,L1,s1,l,m. ⬇[s1, l, m] L ≡ L1 →
+                   ∀L2,s2. ⬇[s2, l, m] L ≡ L2 → L1 = L2.
+#L #L1 #s1 #l #m #H elim H -L -L1 -l -m
+[ #l #m #Hm #L2 #s2 #H elim (drop_inv_atom1 … H) -H //
+| #I #K #V #L2 #s2 #HL12 <(drop_inv_O2 … HL12) -L2 //
+| #I #L #K #V #m #_ #IHLK #L2 #s2 #H
+  lapply (drop_inv_drop1 … H) -H /2 width=2 by/
+| #I #L #K1 #T #V1 #l #m #_ #HVT1 #IHLK1 #X #s2 #H
+  elim (drop_inv_skip1 … H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
+  >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
+  >(IHLK1 … HLK2) -IHLK1 -HLK2 //
+]
+qed-.
+
+(* Basic_1: was: drop_conf_ge *)
+theorem drop_conf_ge: ∀L,L1,s1,l1,m1. ⬇[s1, l1, m1] L ≡ L1 →
+                      ∀L2,s2,m2. ⬇[s2, 0, m2] L ≡ L2 → l1 + m1 ≤ m2 →
+                      ⬇[s2, 0, m2 - m1] L1 ≡ L2.
+#L #L1 #s1 #l1 #m1 #H elim H -L -L1 -l1 -m1 //
+[ #l #m #_ #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
+  #H #Hm destruct
+  @drop_atom #H >Hm // (**) (* explicit constructor *)
+| #I #L #K #V #m #_ #IHLK #L2 #s2 #m2 #H #Hm2
+  lapply (drop_inv_drop1_lt … H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
+  <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
+| #I #L #K #V1 #V2 #l #m #_ #_ #IHLK #L2 #s2 #m2 #H #Hlmm2
+  lapply (transitive_le 1 … Hlmm2) // #Hm2
+  lapply (drop_inv_drop1_lt … H ?) -H // -Hm2 #HL2
+  lapply (transitive_le (1+m) … Hlmm2) // #Hmm2
+  @drop_drop_lt >minus_minus_comm /3 width=1 by lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem drop_conf_be: ∀L0,L1,s1,l1,m1. ⬇[s1, l1, m1] L0 ≡ L1 →
+                      ∀L2,m2. ⬇[m2] L0 ≡ L2 → l1 ≤ m2 → m2 ≤ l1 + m1 →
+                      ∃∃L. ⬇[s1, 0, l1 + m1 - m2] L2 ≡ L & ⬇[l1] L1 ≡ L.
+#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #m2 #H #Hl1 #_ elim (drop_inv_atom1 … H) -H #H #Hm2 destruct
+  >(Hm2 ?) in Hl1; // -Hm2 #Hl1 <(le_n_O_to_eq … Hl1) -l1
+  /4 width=3 by drop_atom, ex2_intro/
+| normalize #I #L #V #L2 #m2 #HL2 #_ #Hm2
+  lapply (le_n_O_to_eq … Hm2) -Hm2 #H destruct
+  lapply (drop_inv_O2 … HL2) -HL2 #H destruct /2 width=3 by drop_pair, ex2_intro/
+| normalize #I #L0 #K0 #V1 #m1 #HLK0 #IHLK0 #L2 #m2 #H #_ #Hm21
+  lapply (drop_inv_O1_pair1 … H) -H * * #Hm2 #HL20
+  [ -IHLK0 -Hm21 destruct <minus_n_O /3 width=3 by drop_drop, ex2_intro/
+  | -HLK0 <minus_le_minus_minus_comm //
+    elim (IHLK0 … HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
+  ]
+| #I #L0 #K0 #V0 #V1 #l1 #m1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #m2 #H #Hl1m2 #Hm2lm1
+  elim (le_inv_plus_l … Hl1m2) #_ #Hm2
+  <minus_le_minus_minus_comm //
+  lapply (drop_inv_drop1_lt … H ?) -H // #HL02
+  elim (IHLK0 … HL02) -L0 /3 width=3 by drop_drop, monotonic_pred, ex2_intro/
+]
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem drop_conf_le: ∀L0,L1,s1,l1,m1. ⬇[s1, l1, m1] L0 ≡ L1 →
+                      ∀L2,s2,m2. ⬇[s2, 0, m2] L0 ≡ L2 → m2 ≤ l1 →
+                      ∃∃L. ⬇[s2, 0, m2] L1 ≡ L & ⬇[s1, l1 - m2, m1] L2 ≡ L.
+#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H elim (drop_inv_atom1 … H) -H
+  #H #Hm2 #_ destruct /4 width=3 by drop_atom, ex2_intro/
+| #I #K0 #V0 #L2 #s2 #m2 #H1 #H2
+  lapply (le_n_O_to_eq … H2) -H2 #H destruct
+  lapply (drop_inv_pair1 … H1) -H1 #H destruct /2 width=3 by drop_pair, ex2_intro/
+| #I #K0 #K1 #V0 #m1 #HK01 #_ #L2 #s2 #m2 #H1 #H2
+  lapply (le_n_O_to_eq … H2) -H2 #H destruct
+  lapply (drop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by drop_drop, ex2_intro/
+| #I #K0 #K1 #V0 #V1 #l1 #m1 #HK01 #HV10 #IHK01 #L2 #s2 #m2 #H #Hm2l1
+  elim (drop_inv_O1_pair1 … H) -H *
+  [ -IHK01 -Hm2l1 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
+  | -HK01 -HV10 #Hm2 #HK0L2
+    elim (IHK01 … HK0L2) -IHK01 -HK0L2 /2 width=1 by monotonic_pred/
+    >minus_le_minus_minus_comm /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+]
+qed-.
+
+(* Note: with "s2", the conclusion parameter is "s1 ∨ s2" *)
+(* Basic_1: was: drop_trans_ge *)
+theorem drop_trans_ge: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
+                       ∀L2,m2. ⬇[m2] L ≡ L2 → l1 ≤ m2 → ⬇[s1, 0, m1 + m2] L1 ≡ L2.
+#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
+  #H #Hm2 destruct /4 width=1 by drop_atom, eq_f2/
+| /2 width=1 by drop_gen/
+| /3 width=1 by drop_drop/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #L #m2 #H #Hlm2
+  lapply (lt_to_le_to_lt 0 … Hlm2) // #Hm2
+  lapply (lt_to_le_to_lt … (m + m2) Hm2 ?) // #Hmm2
+  lapply (drop_inv_drop1_lt … H ?) -H // #HL2
+  @drop_drop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
+]
+qed.
+
+(* Basic_1: was: drop_trans_le *)
+theorem drop_trans_le: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
+                       ∀L2,s2,m2. ⬇[s2, 0, m2] L ≡ L2 → m2 ≤ l1 →
+                       ∃∃L0. ⬇[s2, 0, m2] L1 ≡ L0 & ⬇[s1, l1 - m2, m1] L0 ≡ L2.
+#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
+  #H #Hm2 destruct /4 width=3 by drop_atom, ex2_intro/
+| #I #K #V #L2 #s2 #m2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+  #H destruct /2 width=3 by drop_pair, ex2_intro/
+| #I #L1 #L2 #V #m #_ #IHL12 #L #s2 #m2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+  #H destruct elim (IHL12 … HL2) -IHL12 -HL2 //
+  #L0 #H #HL0 lapply (drop_inv_O2 … H) -H #H destruct
+  /3 width=5 by drop_pair, drop_drop, ex2_intro/
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV12 #IHL12 #L #s2 #m2 #H #Hm2l
+  elim (drop_inv_O1_pair1 … H) -H *
+  [ -Hm2l -IHL12 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
+  | -HL12 -HV12 #Hm2 #HL2
+    elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by drop_drop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
+  ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma d_liftable_llstar: ∀R. d_liftable R → ∀d. d_liftable (llstar … R d).
+#R #HR #d #K #T1 #T2 #H @(lstar_ind_r … d T2 H) -d -T2
+[ #L #s #l #m #_ #U1 #HTU1 #U2 #HTU2 -HR -K
+  >(lift_mono … HTU2 … HTU1) -T1 -U2 -l -m //
+| #d #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
+  elim (lift_total T l m) /3 width=12 by lstar_dx/
+]
+qed-.
+
+(* Basic_1: was: drop_conf_lt *)
+lemma drop_conf_lt: ∀L,L1,s1,l1,m1. ⬇[s1, l1, m1] L ≡ L1 →
+                    ∀I,K2,V2,s2,m2. ⬇[s2, 0, m2] L ≡ K2.ⓑ{I}V2 →
+                    m2 < l1 → let l ≝ l1 - m2 - 1 in
+                    ∃∃K1,V1. ⬇[s2, 0, m2] L1 ≡ K1.ⓑ{I}V1 &
+                             ⬇[s1, l, m1] K2 ≡ K1 & ⬆[l, m1] V1 ≡ V2.
+#L #L1 #s1 #l1 #m1 #H1 #I #K2 #V2 #s2 #m2 #H2 #Hm2l1
+elim (drop_conf_le … H1 … H2) -L /2 width=2 by lt_to_le/ #K #HL1K #HK2
+elim (drop_inv_skip1 … HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/
+#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+lemma drop_trans_lt: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
+                     ∀I,L2,V2,s2,m2. ⬇[s2, 0, m2] L ≡ L2.ⓑ{I}V2 →
+                     m2 < l1 → let l ≝ l1 - m2 - 1 in
+                     ∃∃L0,V0. ⬇[s2, 0, m2] L1 ≡ L0.ⓑ{I}V0 &
+                              ⬇[s1, l, m1] L0 ≡ L2 & ⬆[l, m1] V2 ≡ V0.
+#L1 #L #s1 #l1 #m1 #HL1 #I #L2 #V2 #s2 #m2 #HL2 #Hl21
+elim (drop_trans_le … HL1 … HL2) -L /2 width=1 by lt_to_le/ #L0 #HL10 #HL02
+elim (drop_inv_skip2 … HL02) -HL02 /2 width=1 by lt_plus_to_minus_r/ #L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma drop_trans_ge_comm: ∀L1,L,L2,s1,l1,m1,m2.
+                          ⬇[s1, l1, m1] L1 ≡ L → ⬇[m2] L ≡ L2 → l1 ≤ m2 →
+                          ⬇[s1, 0, m2 + m1] L1 ≡ L2.
+#L1 #L #L2 #s1 #l1 #m1 #m2
+>commutative_plus /2 width=5 by drop_trans_ge/
+qed.
+
+lemma drop_conf_div: ∀I1,L,K,V1,m1. ⬇[m1] L ≡ K.ⓑ{I1}V1 →
+                     ∀I2,V2,m2. ⬇[m2] L ≡ K.ⓑ{I2}V2 →
+                     ∧∧ m1 = m2 & I1 = I2 & V1 = V2.
+#I1 #L #K #V1 #m1 #HLK1 #I2 #V2 #m2 #HLK2
+elim (le_or_ge m1 m2) #Hm
+[ lapply (drop_conf_ge … HLK1 … HLK2 ?)
+| lapply (drop_conf_ge … HLK2 … HLK1 ?)
+] -HLK1 -HLK2 // #HK
+lapply (drop_fwd_length_minus2 … HK) #H
+elim (discr_minus_x_xy … H) -H
+[1,3: normalize <plus_n_Sm #H destruct ]
+#H >H in HK; #HK
+lapply (drop_inv_O2 … HK) -HK #H destruct
+lapply (inv_eq_minus_O … H) -H /3 width=1 by le_to_le_to_eq, and3_intro/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → i < l → |L| ≤ i.
+#L #K #s #l #m #i #HLK #HK #Hl elim (lt_or_ge i (|L|)) //
+#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
+elim (drop_conf_lt … HLK … HLK0) // -HLK -HLK0 -Hl
+#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_lreq.ma
new file mode 100644 (file)
index 0000000..ad39fd7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/drop_lreq.ma
@@ -0,0 +1,92 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/lreq_lreq.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+definition dedropable_sn: predicate (relation lenv) ≝
+                          λR. ∀L1,K1,s,l,m. ⬇[s, l, m] L1 ≡ K1 → ∀K2. R K1 K2 →
+                          ∃∃L2. R L1 L2 & ⬇[s, l, m] L2 ≡ K2 & L1 ⩬[l, m] L2.
+
+(* Properties on equivalence ************************************************)
+
+lemma lreq_drop_trans_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+                          ∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
+                          l ≤ i → i < l + m →
+                          ∃∃K1. K1 ⩬[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #J #K2 #W #s #i #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
+  elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #m #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+  [ #_ destruct >ypred_succ
+    /2 width=3 by drop_pair, ex2_intro/
+  | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+    #H <H -H #H lapply (ylt_inv_succ … H) -H
+    #Him elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
+    >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
+  ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hli
+  elim (yle_inv_succ1 … Hli) -Hli
+  #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
+  #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+  #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+  /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
+]
+qed-.
+
+lemma lreq_drop_conf_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+                         ∀I,K1,W,s,i. ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W →
+                         l ≤ i → i < l + m →
+                         ∃∃K2. K1 ⩬[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W.
+#L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #Hilm
+elim (lreq_drop_trans_be … (lreq_sym … HL12) … HLK1) // -L1 -Hli -Hilm
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
+
+lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
+                  ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2.
+#K2 #i @(nat_ind_plus … i) -i
+[ /3 width=3 by lreq_O2, ex2_intro/
+| #i #IHi #Y #Hi elim (drop_O1_lt (Ⓕ) Y 0) //
+  #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
+  normalize in Hi; elim (IHi L1) -IHi
+  /3 width=5 by drop_drop, lreq_pair, injective_plus_l, ex2_intro/
+]
+qed-.
+
+lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
+#R #HR #L1 #K1 #s #l #m #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
+  /3 width=4 by inj, ex3_intro/
+| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
+  /3 width=6 by lreq_trans, step, ex3_intro/
+]
+qed-.
+
+(* Inversion lemmas on equivalence ******************************************)
+
+lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
+#i @(nat_ind_plus … i) -i
+[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
+| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
+  lapply (drop_fwd_length … HLK1)
+  <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
+  normalize <plus_n_Sm #H destruct
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/fqu.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fqu.ma
new file mode 100644 (file)
index 0000000..955b259
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fqu.ma
@@ -0,0 +1,90 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/supterm_6.ma".
+include "basic_2A/grammar/cl_weight.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* activate genv *)
+inductive fqu: tri_relation genv lenv term ≝
+| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
+| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
+| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
+| fqu_drop   : ∀G,L,K,T,U,m.
+               ⬇[m+1] L ≡ K → ⬆[0, m+1] T ≡ U → fqu G L U G K T
+.
+
+interpretation
+   "structural successor (closure)"
+   'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_drop_lt: ∀G,L,K,T,U,m. 0 < m →
+                   ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
+#G #L #K #T #U #m #Hm >(plus_minus_m_m m 1) /2 width=3 by fqu_drop/
+qed.
+
+lemma fqu_lref_S_lt: ∀I,G,L,V,i. 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(i-1)⦄.
+/3 width=3 by fqu_drop, drop_drop, lift_lref_ge_minus/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+#G #L #K #T #U #m #HLK #HTU
+lapply (drop_fwd_lw_lt … HLK ?) -HLK // #HKL
+lapply (lift_fwd_tw … HTU) -m #H
+normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
+qed-.
+
+fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                               ∀i. T1 = #i → |L2| < |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[1: normalize //
+|3: #a
+|5: /2 width=4 by drop_fwd_length_lt4/
+] #I #G #L #V #T #j #H destruct
+qed-.
+
+lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ → |L2| < |L1|.
+/2 width=7 by fqu_fwd_length_lref1_aux/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 normalize
+/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x, plus_xSy_x_false/
+#G #L #K #T #U #m #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length … HLK) in H; -L
+/2 width=4 by plus_xySz_x_false/
+qed-.
+
+lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
+#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
+qed-. 
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqu_wf_ind: ∀R:relation3 …. (
+                     ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                               R G1 L1 T1
+                 ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq.ma
new file mode 100644 (file)
index 0000000..0893a0b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq.ma
@@ -0,0 +1,64 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/suptermopt_6.ma".
+include "basic_2A/substitution/fqu.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* activate genv *)
+inductive fquq: tri_relation genv lenv term ≝
+| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
+| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
+| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
+| fquq_drop   : ∀G,L,K,T,U,m.
+                ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → fquq G L U G K T
+.
+
+interpretation
+   "optional structural successor (closure)"
+   'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquq_refl: tri_reflexive … fquq.
+/2 width=3 by fquq_drop/ qed.
+
+lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
+#G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1
+lapply (drop_fwd_lw … HLK1) -HLK1
+lapply (lift_fwd_tw … HTU1) -HTU1
+/2 width=1 by le_plus, le_n/
+qed-.
+
+fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                                ∀i. T1 = #i → |L2| ≤ |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+[ #a #I #G #L #V #T #j #H destruct
+| #G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #i #H destruct
+  /2 width=3 by drop_fwd_length_le4/
+]
+qed-.
+
+lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
+/2 width=7 by fquq_fwd_length_lref1_aux/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq_alt.ma
new file mode 100644 (file)
index 0000000..57d9332
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/fquq_alt.ma
@@ -0,0 +1,59 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/suptermoptalt_6.ma".
+include "basic_2A/substitution/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* alternative definition of fquq *)
+definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
+
+interpretation
+   "optional structural successor (closure) alternative"
+   'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquqa_refl: tri_reflexive … fquqa.
+// qed.
+
+lemma fquqa_drop: ∀G,L,K,T,U,m.
+                  ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
+#G #L #K #T #U #m #HLK #HTU elim (eq_or_gt m)
+/3 width=5 by fqu_drop_lt, or_introl/ #H destruct
+>(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
+qed.
+
+(* Main properties **********************************************************)
+
+theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
+* #H1 #H2 #H3 destruct //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
+/2 width=1 by or_introl/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma
new file mode 100644 (file)
index 0000000..b11b0af
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget.ma
@@ -0,0 +1,81 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rdrop_3.ma".
+include "basic_2A/grammar/genv.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+inductive gget (m:nat): relation genv ≝
+| gget_gt: ∀G. |G| ≤ m → gget m G (⋆)
+| gget_eq: ∀G. |G| = m + 1 → gget m G G
+| gget_lt: ∀I,G1,G2,V. m < |G1| → gget m G1 G2 → gget m (G1. ⓑ{I} V) G2
+.
+
+interpretation "global reading"
+   'RDrop m G1 G2 = (gget m G1 G2).
+
+(* basic inversion lemmas ***************************************************)
+
+lemma gget_inv_gt: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| ≤ m → G2 = ⋆.
+#G1 #G2 #m * -G1 -G2 //
+[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
+  lapply (le_plus_to_le_r … 0 H) -H #H
+  lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ #H2
+  lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+  lapply (lt_plus_to_lt_l … 0 H) -H #H
+  elim (lt_zero_false … H)
+]
+qed-.
+
+lemma gget_inv_eq: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| = m + 1 → G1 = G2.
+#G1 #G2 #m * -G1 -G2 //
+[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
+  lapply (le_plus_to_le_r … 0 H) -H #H
+  lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ normalize #H2
+  <(injective_plus_l … H2) in H1; -H2 #H
+  elim (lt_refl_false … H)
+]
+qed-.
+
+fact gget_inv_lt_aux: ∀I,G,G1,G2,V,m. ⬇[m] G ≡ G2 → G = G1. ⓑ{I} V →
+                      m < |G1| → ⬇[m] G1 ≡ G2.
+#I #G #G1 #G2 #V #m * -G -G2
+[ #G #H1 #H destruct #H2
+  lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+  lapply (lt_plus_to_lt_l … 0 H) -H #H
+  elim (lt_zero_false … H)
+| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
+  elim (lt_refl_false … H)
+| #J #G #G2 #W #_ #HG2 #H destruct //
+]
+qed-.
+
+lemma gget_inv_lt: ∀I,G1,G2,V,m.
+                    ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2.
+/2 width=5 by gget_inv_lt_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2.
+#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/
+#I #V #G1 * #G2 #HG12
+elim (lt_or_eq_or_gt m (|G1|)) #Hm
+[ /3 width=2 by gget_lt, ex_intro/
+| destruct /3 width=2 by gget_eq, ex_intro/
+| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *)
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget_gget.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget_gget.ma
new file mode 100644 (file)
index 0000000..d8c752e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/gget_gget.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/gget.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+(* Main properties **********************************************************)
+
+theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2.
+#G #G1 #m #H elim H -G -G1
+[ #G #Hm #G2 #H
+  >(gget_inv_gt … H Hm) -H -Hm //
+| #G #Hm #G2 #H
+  >(gget_inv_eq … H Hm) -H -Hm //
+| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H
+  lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1 by/
+]
+qed-.
+
+lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2).
+#G1 #G2 #m
+elim (gget_total m G1) #G #HG1
+elim (eq_genv_dec G G2) #HG2
+[ destruct /2 width=1 by or_introl/
+| @or_intror #HG12
+  lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1 by/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma
new file mode 100644 (file)
index 0000000..63e8275
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift.ma
@@ -0,0 +1,393 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/rlift_4.ma".
+include "basic_2A/grammar/term_weight.ma".
+include "basic_2A/grammar/term_simple.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Basic_1: includes:
+            lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
+*)
+inductive lift: relation4 nat nat term term ≝
+| lift_sort   : ∀k,l,m. lift l m (⋆k) (⋆k)
+| lift_lref_lt: ∀i,l,m. i < l → lift l m (#i) (#i)
+| lift_lref_ge: ∀i,l,m. l ≤ i → lift l m (#i) (#(i + m))
+| lift_gref   : ∀p,l,m. lift l m (§p) (§p)
+| lift_bind   : ∀a,I,V1,V2,T1,T2,l,m.
+                lift l m V1 V2 → lift (l + 1) m T1 T2 →
+                lift l m (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| lift_flat   : ∀I,V1,V2,T1,T2,l,m.
+                lift l m V1 V2 → lift l m T1 T2 →
+                lift l m (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lift_inv_O2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → m = 0 → T1 = T2.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
+qed-.
+
+lemma lift_inv_O2: ∀l,T1,T2. ⬆[l, 0] T1 ≡ T2 → T1 = T2.
+/2 width=4 by lift_inv_O2_aux/ qed-.
+
+fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l,m] ⋆k ≡ T2 → T2 = ⋆k.
+/2 width=5 by lift_inv_sort1_aux/ qed-.
+
+fact lift_inv_lref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T1 = #i →
+                         (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+]
+qed-.
+
+lemma lift_inv_lref1: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 →
+                      (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
+/2 width=3 by lift_inv_lref1_aux/ qed-.
+
+lemma lift_inv_lref1_lt: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → i < l → T2 = #i.
+#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false … Hll)
+qed-.
+
+lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
+#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false … Hll)
+qed-.
+
+fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l,m] §p ≡ T2 → T2 = §p.
+/2 width=5 by lift_inv_gref1_aux/ qed-.
+
+fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
+                         ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
+                         ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
+                                  T2 = ⓑ{a,I} V2. U2.
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #p #l #m #a #I #V1 #U1 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
+]
+qed-.
+
+lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l,m] ⓑ{a,I} V1. U1 ≡ T2 →
+                      ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
+                               T2 = ⓑ{a,I} V2. U2.
+/2 width=3 by lift_inv_bind1_aux/ qed-.
+
+fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
+                         ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
+                         ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
+                                  T2 = ⓕ{I} V2. U2.
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #p #l #m #I #V1 #U1 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l,m] ⓕ{I} V1. U1 ≡ T2 →
+                      ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
+                               T2 = ⓕ{I} V2. U2.
+/2 width=3 by lift_inv_flat1_aux/ qed-.
+
+fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_sort *)
+lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l,m] T1 ≡ ⋆k → T1 = ⋆k.
+/2 width=5 by lift_inv_sort2_aux/ qed-.
+
+fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
+                         (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lref *)
+lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+                      (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
+/2 width=3 by lift_inv_lref2_aux/ qed-.
+
+(* Basic_1: was: lift_gen_lref_lt *)
+lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → i < l → T1 = #i.
+#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l … Hll) -Hll #Hll
+elim (lt_refl_false … Hll)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_false *)
+lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+                         l ≤ i → i < l + m → ⊥.
+#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H *
+[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
+lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
+elim (lt_refl_false … H)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_ge *)
+lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
+#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l … Hll) -Hll #Hll
+elim (lt_refl_false … Hll)
+qed-.
+
+fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+]
+qed-.
+
+lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l,m] T1 ≡ §p → T1 = §p.
+/2 width=5 by lift_inv_gref2_aux/ qed-.
+
+fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
+                         ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
+                         ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
+                                  T1 = ⓑ{a,I} V1. U1.
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #p #l #m #a #I #V2 #U2 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift_gen_bind *)
+lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l,m] T1 ≡ ⓑ{a,I} V2. U2 →
+                      ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
+                               T1 = ⓑ{a,I} V1. U1.
+/2 width=3 by lift_inv_bind2_aux/ qed-.
+
+fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
+                         ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
+                         ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
+                                  T1 = ⓕ{I} V1. U1.
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #p #l #m #I #V2 #U2 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_flat *)
+lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l,m] T1 ≡  ⓕ{I} V2. U2 →
+                      ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
+                               T1 = ⓕ{I} V1. U1.
+/2 width=3 by lift_inv_flat2_aux/ qed-.
+
+lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
+#l #m #J #V elim V -V
+[ * #i #T #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct
+  | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+  | lapply (lift_inv_gref2 … H) -H #H destruct
+  ]
+| * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
+  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
+  ]
+]
+qed-.
+
+(* Basic_1: was: thead_x_lift_y_y *)
+lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I} V. T ≡ T → ⊥.
+#J #T elim T -T
+[ * #i #V #l #m #H
+  [ lapply (lift_inv_sort2 … H) -H #H destruct
+  | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+  | lapply (lift_inv_gref2 … H) -H #H destruct
+  ]
+| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
+  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
+  ]
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l,m] ②{I}V1.U1 ≡ T2 →
+                      ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & T2 = ②{I}V2.U2.
+* [ #a ] #I #T2 #V1 #U1 #l #m #H
+[ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
+|  elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l,m] T1 ≡ ②{I}V2.U2 →
+                      ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & T1 = ②{I}V1.U1.
+* [ #a ] #I #T1 #V2 #U2 #l #m #H
+[ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
+|  elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
+qed-.
+
+lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: lift_lref_gt *)
+lemma lift_lref_ge_minus: ∀l,m,i. l + m ≤ i → ⬆[l, m] #(i - m) ≡ #i.
+#l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %); /3 width=2 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
+qed.
+
+lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + m ≤ i → j = i - m → ⬆[l, m] #j ≡ #i.
+/2 width=1 by lift_lref_ge_minus/ qed-.
+
+(* Basic_1: was: lift_r *)
+lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
+#T elim T -T
+[ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
+| * /2 width=1 by lift_bind, lift_flat/
+]
+qed.
+
+lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
+  #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
+  elim (IHV1 l m) -IHV1 #V2 #HV12
+  [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
+  | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
+  ]
+]
+qed.
+
+(* Basic_1: was: lift_free (right to left) *)
+lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
+                  ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + m1 → m1 ≤ m2 →
+                  ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
+#l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
+[ /3 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
+  lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
+  lapply (transitive_le … (i+m1) Hl21 ?) /2 width=1 by monotonic_le_plus_l/ -Hl21 #Hl21
+  >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
+| /3 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+  elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+  elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+  elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+  elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Basic_1: was only: dnf_dec2 dnf_dec *)
+lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l,m] T1 ≡ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
+  elim (lt_or_ge i l) #Hli
+  [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
+  | elim (lt_or_ge i (l + m)) #Hilm
+    [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
+    | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
+    ]
+  ]
+| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
+  [ elim (IHV2 l m) -IHV2
+    [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
+      [ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
+      | -V1 #HT2 @or_intror * #X #H
+        elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
+      ]
+    | -IHT2 #HV2 @or_intror * #X #H
+      elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
+    ]
+  | elim (IHV2 l m) -IHV2
+    [ * #V1 #HV12 elim (IHT2 l m) -IHT2
+      [ * #T1 #HT12 /4 width=2 by lift_flat, ex_intro, or_introl/
+      | -V1 #HT2 @or_intror * #X #H
+        elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
+      ]
+    | -IHT2 #HV2 @or_intror * #X #H
+      elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
+    ]
+  ]
+]
+qed.
+
+(* Basic_1: removed theorems 7:
+            lift_head lift_gen_head
+            lift_weight_map lift_weight lift_weight_add lift_weight_add_O
+            lift_tlt_dx
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma
new file mode 100644 (file)
index 0000000..bbc747a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift.ma
@@ -0,0 +1,217 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Main properties ***********************************************************)
+
+(* Basic_1: was: lift_inj *)
+theorem lift_inj: ∀l,m,T1,U. ⬆[l,m] T1 ≡ U → ∀T2. ⬆[l,m] T2 ≡ U → T1 = T2.
+#l #m #T1 #U #H elim H -l -m -T1 -U
+[ #k #l #m #X #HX
+  lapply (lift_inv_sort2 … HX) -HX //
+| #i #l #m #Hil #X #HX
+  lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #l #m #Hli #X #HX
+  lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_le_plus_l/
+| #p #l #m #X #HX
+  lapply (lift_inv_gref2 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lift *)
+theorem lift_div_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                     ∀l2,m2,T2. ⬆[l2 + m1, m2] T2 ≡ T →
+                     l1 ≤ l2 →
+                     ∃∃T0. ⬆[l1, m1] T0 ≡ T2 & ⬆[l2, m2] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
+  lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+  lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
+  lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+  elim (lift_inv_lref2 … Hi) -Hi * #Hil2 #H destruct
+  [ -Hl12 lapply (lt_plus_to_lt_l … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+  | -Hil1 >plus_plus_comm_23 in Hil2; #H lapply (le_plus_to_le_r … H) -H #H
+    elim (le_inv_plus_l … H) -H #Hilm2 #Hm2i
+    lapply (transitive_le … Hl12 Hilm2) -Hl12 #Hl12
+    >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
+    /4 width=3 by lift_lref_ge, ex2_intro/
+  ]
+| #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
+  lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
+| #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
+  lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+  elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+  >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
+  lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+  elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+  elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                     ∀m,m2,T2. ⬆[l1 + m, m2] T2 ≡ T →
+                     m ≤ m1 → m1 ≤ m + m2 →
+                     ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+  >(lift_inv_lref2_lt … H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+  elim (lt_or_ge (i+m1) (l1+m+m2)) #Him1l1m2
+  [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
+  | >(lift_inv_lref2_ge … H ?) -H //
+    lapply (le_plus_to_minus … Him1l1m2) #Hl1m21i
+    elim (le_inv_plus_l … Him1l1m2) -Him1l1m2 #Hl1m12 #Hm2im1
+    @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ] -Hl1m12
+    @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
+  ]
+| #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
+  elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+  elim (IHV1 … HV2) -V // >plus_plus_comm_23 in HT2; #HT2
+  elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
+| #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
+  elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+  elim (IHV1 … HV2) -V //
+  elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
+#l #m #T #U1 #H elim H -l -m -T -U1
+[ #k #l #m #X #HX
+  lapply (lift_inv_sort1 … HX) -HX //
+| #i #l #m #Hil #X #HX
+  lapply (lift_inv_lref1_lt … HX ?) -HX //
+| #i #l #m #Hli #X #HX
+  lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #p #l #m #X #HX
+  lapply (lift_inv_gref1 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_free (left to right) *)
+theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
+                       l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
+  >(lift_inv_sort1 … HT2) -HT2 //
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
+  lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
+  lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
+  lapply (lift_inv_lref1_ge … HT2 ?) -HT2
+  [ @(transitive_le … Hl21 ?) -Hl21 /2 width=1 by monotonic_le_plus_l/
+  | -Hl21 /2 width=1 by lift_lref_ge/
+  ]
+| #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
+  >(lift_inv_gref1 … HT2) -HT2 //
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
+  elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct 
+  lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+  lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
+  elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+  lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+  lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
+]
+qed.
+
+(* Basic_1: was: lift_d (right to left) *)
+theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
+                       ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
+  >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+  lapply (lt_to_le_to_lt … (l1+m2) Hil1 ?) // #Him2
+  elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
+  lapply (transitive_le … Hl21 Hil1) -Hl21 #Hil2
+  lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
+  >plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
+  >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
+  elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+  elim (IHV12 … HV20) -IHV12 -HV20 //
+  elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
+  elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+  elim (IHV12 … HV20) -IHV12 -HV20 //
+  elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Basic_1: was: lift_d (left to right) *)
+theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+                       ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
+                       ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
+  >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
+  lapply (lt_to_le_to_lt … (l1+m1) Hil1 ?) // #Hil1m
+  lapply (lt_to_le_to_lt … (l2-m1) Hil1 ?) /2 width=1 by le_plus_to_minus_r/ #Hil2m
+  lapply (lt_to_le_to_lt … Hil1m Hlml) -Hil1m -Hlml #Hil2
+  lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+  elim (lift_inv_lref1 … HX) -HX * #Himl #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
+  >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
+  elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+  elim (IHV12 … HV20) -IHV12 -HV20 //
+  elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
+  <plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
+  elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+  elim (IHV12 … HV20) -IHV12 -HV20 //
+  elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
+                    ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
+#T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
+elim (lift_total T1 0 m2) #T0 #HT10
+elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
+lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
+qed.
+
+lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
+                    m1 ≤ m2 → ⬆[l + m1, m2 - m1] T1 ≡ T2.
+#T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
+elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
+>(lift_mono … H … HT1) -T //
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift_vector.ma
new file mode 100644 (file)
index 0000000..c7e9739
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_lift_vector.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift_lift.ma".
+include "basic_2A/substitution/lift_vector.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+(* Main properties ***********************************************************)
+
+theorem liftv_mono: ∀Ts,U1s,l,m. ⬆[l,m] Ts ≡ U1s →
+                    ∀U2s:list term. ⬆[l,m] Ts ≡ U2s → U1s = U2s.
+#Ts #U1s #l #m #H elim H -Ts -U1s
+[ #U2s #H >(liftv_inv_nil1 … H) -H //
+| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
+  elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
+  >(lift_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_neg.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_neg.ma
new file mode 100644 (file)
index 0000000..52f8c28
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_neg.ma
@@ -0,0 +1,67 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Properties on negated basic relocation ***********************************)
+
+lemma nlift_lref_be_SO: ∀X,i. ⬆[i, 1] X ≡ #i → ⊥.
+/2 width=7 by lift_inv_lref2_be/ qed-.
+
+lemma nlift_bind_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
+                     ∀a,I,U. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
+#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_bind_dx: ∀U,l,m. (∀T. ⬆[l+1, m] T ≡ U → ⊥) →
+                     ∀a,I,W. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
+#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
+                     ∀I,U. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
+#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_dx: ∀U,l,m. (∀T. ⬆[l, m] T ≡ U → ⊥) →
+                     ∀I,W. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
+#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+(* Inversion lemmas on negated basic relocation *****************************)
+
+lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⬆[i, 1] X ≡ #j → ⊥) → j = i.
+#i #j elim (lt_or_eq_or_gt i j) // #Hij #H
+[ elim (H (#(j-1))) -H /2 width=1 by lift_lref_ge_minus/
+| elim (H (#j)) -H /2 width=1 by lift_lref_lt/
+]
+qed-.
+
+lemma nlift_inv_bind: ∀a,I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥) →
+                      (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l+1, m] T ≡ U → ⊥).
+#a #I #W #U #l #m #H elim (is_lift_dec W l m)
+[ * /4 width=2 by lift_bind, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+qed-.
+
+lemma nlift_inv_flat: ∀I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥) →
+                      (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l, m] T ≡ U → ⊥).
+#I #W #U #l #m #H elim (is_lift_dec W l m)
+[ * /4 width=2 by lift_flat, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_vector.ma
new file mode 100644 (file)
index 0000000..cb80a68
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lift_vector.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/term_vector.ma".
+include "basic_2A/substitution/lift.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+inductive liftv (l,m:nat) : relation (list term) ≝
+| liftv_nil : liftv l m (◊) (◊)
+| liftv_cons: ∀T1s,T2s,T1,T2.
+              ⬆[l, m] T1 ≡ T2 → liftv l m T1s T2s →
+              liftv l m (T1 @ T1s) (T2 @ T2s)
+.
+
+interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact liftv_inv_nil1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
+#T1s #T2s #l #m * -T1s -T2s //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed-.
+
+lemma liftv_inv_nil1: ∀T2s,l,m. ⬆[l, m] ◊ ≡ T2s → T2s = ◊.
+/2 width=5 by liftv_inv_nil1_aux/ qed-.
+
+fact liftv_inv_cons1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s →
+                          ∀U1,U1s. T1s = U1 @ U1s →
+                          ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
+                                    T2s = U2 @ U2s.
+#T1s #T2s #l #m * -T1s -T2s
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 @ U1s ≡ T2s →
+                       ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
+                                 T2s = U2 @ U2s.
+/2 width=3 by liftv_inv_cons1_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma liftv_total: ∀l,m. ∀T1s:list term. ∃T2s. ⬆[l, m] T1s ≡ T2s.
+#l #m #T1s elim T1s -T1s
+[ /2 width=2 by liftv_nil, ex_intro/
+| #T1 #T1s * #T2s #HT12s
+  elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn.ma
new file mode 100644 (file)
index 0000000..18adaac
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn.ma
@@ -0,0 +1,89 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/grammar/lenv_length.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+inductive lpx_sn (R:relation3 lenv term term): relation lenv ≝
+| lpx_sn_atom: lpx_sn R (⋆) (⋆)
+| lpx_sn_pair: ∀I,K1,K2,V1,V2.
+               lpx_sn R K1 K2 → R K1 V1 V2 →
+               lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
+#R #HR #L elim L -L /2 width=1 by lpx_sn_atom, lpx_sn_pair/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_sn_inv_atom1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom1: ∀R,L2. lpx_sn R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
+
+fact lpx_sn_inv_pair1_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+                           ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair1: ∀R,I,K1,V1,L2. lpx_sn R (K1. ⓑ{I} V1) L2 →
+                        ∃∃K2,V2. lpx_sn R K1 K2 & R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
+
+fact lpx_sn_inv_atom2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_sn_inv_atom2: ∀R,L1. lpx_sn R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
+
+fact lpx_sn_inv_pair2_aux: ∀R,L1,L2. lpx_sn R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+                           ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lpx_sn_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn R L1 (K2. ⓑ{I} V2) →
+                        ∃∃K1,V1. lpx_sn R K1 K2 & R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
+
+lemma lpx_sn_inv_pair: ∀R,I1,I2,L1,L2,V1,V2.
+                       lpx_sn R (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) →
+                       ∧∧ lpx_sn R L1 L2 & R L1 V1 V2 & I1 = I2.
+#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lpx_sn_inv_pair1 … H) -H
+#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_alt.ma
new file mode 100644 (file)
index 0000000..94c112b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_alt.ma
@@ -0,0 +1,125 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop.ma".
+include "basic_2A/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* alternative definition of lpx_sn *)
+definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
+                       λR,L1,L2. |L1| = |L2| ∧
+                       (∀I1,I2,K1,K2,V1,V2,i.
+                          ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                          I1 = I2 ∧ R K1 V1 V2
+                       ).
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
+#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_sn/
+qed-.
+
+lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
+                            ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I1 #L2 #K1 #V1 #H elim H -H
+#H #IH elim (length_inv_pos_sn … H) -H
+#I2 #K2 #V2 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by drop_drop, conj/
+qed-.
+
+lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
+#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_dx/
+qed-.
+
+lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
+                            ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I2 #L1 #K2 #V2 #H elim H -H
+#H #IH elim (length_inv_pos_dx … H) -H
+#I1 #K1 #V1 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
+/2 width=1 by drop_drop, conj/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
+#R @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1
+#H destruct
+qed.
+
+lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
+                       lpx_sn_alt R L1 L2 → R L1 V1 V2 →
+                       lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
+#HL12 #IH @conj normalize //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
+[ #HLK1 #HLK2
+  lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct
+  lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct
+  /2 width=1 by conj/
+| -HL12 -HV12 /3 width=6 by drop_inv_drop1/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
+#R #L1 elim L1 -L1
+[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
+| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
+  #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
+]
+qed-.
+
+(* alternative definition of lpx_sn *****************************************)
+
+lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+                        (∀I1,I2,K1,K2,V1,V2,i.
+                           ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           I1 = I2 ∧ R K1 V1 V2
+                        ) → lpx_sn R L1 L2.
+/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
+
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
+                      |L1| = |L2| ∧
+                      ∀I1,I2,K1,K2,V1,V2,i.
+                      ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+                      I1 = I2 ∧ R K1 V1 V2.
+#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H elim H -H /3 width=4 by conj/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_drop.ma
new file mode 100644 (file)
index 0000000..d97a5d3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_drop.ma
@@ -0,0 +1,104 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_lreq.ma".
+include "basic_2A/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on dropping ****************************************************)
+
+lemma lpx_sn_drop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
+                        ∀I,K1,V1,i. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
+                        ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
+  [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
+  | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+    /3 width=5 by drop_drop_lt, ex3_2_intro/
+  ]
+]
+qed-.
+
+lemma lpx_sn_drop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
+                         ∀I,K2,V2,i. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+                         ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
+  [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
+  | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+    /3 width=5 by drop_drop_lt, ex3_2_intro/
+  ]
+]
+qed-.
+
+lemma lpx_sn_deliftable_dropable: ∀R. d_deliftable_sn R → dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 … H) -H
+  /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+  #L2 #V2 #HL12 #HV12 #H destruct
+  /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+  #L2 #V2 #HL12 #HV12 #H destruct
+  elim (IHLK1 … HL12) -L1 /3 width=3 by drop_drop, ex2_intro/
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
+  elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+  elim (HR … HV12 … HLK1 … HWV1) -V1
+  elim (IHLK1 … HL12) -L1 /3 width=5 by drop_skip, lpx_sn_pair, ex2_intro/
+]
+qed-.
+
+lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
+                                  d_liftable R → dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 … H) -H
+  /4 width=4 by drop_atom, lpx_sn_atom, ex3_intro/
+| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+  #K2 #V2 #HK12 #HV12 #H destruct
+  lapply (lpx_sn_fwd_length … HK12)
+  #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
+  /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
+  @lreq_O2 normalize //
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
+  /3 width=5 by drop_drop, lreq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
+  elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+  elim (lift_total W2 l m) #V2 #HWV2
+  lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+  elim (IHLK1 … HK12) -K1
+  /3 width=6 by drop_skip, lreq_succ, lpx_sn_pair, ex3_intro/
+]
+qed-.
+
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,l,m. ⬇[s, l, m] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+                          l = 0 → ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #l #m #H elim H -L2 -K2 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom2 … H) -H 
+  /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+  #K1 #V1 #HK12 #HV12 #H destruct
+  /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #m #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+  #L1 #V1 #HL12 #HV12 #H destruct
+  elim (IHLK2 … HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_
+  <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
+/2 width=5 by lpx_sn_dropable_aux/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_lpx_sn.ma
new file mode 100644 (file)
index 0000000..2df0d37
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_lpx_sn.ma
@@ -0,0 +1,48 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
+                             ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+                             ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
+                             ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
+                              ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
+                              ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
+#R #HR #L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
+qed-.
+
+theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+                     confluent2 … (lpx_sn R1) (lpx_sn R2).
+#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #x #IH *
+[ #_ #X1 #H1 #X2 #H2 -x
+  >(lpx_sn_inv_atom1 … H1) -X1
+  >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
+| #L0 #I #V0 #Hx #X1 #H1 #X2 #H2 destruct
+  elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+  elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+  elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+  elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_tc.ma
new file mode 100644 (file)
index 0000000..61bcb50
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lpx_sn_tc.ma
@@ -0,0 +1,119 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+                           ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+                           ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+                      ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+                      ∀V1,V2. LTC … R L1 V1 V2 →
+                      TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
+                            ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+                            TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                           ∀I,L1,K2,V2. TC  … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+                           ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+  elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+  lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                     ∀S:relation lenv.
+                     S (⋆) (⋆) → (
+                        ∀I,K1,K2,V1,V2.
+                        TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+                        S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+                     ) →
+                     ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+  elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+  #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                              ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+                              ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+                              ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                           ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+                           ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                                ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+                                lpx_sn (LTC … R) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+  >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby.ma
new file mode 100644 (file)
index 0000000..0aab792
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby.ma
@@ -0,0 +1,237 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+include "basic_2A/notation/relations/lrsubeq_4.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
+| lsuby_atom: ∀L,l,m. lsuby l m L (⋆)
+| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
+              lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
+              lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
+              lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+.
+
+interpretation
+  "local environment refinement (extended substitution)"
+  'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m →
+                     L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
+#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l →
+                     L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
+qed.
+
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+                      ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
+#L elim L -L //
+#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
+#Hl destruct /2 width=1 by lsuby_succ/
+#m elim (ynat_cases … m) [| * #x ]
+#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
+qed.
+
+lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, yinj 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
+[ #l #H elim (le_plus_xSy_O_false … H)
+| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r … H) -H #HL12
+ elim (ynat_cases l) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
+]
+qed.
+
+lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
+#l #m #L1 #L2 #H elim H -l -m -L1 -L2
+[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l … H)
+  /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l … H)
+  /3 width=1 by lsuby_succ/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
+]
+qed-.
+
+lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
+/2 width=5 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
+                          L2 = ⋆ ∨
+                          ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+  /3 width=5 by ex2_3_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
+                       L2 = ⋆ ∨
+                       ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by lsuby_inv_zero1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
+                          L2 = ⋆ ∨
+                          ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
+  /3 width=4 by ex2_2_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
+                       L2 = ⋆ ∨
+                       ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V.
+/2 width=6 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
+                          L2 = ⋆ ∨
+                          ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
+  /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
+                       L2 = ⋆ ∨
+                       ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by lsuby_inv_succ1_aux/ qed-.
+
+fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
+                          ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
+  /2 width=5 by ex2_3_intro/
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
+                       ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=9 by lsuby_inv_zero2_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
+                          ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
+  /2 width=4 by ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
+                       ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V.
+/2 width=6 by lsuby_inv_pair2_aux/ qed-.
+
+fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                          ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
+                          ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W2 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
+  /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
+                       ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5 by lsuby_inv_succ2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize /2 width=1 by le_S_S/
+qed-.
+
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+                           ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+                           l ≤ i → i < l + m →
+                           ∃∃I1,K1. K1 ⊆[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #s #i #H
+  elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
+  elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+  [ #_ destruct -I2 >ypred_succ
+    /2 width=4 by drop_pair, ex2_2_intro/
+  | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+    #H <H -H #H lapply (ylt_inv_succ … H) -H
+    #Him elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
+    >yminus_succ <yminus_inj /3 width=4 by drop_drop_lt, ex2_2_intro/
+  ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli
+  elim (yle_inv_succ1 … Hli) -Hli
+  #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
+  #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+  #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+  /4 width=4 by ylt_O, drop_drop_lt, ex2_2_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma
new file mode 100644 (file)
index 0000000..8e45e98
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/substitution/lsuby_lsuby.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: ∀l,m. Transitive … (lsuby l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 … H) -H
+  #H destruct //
+| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
+  * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
+  * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
+  * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma
new file mode 100644 (file)
index 0000000..20cce2f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas.ma
@@ -0,0 +1,190 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/statictypestar_6.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/drop.ma".
+include "basic_2A/static/sh.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* activate genv *)
+inductive lstas (h): nat → relation4 genv lenv term term ≝
+| lstas_sort: ∀G,L,d,k. lstas h d G L (⋆k) (⋆((next h)^d k))
+| lstas_ldef: ∀G,L,K,V,W,U,i,d. ⬇[i] L ≡ K.ⓓV → lstas h d G K V W →
+              ⬆[0, i+1] W ≡ U → lstas h d G L (#i) U
+| lstas_zero: ∀G,L,K,W,V,i. ⬇[i] L ≡ K.ⓛW → lstas h 0 G K W V →
+              lstas h 0 G L (#i) (#i)
+| lstas_succ: ∀G,L,K,W,V,U,i,d. ⬇[i] L ≡ K.ⓛW → lstas h d G K W V →
+              ⬆[0, i+1] V ≡ U → lstas h (d+1) G L (#i) U
+| lstas_bind: ∀a,I,G,L,V,T,U,d. lstas h d G (L.ⓑ{I}V) T U →
+              lstas h d G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| lstas_appl: ∀G,L,V,T,U,d. lstas h d G L T U → lstas h d G L (ⓐV.T) (ⓐV.U)
+| lstas_cast: ∀G,L,W,T,U,d. lstas h d G L T U → lstas h d G L (ⓝW.T) U
+.
+
+interpretation "nat-iterated static type assignment (term)"
+   'StaticTypeStar h G L d T U = (lstas h d G L T U).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lstas_inv_sort1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀k0. T = ⋆k0 →
+                          U = ⋆((next h)^d k0).
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #k0 #H destruct //
+| #G #L #K #V #W #U #i #d #_ #_ #_ #k0 #H destruct
+| #G #L #K #W #V #i #_ #_ #k0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #W #T #U #d #_ #k0 #H destruct
+qed-.
+
+(* Basic_1: was just: sty0_gen_sort *)
+lemma lstas_inv_sort1: ∀h,G,L,X,k,d. ⦃G, L⦄ ⊢ ⋆k •*[h, d] X → X = ⋆((next h)^d k).
+/2 width=5 by lstas_inv_sort1_aux/
+qed-.
+
+fact lstas_inv_lref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀j. T = #j → ∨∨
+                          (∃∃K,V,W. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W &
+                                    ⬆[0, j+1] W ≡ U
+                          ) |
+                          (∃∃K,W,V. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                    U = #j & d = 0
+                          ) |
+                          (∃∃K,W,V,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V &
+                                       ⬆[0, j+1] V ≡ U & d = d0+1
+                          ).
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #j #H destruct
+| #G #L #K #V #W #U #i #d #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
+| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/
+| #G #L #K #W #V #U #i #d #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/ 
+| #a #I #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #W #T #U #d #_ #j #H destruct
+]
+qed-.
+
+lemma lstas_inv_lref1: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d] X → ∨∨
+                       (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W &
+                                 ⬆[0, i+1] W ≡ X
+                       ) |
+                       (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                 X = #i & d = 0
+                       ) |                      
+                       (∃∃K,W,V,d0. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V &
+                                    ⬆[0, i+1] V ≡ X & d = d0+1
+                       ).
+/2 width=3 by lstas_inv_lref1_aux/
+qed-.
+
+lemma lstas_inv_lref1_O: ∀h,G,L,X,i. ⦃G, L⦄ ⊢ #i •*[h, 0] X →
+                         (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, 0] W &
+                                   ⬆[0, i+1] W ≡ X
+                         ) ∨
+                         (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                   X = #i
+                         ).
+#h #G #L #X #i #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#K #W #V #d #_ #_ #_ <plus_n_Sm #H destruct
+qed-.
+
+(* Basic_1: was just: sty0_gen_lref *)
+lemma lstas_inv_lref1_S: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d+1] X →
+                         (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d+1] W &
+                                   ⬆[0, i+1] W ≡ X
+                         ) ∨                      
+                         (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d] V &
+                                   ⬆[0, i+1] V ≡ X
+                         ).
+#h #G #L #X #i #d #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#K #W #V #_ #_ #_ <plus_n_Sm #H destruct
+qed-.
+
+fact lstas_inv_gref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀p0. T = §p0 → ⊥.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #p0 #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #p0 #H destruct
+| #G #L #K #W #V #i #_ #_ #p0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #W #T #U #d #_ #p0 #H destruct
+qed-.
+
+lemma lstas_inv_gref1: ∀h,G,L,X,p,d. ⦃G, L⦄ ⊢ §p •*[h, d] X → ⊥.
+/2 width=9 by lstas_inv_gref1_aux/
+qed-.
+
+fact lstas_inv_bind1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
+                          ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •*[h, d] Z & U = ⓑ{b,J}Y.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #b #J #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #U #d #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #d #_ #b #J #X #Y #H destruct
+]
+qed-.
+
+(* Basic_1: was just: sty0_gen_bind *)
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,d. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, d] X →
+                       ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, d] U & X = ⓑ{a,I}V.U.
+/2 width=3 by lstas_inv_bind1_aux/
+qed-.
+
+fact lstas_inv_appl1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀X,Y. T = ⓐY.X →
+                          ∃∃Z. ⦃G, L⦄ ⊢ X •*[h, d] Z & U = ⓐY.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #W #T #U #d #_ #X #Y #H destruct
+]
+qed-.
+
+(* Basic_1: was just: sty0_gen_appl *)
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,d. ⦃G, L⦄ ⊢ ⓐV.T •*[h, d] X →
+                       ∃∃U. ⦃G, L⦄ ⊢ T •*[h, d] U & X = ⓐV.U.
+/2 width=3 by lstas_inv_appl1_aux/
+qed-.
+
+fact lstas_inv_cast1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀X,Y. T = ⓝY.X →
+                          ⦃G, L⦄ ⊢ X •*[h, d] U.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #W #T #U #d #HTU #X #Y #H destruct //
+]
+qed-.
+
+(* Basic_1: was just: sty0_gen_cast *)
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,d. ⦃G, L⦄ ⊢ ⓝW.T •*[h, d] U → ⦃G, L⦄ ⊢ T •*[h, d] U.
+/2 width=4 by lstas_inv_cast1_aux/
+qed-.
+
+(* Basic_1: removed theorems 7:
+            sty1_abbr sty1_appl sty1_bind sty1_cast2
+            sty1_correct sty1_lift sty1_trans
+*)

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_aaa.ma
new file mode 100644 (file)
index 0000000..9345eb7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_aaa.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/aaa_lift.ma".
+include "basic_2A/unfold/lstas_lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_lstas: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀d.
+                 ∃∃U. ⦃G, L⦄ ⊢ T •*[h, d] U & ⦃G, L⦄ ⊢ U ⁝ A.
+#h #G #L #T #A #H elim H -G -L -T -A
+[ /2 width=3 by ex2_intro/
+| * #G #L #K #V #B #i #HLK #HV #IHV #d
+  [ elim (IHV d) -IHV #W
+    elim (lift_total W 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=10 by lstas_ldef, aaa_lift, ex2_intro/
+  | @(nat_ind_plus … d) -d
+    [ elim (IHV 0) -IHV /3 width=7 by lstas_zero, aaa_lref, ex2_intro/
+    | #d #_ elim (IHV d) -IHV #W
+      elim (lift_total W 0 (i+1))
+      lapply (drop_fwd_drop2 … HLK)
+      /3 width=10 by lstas_succ, aaa_lift, ex2_intro/
+    ]
+  ]
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
+  /3 width=7 by lstas_bind, aaa_abbr, ex2_intro/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
+  /3 width=6 by lstas_bind, aaa_abst, ex2_intro/
+| #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
+  /3 width=6 by lstas_appl, aaa_appl, ex2_intro/
+| #G #L #W #T #A #HW #_ #_ #IHT #d elim (IHT d) -IHT
+  /3 width=3 by lstas_cast, aaa_cast, ex2_intro/
+]
+qed-.
+
+lemma lstas_aaa_conf: ∀h,G,L,d. Conf3 … (aaa G L) (lstas h d G L).
+#h #G #L #d #A #T #HT #U #HTU
+elim (aaa_lstas h … HT d) -HT #X #HTX
+lapply (lstas_mono … HTX … HTU) -T //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_da.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_da.ma
new file mode 100644 (file)
index 0000000..d3503d1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_da.ma
@@ -0,0 +1,95 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/static/da_da.ma".
+include "basic_2A/unfold/lstas_lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+lemma da_lstas: ∀h,g,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 → ∀d2.
+                ∃∃U. ⦃G, L⦄ ⊢ T •*[h, d2] U & ⦃G, L⦄ ⊢ U ▪[h, g] d1-d2.
+#h #g #G #L #T #d1 #H elim H -G -L -T -d1
+[ /4 width=3 by da_sort, deg_iter, ex2_intro/
+| #G #L #K #V #i #d1 #HLK #_ #IHV #d2
+  elim (IHV d2) -IHV #W
+  elim (lift_total W 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK)
+  /3 width=10 by lstas_ldef, da_lift, ex2_intro/
+| #G #L #K #W #i #d1 #HLK #HW #IHW #d2 @(nat_ind_plus … d2) -d2
+  [ elim (IHW 0) -IHW /3 width=6 by lstas_zero, da_ldec, ex2_intro/
+  | #d #_ elim (IHW d) -IHW #V
+    elim (lift_total V 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=10 by lstas_succ, da_lift, ex2_intro/
+  ]
+| #a #I #G #L #V #T #d1 #_ #IHT #d2 elim (IHT … d2) -IHT
+  /3 width=6 by lstas_bind, da_bind, ex2_intro/
+| * #G #L #V #T #d1 #_ #IHT #d2 elim (IHT … d2) -IHT
+  /3 width=5 by lstas_appl, lstas_cast, da_flat, ex2_intro/
+]
+qed-.
+
+lemma lstas_da_conf: ∀h,g,G,L,T,U,d2. ⦃G, L⦄ ⊢ T •*[h, d2] U →
+                     ∀d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 → ⦃G, L⦄ ⊢ U ▪[h, g] d1-d2.
+#h #g #G #L #T #U #d2 #HTU #d1 #HT
+elim (da_lstas … HT d2) -HT #X #HTX
+lapply (lstas_mono … HTX … HTU) -T //
+qed-.
+
+(* inversion lemmas on degree assignment for terms **************************)
+
+lemma lstas_inv_da: ∀h,g,G,L,T,U,d2. ⦃G, L⦄ ⊢ T •*[h, d2] U →
+                    ∃∃d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 & ⦃G, L⦄ ⊢ U ▪[h, g] d1-d2.
+#h #g #G #L #T #U #d2 #H elim H -G -L -T -U -d2
+[ #G #L #d2 #k elim (deg_total h g k) /4 width=3 by da_sort, deg_iter, ex2_intro/
+| #G #L #K #V #W #U #i #d2 #HLK #_ #HWU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex2_intro/
+| #G #L #K #W #V #i #HLK #_ * /3 width=6 by da_ldec, ex2_intro/
+| #G #L #K #W #V #U #i #d2 #HLK #_ #HVU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldec, ex2_intro/
+| #a #I #G #L #V #T #U #d2 #_ * /3 width=3 by da_bind, ex2_intro/
+| #G #L #V #T #U #d2 #_ * /3 width=3 by da_flat, ex2_intro/
+| #G #L #W #T #U #d2 #_ * /3 width=3 by da_flat, ex2_intro/
+]
+qed-.
+
+lemma lstas_inv_da_ge: ∀h,G,L,T,U,d2,d. ⦃G, L⦄ ⊢ T •*[h, d2] U →
+                       ∃∃g,d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 & ⦃G, L⦄ ⊢ U ▪[h, g] d1-d2 & d ≤ d1.
+#h #G #L #T #U #d2 #d #H elim H -G -L -T -U -d2
+[ /4 width=5 by da_sort, deg_iter, ex3_2_intro/
+| #G #L #K #V #W #U #i #d2 #HLK #_ #HWU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex3_2_intro/
+| #G #L #K #W #V #i #HLK #_ * 
+  #g #d1 #HW #HV #Hd1 /4 width=6 by da_ldec, lt_to_le, le_S_S, ex3_2_intro/
+| #G #L #K #W #V #U #i #d2 #HLK #_ #HVU *
+  lapply (drop_fwd_drop2 … HLK)
+  /4 width=10 by da_lift, da_ldec, lt_to_le, le_S_S, ex3_2_intro/
+| #a #I #G #L #V #T #U #d2 #_ * /3 width=5 by da_bind, ex3_2_intro/
+| #G #L #V #T #U #d2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+| #G #L #W #T #U #d2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_refl_pos: ∀h,G,L,T,d. ⦃G, L⦄ ⊢ T •*[h, d+1] T → ⊥.
+#h #G #L #T #d2 #H elim (lstas_inv_da_ge … (d2+1) H) -H
+#g #d1 #HT1 #HT12 #Hd21 lapply (da_mono … HT1 … HT12) -h -G -L -T
+#H elim (discr_x_minus_xy … H) -H
+[ #H destruct /2 width=3 by le_plus_xSy_O_false/
+| -d1 <plus_n_Sm #H destruct 
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lift.ma
new file mode 100644 (file)
index 0000000..d6114a6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lift.ma
@@ -0,0 +1,183 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/substitution/drop_drop.ma".
+include "basic_2A/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was just: sty0_lift *)
+lemma lstas_lift: ∀h,G,d. d_liftable (lstas h G d).
+#h #G #d #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -d
+[ #G #L1 #d #k #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
+  >(lift_inv_sort1 … H1) -X1
+  >(lift_inv_sort1 … H2) -X2 //
+| #G #L1 #K1 #V1 #W1 #W #i #d #HLK1 #_ #HW1 #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by lstas_ldef/
+  | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+    lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/
+  ]
+| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
+  >(lift_mono … HWU2 … H) -U2
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (lift_total W1 (l-i-1) m) #W2 #HW12
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=10 by lstas_zero/
+  | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=10 by lstas_zero, drop_inv_gen/
+  ]
+| #G #L1 #K1 #W1 #V1 #W #i #d #HLK1 #_ #HW1 #IHWV1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hil #H destruct
+  [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
+    /3 width=9 by lstas_succ/
+  | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+    lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/
+  ]
+| #a #I #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
+  elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+  elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/
+| #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
+  elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+  elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_appl/
+| #G #L1 #W1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X #H #U2 #HU12
+  elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/
+]
+qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma lstas_inv_lift1: ∀h,G,d. d_deliftable_sn (lstas h G d).
+#h #G #d #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -d
+[ #G #L2 #d #k #L1 #s #l #m #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #W2 #W #i #d #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HVW2 | -IHVW2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
+    elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
+    elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_ldef, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+    elim (le_inv_plus_l … Hil) -Hil #Hlim #mi
+    elim (lift_split … HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
+    #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_ldef, le_S, ex2_intro/
+  ]
+| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+    elim (IHWV2 … HK21 … HW12) -K2
+    /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2
+    /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/
+  ]
+| #G #L2 #K2 #W2 #V2 #W #i #d #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+    elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1
+    elim (lift_trans_le … HV12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_succ, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+    elim (le_inv_plus_l … Hil) -Hil #Hlim #mi
+    elim (lift_split … HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
+    #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_succ, le_S, ex2_intro/
+  ]
+| #a #I #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_split_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → ∀d1,d2. d = d1 + d2 →
+                       ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
+[ #G #L #d #k #d1 #d2 #H destruct
+  >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/
+| #G #L #K #V1 #V2 #U2 #i #d #HLK #_ #VU2 #IHV12 #d1 #d2 #H destruct
+  elim (IHV12 d1 d2) -IHV12 // #V
+  elim (lift_total V 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK)
+  /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/
+| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #d1 #d2 #H
+  elim (zero_eq_plus … H) -H #H1 #H2 destruct
+  /3 width=5 by lstas_zero, ex2_intro/
+| #G #L #K #W1 #W2 #U2 #i #d #HLK #HW12 #HWU2 #IHW12 #d1 @(nat_ind_plus … d1) -d1
+  [ #d2 normalize #H destruct
+    elim (IHW12 0 d) -IHW12 //
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=8 by lstas_succ, lstas_zero, ex2_intro/
+  | #d1 #_ #d2 <plus_plus_comm_23 #H lapply (injective_plus_l … H) -H #H
+    elim (IHW12 … H) -d #W
+    elim (lift_total W 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=12 by lstas_lift, lstas_succ, ex2_intro/
+  ]
+| #a #I #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
+  elim (IHTU … H) -d /3 width=3 by lstas_bind, ex2_intro/
+| #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
+  elim (IHTU … H) -d /3 width=3 by lstas_appl, ex2_intro/
+| #G #L #W #T #U #d #_ #IHTU #d1 #d2 #H
+  elim (IHTU … H) -d /3 width=3 by lstas_cast, ex2_intro/
+]
+qed-.
+
+lemma lstas_split: ∀h,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*[h, d1 + d2] T2 →
+                   ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+/2 width=3 by lstas_split_aux/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_lstas: ∀h,G,L,T,T1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 →
+                   ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+#h #G #L #T #T1 #d1 #H elim H -G -L -T -T1 -d1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V #V1 #U1 #i #d1 #HLK #_ #HVU1 #IHV1 #d2
+  elim (IHV1 d2) -IHV1 #V2
+  elim (lift_total V2 0 (i+1))
+  /3 width=7 by ex_intro, lstas_ldef/
+| #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #d2
+  @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+  #d2 #_ elim (IHW1 d2) -IHW1 #W2
+  elim (lift_total W2 0 (i+1))
+  /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #_ #IHW1 #d2
+  @(nat_ind_plus … d2) -d2
+  [ elim (IHW1 0) -IHW1 /3 width=5 by lstas_zero, ex_intro/
+  | #d2 #_ elim (IHW1 d2) -IHW1
+    #W2 elim (lift_total W2 0 (i+1)) /3 width=7 by ex_intro, lstas_succ/
+  ]
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_cast, ex_intro/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_llpx_sn.ma
new file mode 100644 (file)
index 0000000..07ee72b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_llpx_sn.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/multiple/llpx_sn_drop.ma".
+include "basic_2A/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on lazy sn pointwise extensions *******************************)
+
+lemma lstas_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R →
+                          ∀h,G,d. s_r_confluent1 … (lstas h d G) (llpx_sn R 0).
+#R #H1R #H2R #h #G #d #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2 -d
+[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
+| #G #Ls #Ks #V1s #V2s #W2s #i #d #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+  #Kd #V1l #HLKd #HV1s #HV1sd
+  lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+  lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+  @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| //
+| #G #Ls #Ks #V1s #V2s #W2s #i #d #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+  #Kd #V1l #HLKd #HV1s #HV1sd
+  lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
+  lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
+  @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+| #a #I #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
+  /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
+| #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+| #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
+  /3 width=1 by llpx_sn_flat/
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lstas.ma
new file mode 100644 (file)
index 0000000..9433915
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/lstas_lstas.ma
@@ -0,0 +1,124 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/unfold/lstas_lift.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Main properties **********************************************************)
+
+theorem lstas_trans: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                     ∀T2,d2. ⦃G, L⦄ ⊢ T •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*[h, d1+d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ #G #L #d1 #k #X #d2 #H >(lstas_inv_sort1 … H) -X
+  <iter_plus /2 width=1 by lstas_sort/
+| #G #L #K #V1 #V #U #i #d1 #HLK #_ #HVU #IHV1 #U2 #d2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H0
+  elim (lstas_inv_lift1 … HU2 … H0 … HVU)
+  /3 width=6 by lstas_ldef/
+| //
+| #G #L #K #W1 #W #U #i #d1 #HLK #_ #HWU #IHW1 #U2 #d2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H0
+  elim (lstas_inv_lift1 … HU2 … H0 … HWU)
+  /3 width=6 by lstas_succ/
+| #a #I #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
+  elim (lstas_inv_bind1 … H) -H #T2 #HT2 #H destruct
+  /3 width=1 by lstas_bind/
+| #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
+  elim (lstas_inv_appl1 … H) -H #T2 #HT2 #H destruct
+  /3 width=1 by lstas_appl/
+| /3 width=1 by lstas_cast/
+]
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem lstas_mono: ∀h,G,L,d. singlevalued … (lstas h d G L).
+#h #G #L #d #T #T1 #H elim H -G -L -T -T1 -d
+[ #G #L #d #k #X #H >(lstas_inv_sort1 … H) -X //
+| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H
+  elim (lstas_inv_lref1 … H) -H *
+  #K0 #V0 #W0 [3: #d0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+  #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct
+  /2 width=5 by lift_mono/
+| #G #L #K #W #W1 #i #HLK #_ #_ #X #H
+  elim (lstas_inv_lref1_O … H) -H *
+  #K0 #V0 #W0 #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct //
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H
+  elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+  #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct
+  /2 width=5 by lift_mono/
+| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H
+  elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #V #T #U1 #d #_ #IHTU1 #X #H
+  elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H
+  lapply (lstas_inv_cast1 … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was just: sty0_correct *)
+lemma lstas_correct: ∀h,G,L,T1,T,d1. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                     ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2
+  elim (IHV1 d2) -IHV1 #V2
+  elim (lift_total V2 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK) -HLK
+  /3 width=11 by ex_intro, lstas_lift/
+| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2
+  @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+  #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2
+  lapply (lstas_trans … HW1 … HW2) -W
+  elim (lift_total W2 0 (i+1))
+  /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2
+  elim (IHW1 d2) -IHW1 #W2
+  elim (lift_total W2 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK) -HLK
+  /3 width=11 by ex_intro, lstas_lift/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+  elim (IHTU d2) -IHTU /2 width=2 by ex_intro/
+]
+qed-.
+
+(* more main properties *****************************************************)
+
+theorem lstas_conf_le: ∀h,G,L,T,U1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] U1 →
+                       ∀U2,d2. d1 ≤ d2 → ⦃G, L⦄ ⊢ T •*[h, d2] U2 →
+                       ⦃G, L⦄ ⊢ U1 •*[h, d2-d1] U2.
+#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12
+>(plus_minus_m_m … Hd12) in ⊢ (%→?); -Hd12 >commutative_plus #H
+elim (lstas_split … H) -H #U #HTU
+>(lstas_mono … HTU … HTU1) -T //
+qed-.
+
+theorem lstas_conf: ∀h,G,L,T0,T1,d1. ⦃G, L⦄ ⊢ T0 •*[h, d1] T1 →
+                    ∀T2,d2. ⦃G, L⦄ ⊢ T0 •*[h, d2] T2 →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T2 •*[h, d1] T.
+#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02
+elim (lstas_lstas … HT01 (d1+d2)) #T #HT0
+lapply (lstas_conf_le … HT01 … HT0) // -HT01 <minus_plus_m_m_commutative
+lapply (lstas_conf_le … HT02 … HT0) // -T0 <minus_plus_m_m
+/2 width=3 by ex2_intro/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/unfold/unfold.ma b/matita/matita/contribs/lambdadelta/basic_2A/unfold/unfold.ma
new file mode 100644 (file)
index 0000000..063d9c7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2A/unfold/unfold.ma
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2A/notation/relations/unfold_4.ma".
+include "basic_2A/grammar/lenv_append.ma".
+include "basic_2A/grammar/genv.ma".
+include "basic_2A/substitution/drop.ma".
+
+(* CONTEXT-SENSITIVE UNFOLD FOR TERMS ***************************************)
+
+(* activate genv *)
+inductive unfold: relation4 genv lenv term lenv ≝
+| unfold_sort: ∀G,L,k. unfold G L (⋆k) L
+| unfold_lref: ∀I,G,L1,L2,K1,K2,V,i. ⬇[i] L1 ≡ K1. ⓑ{I}V →
+               unfold G K1 V K2 → ⬇[Ⓣ, |L2|, i] L2 ≡ K2 →
+               unfold G L1 (#i) (L1@@L2)
+| unfold_bind: ∀a,I,G,L1,L2,V,T.
+               unfold G (L1.ⓑ{I}V) T L2 → unfold G L1 (ⓑ{a,I}V.T) L2
+| unfold_flat: ∀I,G,L1,L2,V,T.
+               unfold G L1 T L2 → unfold G L1 (ⓕ{I}V.T) L2
+.
+
+interpretation "context-sensitive unfold (term)"
+   'Unfold G L1 T L2 = (unfold G L1 T L2).

diff --git a/matita/matita/contribs/lambdadelta/bin/Makefile.common b/matita/matita/contribs/lambdadelta/bin/Makefile.common
new file mode 100644 (file)
index 0000000..7f95f4a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/Makefile.common
@@ -0,0 +1,36 @@
+H=@
+
+OCAMLOPTIONS = -linkpkg -package \"$(REQUIRES)\" $(CAMLOPTIONS)
+OCAMLFIND = OCAMLPATH=$(OCAMLPATH) ocamlfind
+OCAMLC = $(OCAMLFIND) ocamlc -g $(OCAMLOPTIONS)
+OCAMLOPT = $(OCAMLFIND) opt $(OCAMLOPTIONS)
+
+CAMLP_FEATURES = $(F:%=-D%)
+
+AMLS = $(wildcard *.aml)
+
+define BUILD_TEMPLATE
+$(1).all:
+       @echo "  OCAMLBUILD $(1)"
+       $(H)ocamlbuild $$(BUILDOPTIONS) -ocamlc "$$(OCAMLC)" -ocamlopt "$$(OCAMLOPT)" -yaccflags "-v" -pp "$$(CAMLP)" $(1)
+
+.PHONY: $(1).all
+endef
+
+all:: $(AMLS:%.aml=%.ml) $(EXECS:%=%.native.all)
+
+byte:: $(AMLS:%.aml=%.ml) $(EXECS:%=%.byte.all)
+
+$(foreach EXEC, $(EXECS), $(eval $(call BUILD_TEMPLATE,$(EXEC:%=%.native))))
+
+$(foreach EXEC, $(EXECS), $(eval $(call BUILD_TEMPLATE,$(EXEC:%=%.byte))))
+
+clean::
+       @echo "  OCAMLBUILD -clean"
+       $(H)ocamlbuild -clean
+       $(H)$(RM) $(AMLS:%.aml=%.ml) *~
+
+.PHONY: all clean
+
+%.ml: %.aml
+       $(H)$(ALPHA) < $< > $@

diff --git a/matita/matita/contribs/lambdadelta/bin/a.ml b/matita/matita/contribs/lambdadelta/bin/a.ml
new file mode 100644 (file)
index 0000000..4f310c8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/a.ml
@@ -0,0 +1,17 @@
+let f = "0123456789abcdef"
+
+let r, g, b = 1.0, 0.5, 0.0
+
+let h = 1. /. 2.
+
+let mk_h x = x +. (1. -. x) *. h
+
+let rr, gg, bb = mk_h r, mk_h g, mk_h b 
+
+let mk_f x = 
+   let x = int_of_float x in
+   print_char f.[x / 16]; print_char f.[x mod 16]  
+
+let _ = 
+   mk_f (rr *. 255.); mk_f (gg *. 255.); mk_f (bb *. 255.);
+   print_newline ()

diff --git a/matita/matita/contribs/lambdadelta/bin/hls.ml b/matita/matita/contribs/lambdadelta/bin/hls.ml
new file mode 100644 (file)
index 0000000..d796a4a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/hls.ml
@@ -0,0 +1,57 @@
+let cols =
+   try int_of_string (Sys.getenv "COLUMNS")
+   with Not_found -> failwith "environment variable COLUMNS not visible"
+
+let hl = ref []
+
+let normal = "\x1B[0m"
+
+let color = "\x1B[32m"
+
+let add s =
+   if s = "" then false else
+   begin hl := s :: !hl; true end
+
+let rec read ich =
+   if Scanf.fscanf ich "%s " add then read ich
+
+let length l s = max l (String.length s)
+
+let split s =
+try 
+   let i = String.rindex s '.' in
+   if i = 0 then s, "" else
+   String.sub s 0 i, String.sub s i (String.length s - i)    
+with Not_found -> s, ""
+
+let compare s1 s2 =
+   let n1, e1 = split s1 in
+   let n2, e2 = split s2 in
+   let e = String.compare e1 e2 in
+   if e = 0 then String.compare n1 n2 else e
+
+let write l c s =
+   let pre, post = if List.mem s !hl then color, normal else "", "" in
+   let spc = String.make (l - String.length s) ' ' in
+   let bol, ret =
+       if c = 0 || c = cols then "", l else
+       if c + l < cols then " ", c + succ l else "\n", l
+   in
+   Printf.printf "%s%s%s%s%s" bol pre s post spc;
+   ret
+
+let process fname =
+   let ich = open_in fname in
+   read ich; close_in ich
+
+let help = ""
+
+let main =
+   Arg.parse [] process help;
+   let files = Sys.readdir "." in
+   let l = Array.fold_left length 0 files in
+   if cols < l then failwith "window too small";
+   Array.fast_sort compare files;
+   let c = Array.fold_left (write l) 0 files in
+   if 0 < c && c < cols then print_newline ();
+

diff --git a/matita/matita/contribs/lambdadelta/bin/index/Makefile b/matita/matita/contribs/lambdadelta/bin/index/Makefile
new file mode 100644 (file)
index 0000000..faf8860
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/index/Makefile
@@ -0,0 +1,10 @@
+EXECS = index
+
+REQUIRES = unix
+
+include ../Makefile.common
+
+test:
+#      @$(MAKE) --no-print-directory -C ../../ www
+
+.PHONY: test

diff --git a/matita/matita/contribs/lambdadelta/bin/index/index.ml b/matita/matita/contribs/lambdadelta/bin/index/index.ml
new file mode 100644 (file)
index 0000000..9496cc7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/index/index.ml
@@ -0,0 +1,123 @@
+module KF = Filename
+module KP = Printf
+module KU = Unix
+
+type status = {
+(* base directory *)
+  bd: string;
+(* input prefix *)
+  ip: string;
+(* output prefix *)
+  op: string;
+(* current path *)
+  cp: string list
+}
+
+let initial_status = {
+  bd = ""; ip = ""; op = "";
+  cp = [];
+}
+
+let imp_st = ref initial_status
+
+let i_ext = ".ld.ldw.xml"
+let o_ext = ".ld.html"
+
+let concats l =
+  List.fold_left KF.concat "" l
+
+let concat st dname = {st with
+  ip = KF.concat st.ip dname; op = KF.concat st.op dname;
+}
+
+let normalize dname =
+  if dname = KF.current_dir_name then "" else dname
+
+let mk_rlink s_to s_body =
+  KP.sprintf "<rlink to=\"%s\">%s</rlink>" s_to s_body
+
+let out_entry st dname och dirs name =
+  let iname = concats [st.bd; st.ip; dname; name] in
+  let stats = KU.lstat iname in
+  match stats.KU.st_kind with
+  | KU.S_REG when KF.check_suffix name i_ext ->
+    let base = KF.chop_suffix name i_ext in 
+    let oname = concats [st.bd; st.op; dname; base^o_ext] in
+    KP.fprintf och "    <file class=\"global emph\" type=\"&#x1F5CF;\" to=\"%s\" name=\"%s.ld\"/>\n" oname base;
+    dirs
+  | KU.S_DIR ->
+    let oname = concats [st.bd; st.op; dname; name] in
+    KP.fprintf och "    <file class=\"alpha emph\" type=\"&#x1F5C1;\" to=\"%s\" name=\"%s/\"/>\n" oname name;
+    name :: dirs
+  | _        ->
+    dirs
+
+let mk_path st och =
+  let path = String.concat "/" (List.rev st.cp) in
+  KP.fprintf och "    Contents of %s/\n" path
+
+let list_dir st dname och =
+  let iname = concats [st.bd; st.ip; dname] in
+  let dir = Sys.readdir iname in
+  Array.sort String.compare dir;
+  KP.fprintf och "   <index>\n";
+  let dirs = Array.fold_left (out_entry st dname och) [] dir in
+  KP.fprintf och "   </index>\n";
+  dirs
+
+let out_index st dname och =
+  KP.fprintf och "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n\n";
+  KP.fprintf och "<page xmlns=\"http://lambdadelta.info/\"\n";
+  KP.fprintf och "      description=\"\\lambda\\delta home page\"\n";
+  KP.fprintf och "      title=\"\\lambda\\delta home page\"\n";
+  KP.fprintf och "      head=\"λδ digital library (LDDL)\"\n";
+  KP.fprintf och ">\n";
+  KP.fprintf och "  <sitemap name=\"sitemap\"/>\n";
+  KP.fprintf och "  <section5 name=\"index\">Index</section5>\n";
+  KP.fprintf och "  <subsection name=\"path\">\n";
+  mk_path st och;
+  KP.fprintf och "  </subsection>\n";
+  KP.fprintf och "  <body>\n";
+  let dirs = list_dir st dname och in
+  KP.fprintf och "  </body>\n";
+  KP.fprintf och "  <footer><img label=\"helena\"/></footer>\n";
+  KP.fprintf och "</page>\n";
+  dirs
+
+let rec out_dir st dname =
+  let s_to, s_body =
+    if dname = ""
+    then concats [st.bd; st.op], "ld:" 
+    else concats [st.bd; st.op; dname], dname
+  in
+  let st = {st with cp = mk_rlink s_to s_body :: st.cp} in
+  let oname = concats [st.bd; st.ip; dname; "index.ldw.xml"] in
+  let och = open_out oname in
+  let dirs = out_index st dname och in
+  close_out och;
+  let map st = out_dir (concat st dname) in
+  List.iter (map st) dirs
+
+let help_b = "<dir>  Set this base directory"
+let help_i = "<dir>  Set this input prefix"
+let help_o = "<dir>  Set this output prefix"
+let help = "Usage: index [ -bio <dir> | <dir> ]*"
+
+let set_b bd =
+  imp_st := {!imp_st with bd = normalize bd}
+
+let set_i ip =
+  imp_st := {!imp_st with ip = normalize ip}
+
+let set_o op =
+  imp_st := {!imp_st with op = normalize op}
+
+let process dname =
+  out_dir !imp_st (normalize dname)
+
+let main =
+  Arg.parse [
+    "-b", Arg.String set_b, help_b;
+    "-i", Arg.String set_i, help_i;
+    "-o", Arg.String set_o, help_o;
+  ] process help

diff --git a/matita/matita/contribs/lambdadelta/bin/inline/Makefile b/matita/matita/contribs/lambdadelta/bin/inline/Makefile
new file mode 100644 (file)
index 0000000..60ad8b7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/inline/Makefile
@@ -0,0 +1,12 @@
+EXECS = inline
+
+REQUIRES =
+
+include ../Makefile.common
+
+test:
+       @./inline.native -p ../lambdadelta/*/deps.txt > deps.txt
+       @../matitadep/matitadep.native -c ../lambdadelta/.depend deps.txt > redundant.txt
+       @./inline.native -i -b ../lambdadelta redundant.txt ../lambdadelta/*/deps.txt
+
+.PHONY: test

diff --git a/matita/matita/contribs/lambdadelta/bin/inline/inline.ml b/matita/matita/contribs/lambdadelta/bin/inline/inline.ml
new file mode 100644 (file)
index 0000000..905f7ec
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/inline/inline.ml
@@ -0,0 +1,162 @@
+module Deps  = Set.Make(String)
+module Table = Map.Make(String)
+
+let opt_map f = function
+  | None   -> None
+  | Some a -> Some (f a)
+
+let rec filename_split l s =
+  let dir, base = Filename.dirname s, Filename.basename s in
+  if dir = Filename.current_dir_name then base::l else filename_split (base::l) dir
+
+let filename_concat l =
+  String.concat Filename.dir_sep l
+
+let relative s =
+  match filename_split [] s with
+  | "cic:" :: "matita" :: "lambdadelta" :: tl -> List.rev tl
+  | _                                         -> []
+
+let to_string l =
+  filename_concat (List.rev l)
+
+let table = ref (Table.empty: Deps.t Table.t) 
+
+let add src dep =
+  let deps = match Table.find_opt src !table with
+  | None      -> Deps.singleton dep
+  | Some deps -> Deps.add dep deps
+  in
+  table := Table.add src deps !table
+
+let split_or s =
+  let map m = Printf.sprintf "or_%u" m in 
+  try Scanf.sscanf s "or%u" map
+  with Scanf.Scan_failure _ | End_of_file -> ""
+
+let split_and s =
+  let map m = Printf.sprintf "and_%u" m in 
+  try Scanf.sscanf s "and%u" map
+  with Scanf.Scan_failure _ | End_of_file -> ""
+
+let split_ex s =
+  let map m n = Printf.sprintf "ex_%u_%u" m n in 
+  try Scanf.sscanf s "ex%u=%u" map
+  with Scanf.Scan_failure _ | End_of_file -> ""
+
+let split_ex1 s =
+  let map m = Printf.sprintf "ex_%u_1" m in 
+  try Scanf.sscanf s "ex%u" map
+  with Scanf.Scan_failure _ | End_of_file -> ""
+
+let map_deps s1 s2 =
+  match relative s2 with
+  | [b2;"xoa";"xoa";"ground_2"] ->
+    let r1 = List.tl (relative s1) in
+    let r1 = to_string r1 in
+    let b2 = Filename.remove_extension b2 in
+(* '_' is accepted (and ignored) within integer literals *)
+    let b2 = String.concat "=" (String.split_on_char '_' b2) in
+    let r2 =
+      let cx = split_ex b2 in
+      let cy = split_ex1 b2 in
+      let ca = split_and b2 in
+      let co = split_or b2 in
+      if cx <> "" then cx else
+      if cy <> "" then cy else
+      if ca <> "" then ca else
+      if co <> "" then co else
+      failwith (Printf.sprintf "unrecognized xoa: %S\n" b2)
+    in
+    if r1 <> "ground_2/xoa/xoa" then add r1 r2
+  | _                           -> ()
+
+let reds = ref []
+
+let map_reds s1 s2 =
+  reds := (s1,s2) :: !reds
+
+let rec read map_deps map_reds ich =
+  let line = input_line ich in
+  begin try Scanf.sscanf line "%S: %S" map_deps
+  with Scanf.Scan_failure _ | End_of_file ->
+    begin try Scanf.sscanf line "%S: redundant %S" map_reds
+    with Scanf.Scan_failure _ | End_of_file ->
+      Printf.eprintf "unknown line: %s.\n" line
+    end
+  end; 
+  read map_deps map_reds ich
+
+let xoadir = ref "ground_2/xoa"
+
+let print_deps () =
+  let map_d src dep =
+    let src = src^".ma" in
+    let dep = Filename.concat !xoadir (dep^".ma") in
+    if List.mem (src,dep) !reds then ()
+    else Printf.printf "%S: %S\n" src dep
+  in
+  let map_t src deps =
+    Deps.iter (map_d src) deps
+  in
+  Table.iter map_t !table
+
+let rec copy xn ich och =
+  if xn = Some 0 then ()
+  else begin
+    Printf.fprintf och "%s\n" (input_line ich); 
+    copy (opt_map pred xn) ich och
+  end
+
+let base_dir = ref ""
+
+let preamble = ref 14
+
+let insert_deps () =
+  let map_d src dep rdeps =
+    let dep = Filename.concat !xoadir (dep^".ma") in
+    if List.mem (src,dep) !reds then rdeps else dep::rdeps
+  in
+  let map_r och rdep =
+    Printf.fprintf och "include %S.\n" rdep;
+  in
+  let map_t src deps =
+    let src = src^".ma" in
+    let rdeps = Deps.fold (map_d src) deps [] in
+    if rdeps <> [] then begin    
+      let ma = Filename.concat !base_dir src in
+      let old = ma^".old" in
+      Sys.rename ma old;
+      let och = open_out ma in
+      let ich = open_in old in
+      copy (Some !preamble) ich och;
+      List.iter (map_r och) (List.rev rdeps);
+      try copy None ich och
+      with End_of_file -> close_in ich; close_out och
+    end 
+  in
+  Table.iter map_t !table
+
+let process fname =
+  let ich = open_in fname in
+  try read map_deps map_reds ich with 
+  | End_of_file -> close_in ich
+
+let help_b = "<dir>  Set this base directory (default: current directory)"
+let help_i = "  Insert the dependences (default: no)"
+let help_l = "<int>  .ma preamble has theese lines (default: 14)"
+let help_p = "  Print the dependences to be inserted (default: no)"
+let help = "inline [ -ip | -b <dir> | -l <int> | <file> ]*"
+
+let print = ref false
+let insert = ref false
+
+let _ =
+  Arg.parse [
+    "-b", Arg.String ((:=) base_dir), help_b;
+    "-l", Arg.Int ((:=) preamble), help_l;
+    "-i", Arg.Set insert, help_i; 
+    "-p", Arg.Set print, help_p; 
+  ] process help;
+  if !print then print_deps ();
+  if !insert then insert_deps ()

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/Makefile b/matita/matita/contribs/lambdadelta/bin/xhtbl/Makefile
new file mode 100644 (file)
index 0000000..c56f2d8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/Makefile
@@ -0,0 +1,10 @@
+EXECS = xhtbl
+
+REQUIRES = str
+
+include ../Makefile.common
+
+test:
+       @$(MAKE) --no-print-directory -C ../../ www
+
+.PHONY: test

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/attr.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/attr.ml
new file mode 100644 (file)
index 0000000..36b3d00
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/attr.ml
@@ -0,0 +1,20 @@
+module L = List
+
+module T = Table
+
+(* true for a row specification *)
+type 'a atom = 'a * bool * int option * int option
+
+type 'a atoms = 'a atom list
+
+let get_attr concat null a y x =
+   let map y x (c, b, x1, x2) = match b, x1, x2 with
+      | _    , None, None       -> c
+      | false, None, Some c2    -> if x <= c2 then c else null
+      | false, Some c1, None    -> if x >= c1 then c else null
+      | false, Some c1, Some c2 -> if x >= c1 && x <= c2 then c else null
+      | true , None, Some r2    -> if y <= r2 then c else null
+      | true , Some r1, None    -> if y >= r1 then c else null
+      | true , Some r1, Some r2 -> if y >= r1 && y <= r2 then c else null
+   in
+   concat (L.map (map y x) a)

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/fold.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/fold.ml
new file mode 100644 (file)
index 0000000..752b06d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/fold.ml
@@ -0,0 +1,25 @@
+module T = Table
+
+type 'a fold_cb = {
+   open_table : 'a -> T.table -> 'a;
+   close_table: 'a -> T.table -> 'a;
+   map_key    : 'a -> T.key -> 'a;
+   open_line  : bool -> 'a -> 'a;
+   close_line : bool -> 'a -> 'a;
+   open_entry : bool -> 'a -> 'a;   
+   close_entry: bool -> 'a -> 'a -> 'a;
+}
+
+let map h g f a b = h a (g (f a) b)
+
+let rec fold_table cb a t =
+   let a = cb.open_table a t in
+   let a = fold_entry cb a t.T.te in
+   cb.close_table a t
+
+and fold_entry cb a = function
+   | T.Key k        -> cb.map_key a k
+   | T.Line (r, ts) ->
+      let a = cb.open_line r a in
+      let a = List.fold_left (map (cb.close_entry r) (fold_table cb) (cb.open_entry r)) a ts in
+      cb.close_line r a

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/matrix.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/matrix.ml
new file mode 100644 (file)
index 0000000..1c65c50
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/matrix.ml
@@ -0,0 +1,63 @@
+module A = Array
+module N = Filename
+
+module T = Table
+
+type cell = {
+   ck: T.text list; (* contents *)
+   cc: T.css;       (* css classes *)
+   cu: T.uri;       (* uri *)
+   cx: T.ext;       (* extension *)
+   cn: T.anchor;    (* named anchor *)   
+   cb: T.border;    (* border *)
+}
+
+type matrix = {
+   r: int;              (* rows *)
+   c: int;              (* columns *)
+   m: cell array array; (* matrix *)
+}
+
+let strand a b = if a = "" then b else a
+
+let empty = {
+   ck = []; cc = []; cu = ""; cx = ""; cn = ""; cb = T.border false;
+}
+
+let make ts = {
+   r = ts.T.rf; c = ts.T.cf;
+   m = A.make_matrix ts.T.rf ts.T.cf empty;
+}
+
+let set_key m y x kl = 
+   m.m.(y).(x) <- {m.m.(y).(x) with ck = kl}
+
+let set_attrs m y x c u e n = 
+   m.m.(y).(x) <- {m.m.(y).(x) with
+      cc = c @ m.m.(y).(x).cc;
+      cu = u ^ m.m.(y).(x).cu;
+      cx = m.m.(y).(x).cx ^ e;
+      cn = strand m.m.(y).(x).cn n;
+   }
+
+let set_west m y x b =
+   let c = m.m.(y).(x) in
+   let cb = {c.cb with T.w = c.cb.T.w || b.T.w} in 
+   m.m.(y).(x) <- {c with cb = cb}
+
+let set_north m y x b =
+   let c = m.m.(y).(x) in
+   let cb = {c.cb with T.n = c.cb.T.n || b.T.n} in 
+   m.m.(y).(x) <- {c with cb = cb}
+
+let set_east m y x b =
+   if x < pred m.c then set_west m y (succ x) {b with T.w = b.T.e} else
+   let c = m.m.(y).(x) in
+   let cb = {c.cb with T.e = c.cb.T.e || b.T.e} in 
+   m.m.(y).(x) <- {c with cb = cb}
+
+let set_south m y x b =
+   if y < pred m.r then set_north m (succ y) x {b with T.n = b.T.s} else
+   let c = m.m.(y).(x) in
+   let cb = {c.cb with T.s = c.cb.T.s || b.T.s} in 
+   m.m.(y).(x) <- {c with cb = cb}

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/options.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/options.ml
new file mode 100644 (file)
index 0000000..21ebec1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/options.ml
@@ -0,0 +1,39 @@
+let output_dir_default = ""
+
+let baseuri_default = ""
+
+let debug_lexer_default = false
+
+let debug_pass_default = false
+
+let pass_default = false
+
+let output_dir = ref output_dir_default
+
+let baseuri = ref baseuri_default
+
+let debug_lexer = ref debug_lexer_default
+
+let d0 = ref debug_pass_default
+
+let d1 = ref debug_pass_default
+
+let d2 = ref debug_pass_default
+
+let e1 = ref debug_pass_default
+
+let e2 = ref debug_pass_default
+
+let p0 = ref pass_default
+
+let p1 = ref pass_default
+
+let p2 = ref pass_default
+
+let clear () =
+   output_dir := output_dir_default;
+   baseuri := baseuri_default;
+   debug_lexer := debug_lexer_default;
+   d0 := debug_pass_default; d1 := debug_pass_default; d2 := debug_pass_default; 
+   e1 := debug_pass_default; e2 := debug_pass_default;
+   p0 := pass_default; p1 := pass_default; p2 := pass_default

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/pass1.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass1.ml
new file mode 100644 (file)
index 0000000..bedd961
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass1.ml
@@ -0,0 +1,88 @@
+module L = List
+
+module T = Table
+module F = Fold
+
+type status = {
+   ts: T.size; (* current dimensions *)
+   tc: T.css;  (* current class *)
+   tu: T.uri;  (* current uri *)
+   tx: T.ext;  (* current extension *)
+}
+
+let empty = {
+   ts = T.no_size; tc = []; tu = ""; tx = ""
+}
+
+let init b ts =
+   if b then
+      {ts with T.ri = max_int; T.ci = 0}
+   else
+      {ts with T.ri = 0; T.ci = max_int}
+
+let combine b ts1 ts2 =
+   if b then    
+      {ts1 with 
+         T.rf = max ts1.T.rf ts2.T.rf; T.ri = min ts1.T.ri ts2.T.ri; 
+        T.cf = ts1.T.cf + ts2.T.cf; T.ci = ts1.T.ci + ts2.T.ci;
+      }
+   else
+      {ts1 with
+         T.cf = max ts1.T.cf ts2.T.cf; T.ci = min ts1.T.ci ts2.T.ci;
+        T.rf = ts1.T.rf + ts2.T.rf; T.ri = ts1.T.ri + ts2.T.ri; 
+      }
+
+let deinit ts = {ts with
+   T.ri = if ts.T.ri = max_int then 0 else ts.T.ri;
+   T.ci = if ts.T.ci = max_int then 0 else ts.T.ci;
+}
+
+(****************************************************************************)
+
+let open_table st t =
+   t.T.tc <- t.T.tc @ st.tc; t.T.tu <- st.tu ^ t.T.tu; t.T.tx <- st.tx ^ t.T.tx; 
+   {st with tc = t.T.tc; tu = t.T.tu; tx = t.T.tx}
+
+let close_table st t =
+   t.T.ts <- st.ts; st
+
+let map_key st k = 
+   let ts = match k, st.ts.T.p with
+      | T.Text _     , _          ->
+         {st.ts with T.rf = 1; T.cf = 1; T.ri = 0; T.ci = 0}
+      | T.Glue None  , _          ->
+         {st.ts with T.rf = 0; T.cf = 0; T.ri = 1; T.ci = 1}
+      | T.Glue Some g, Some false ->
+         {st.ts with T.rf = g; T.cf = 0; T.ri = 0; T.ci = 1}
+      | T.Glue Some g, Some true  ->
+         {st.ts with T.rf = 0; T.cf = g; T.ri = 1; T.ci = 0}
+      | T.Glue Some g, None       ->
+         {st.ts with T.rf = g; T.cf = g; T.ri = 0; T.ci = 0}
+   in
+   {st with ts = ts}
+
+let open_line b st =
+   let ts = init b st.ts in
+   let ts = {ts with T.rf = 0; T.cf = 0} in
+   {st with ts = ts}
+
+let open_entry b st =
+   let ts = {st.ts with T.p = Some b} in
+   {st with ts = ts}
+
+let close_entry b st sst =
+   {st with ts = combine b st.ts sst.ts}
+
+let close_line b st =
+   {st with ts = deinit st.ts}
+
+let cb = {
+   F.open_table = open_table; F.close_table = close_table;   
+   F.open_line = open_line; F.close_line = close_line;
+   F.open_entry = open_entry; F.close_entry = close_entry;
+   F.map_key = map_key;
+}
+
+let process t =
+   let st = F.fold_table cb empty t in
+   st.ts

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/pass2.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass2.ml
new file mode 100644 (file)
index 0000000..549d765
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass2.ml
@@ -0,0 +1,139 @@
+module O = Options
+module T = Table
+module M = Matrix
+module F = Fold
+
+type status = {
+   ts: T.size;   (* current dimensions *)
+   tm: M.matrix; (* current matrix *)
+}
+
+let initial t m = {
+   ts = {t.T.ts with T.ri = 0; T.ci = 0};
+   tm = m;
+}
+
+let resize b sts tts = 
+   if b then begin (* parent is a row *) 
+      if tts.T.rf < sts.T.rf && tts.T.ri = 0 then
+         failwith "underful column";
+      {tts with T.rf = sts.T.rf; T.cf = tts.T.cf + sts.T.ci * tts.T.ci}
+   end else begin (* parent is a column *)
+      if tts.T.cf < sts.T.cf && tts.T.ci = 0 then
+         failwith "underful row";
+      {tts with T.cf = sts.T.cf; T.rf = tts.T.rf + sts.T.ri * tts.T.ri}
+   end
+
+let fill b sts tts =
+   if b then (* parent is a row *) 
+      {sts with T.ri = 
+         let rf, ri = sts.T.rf - tts.T.rf, tts.T.ri in
+        if ri = 0 then 0 else
+        if rf mod ri = 0 then rf / ri else
+        failwith "fracted column"
+      }
+   else (* parent is a column *)
+      {sts with T.ci = 
+         let cf, ci = sts.T.cf - tts.T.cf, tts.T.ci in
+        if ci = 0 then 0 else
+        if cf mod ci = 0 then cf / ci else
+        failwith "fracted row"
+      }
+
+let place b sts tts =
+   if b then (* parent is a row *)
+      {sts with T.x = sts.T.x + tts.T.cf}
+   else (* parent is a column *)
+      {sts with T.y = sts.T.y + tts.T.rf}
+
+let set_key st t = match t.T.te with
+   | T.Key (T.Text sl) -> M.set_key st.tm t.T.ts.T.y t.T.ts.T.x sl  
+   | _                 -> ()
+
+let set_attrs st t =
+   let rec aux y x =
+      if y >= t.T.ts.T.rf then () else
+      if x >= t.T.ts.T.cf then aux (succ y) 0 else begin
+        M.set_attrs st.tm (t.T.ts.T.y + y) (t.T.ts.T.x + x) t.T.tc t.T.tu t.T.tx t.T.tn;
+         aux y (succ x)
+      end
+   in
+   match t.T.te with 
+      | T.Key _ -> aux 0 0 
+      | _       -> ()
+
+let set_borders st t =
+   let rec aux_we y =
+      if y >= t.T.ts.T.rf then () else begin
+        M.set_west st.tm (t.T.ts.T.y + y) t.T.ts.T.x t.T.tb;
+        if t.T.ts.T.cf > 0 then 
+           M.set_east st.tm (t.T.ts.T.y + y) (t.T.ts.T.x + pred t.T.ts.T.cf) t.T.tb;
+         aux_we (succ y)
+      end      
+   in
+   let rec aux_ns x =
+      if x >= t.T.ts.T.cf then () else begin
+        M.set_north st.tm t.T.ts.T.y (t.T.ts.T.x + x) t.T.tb;
+        if t.T.ts.T.rf > 0 then 
+           M.set_south st.tm (t.T.ts.T.y + pred t.T.ts.T.rf) (t.T.ts.T.x + x) t.T.tb;
+         aux_ns (succ x)
+      end      
+   in
+   match t.T.te with 
+      | T.Line (true, _) -> aux_we 0; aux_ns 0
+      | _                -> ()
+
+let print st t = 
+   if !O.e2 then
+      Printf.printf "#%u: (%u+%u, %u+%u) - (%u+%u, %u+%u)\n"
+         t.T.ti 
+         t.T.ts.T.rf t.T.ts.T.ri 
+         t.T.ts.T.cf t.T.ts.T.ci
+         st.ts.T.rf st.ts.T.ri
+         st.ts.T.cf st.ts.T.ci
+
+(****************************************************************************)
+
+let open_table st t =
+   print st t;
+   let ts = match t.T.ts.T.p with
+      | None   ->
+         let ts = fill false st.ts t.T.ts in
+         let ts = fill true ts t.T.ts in
+        t.T.ts <- resize false st.ts t.T.ts;
+         t.T.ts <- resize true st.ts t.T.ts;
+        ts
+      | Some b ->
+         let ts = fill b st.ts t.T.ts in
+        t.T.ts <- resize b st.ts t.T.ts;
+        ts
+   in
+   t.T.ts <- {t.T.ts with T.ri = 0; T.ci = 0; T.x = st.ts.T.x; T.y = st.ts.T.y};
+   let ts = {ts with T.rf = t.T.ts.T.rf; T.cf = t.T.ts.T.cf} in
+   let st = {st with ts = ts} in 
+   print st t; st
+
+let close_table st t =
+   set_key st t; set_attrs st t; set_borders st t; st
+
+let map_key st k = st
+
+let open_line b st = st
+
+let open_entry b st = st
+
+let close_entry b st sst =
+   let ts = place b st.ts sst.ts in
+   {st with ts = ts}
+
+let close_line b st = st
+
+let cb = {
+   F.open_table = open_table; F.close_table = close_table;   
+   F.open_line = open_line; F.close_line = close_line;
+   F.open_entry = open_entry; F.close_entry = close_entry;
+   F.map_key = map_key;
+}
+
+let process t m =
+   let _ = F.fold_table cb (initial t m) t in ()

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/pass3.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass3.ml
new file mode 100644 (file)
index 0000000..d2455a3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/pass3.ml
@@ -0,0 +1,32 @@
+module L = List
+module S = String
+module V = Array
+
+module T = Table
+module M = Matrix
+module A = Attr
+
+type status = {
+   m: M.matrix;
+   c: T.css A.atoms;
+   u: T.uri A.atoms;
+   x: T.ext A.atoms;
+}
+
+let initial c u x m = {
+   m = m; c = c; u = u; x = x
+}
+
+let process_cell st y x c =
+   M.set_attrs st.m y x
+      (A.get_attr L.concat [] st.c y x)
+      (A.get_attr (S.concat "") "" st.u y x)
+      (A.get_attr (S.concat "") "" st.x y x)
+      ""
+
+let process_row st y row =
+   V.iteri (process_cell st y) row
+
+let process css uri ext matrix =
+   let st = initial css uri ext matrix in
+   V.iteri (process_row st) matrix.M.m

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/table.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/table.ml
new file mode 100644 (file)
index 0000000..d3ee13b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/table.ml
@@ -0,0 +1,70 @@
+type css = string list
+
+type uri = string
+
+type ext = string
+
+type anchor = string
+
+type absolute = bool
+
+type size = {
+   y : int;         (* first row *)
+   x : int;         (* first column *)
+   rf: int;         (* finite rows *)
+   cf: int;         (* finite columns *)
+   ri: int;         (* infinite rows *)
+   ci: int;         (* infinite columns *)
+   p : bool option; (* parent kind *)
+}
+
+type border = {
+   n: bool; (* north *)
+   s: bool; (* south *)
+   e: bool; (* east *)
+   w: bool; (* west *)
+}
+
+type text = Plain of string
+          | Link of absolute * string * string
+
+type key = Text of text list
+         | Glue of int option
+
+type table = {
+           tn: anchor; (* named anchor *)
+   mutable tc: css;    (* css classes *)
+   mutable tu: uri;    (* uri *)
+   mutable tx: ext;    (* uri extension *)
+   mutable ts: size;   (* dimension *)
+           tb: border; (* border *)
+           te: entry;  (* contents *)
+          ti: int;    (* table identifier *)
+}
+
+and entry = Key  of key
+          | Line of bool * table list (* true for a row *)
+
+let id =
+   let current = ref 0 in
+   fun () -> incr current; !current
+
+let no_size = {
+   y = 0; x = 0; rf = 0; cf = 0; ri = 0; ci = 0; p = None;
+}
+
+let border b = {
+   n = b; s = b; e = b; w = b;
+}
+
+let mk_key k tc tu tx tn = {
+   ts = no_size; tb = border false; te = Key k;
+   tc = tc; tu = tu; tx = tx; tn = tn;
+   ti = id ();
+}
+
+let mk_line b tl tc tu tx tn = {
+   ts = no_size; tb = border b; te = Line (b, tl);
+   tc = tc; tu = tu; tx = tx; tn = tn;
+   ti = id ();
+}

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/textLexer.mll b/matita/matita/contribs/lambdadelta/bin/xhtbl/textLexer.mll
new file mode 100644 (file)
index 0000000..4b06e4c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/textLexer.mll
@@ -0,0 +1,47 @@
+{
+   module S  = String
+   
+   module O  = Options
+   module TP = TextParser
+   
+   let out s = if !O.debug_lexer then prerr_endline s
+}
+
+let SPC = ['\r' '\n' '\t' ' ']+
+let QT  = "\""
+let NUM = ['0'-'9']+
+
+rule token = parse
+   | SPC      { token lexbuf                    }
+   | QT       { let s = str lexbuf in 
+                out s; TP.TEXT s                }
+   | NUM as s { out s; TP.NUM (int_of_string s) }
+   | "(*"     { block lexbuf; token lexbuf      }
+   | "{"      { out "{"; TP.OC                  }
+   | "}"      { out "}"; TP.CC                  }
+   | "["      { out "["; TP.OB                  }
+   | "]"      { out "]"; TP.CB                  }   
+   | "*"      { out "*"; TP.SR                  }
+   | "^"      { out "^"; TP.CF                  }
+   | "+"      { out "+"; TP.PS                  }
+   | "("      { out "("; TP.OP                  }
+   | ")"      { out ")"; TP.CP                  }   
+   | "@"      { out ")"; TP.AT                  }    
+   | "space"  { out "space"; TP.SPACE           }
+   | "name"   { out "name"; TP.NAME             }   
+   | "table"  { out "table"; TP.TABLE           }
+   | "class"  { out "class"; TP.CSS             }
+   | "uri"    { out "uri"; TP.URI               }
+   | "ext"    { out "ext"; TP.EXT               }
+   | eof      { TP.EOF                          }
+and str = parse
+   | QT       { ""                              }
+   | "\\\\"   { "\\" ^ str lexbuf               }
+   | "\\\""   { "\"" ^ str lexbuf               }
+   | _ as c   { S.make 1 c ^ str lexbuf         }
+and block = parse
+   | "*)"     { ()                              }
+   | "(*"     { block lexbuf; block lexbuf      }
+   | QT       { let _ = str lexbuf in
+                block lexbuf                    }
+   | _        { block lexbuf                    }

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/textParser.mly b/matita/matita/contribs/lambdadelta/bin/xhtbl/textParser.mly
new file mode 100644 (file)
index 0000000..9072c2b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/textParser.mly
@@ -0,0 +1,147 @@
+%{
+
+module S = Str
+module L = List
+module T = Table
+
+let split s =
+   S.split (S.regexp "[ \r\n\t]+") s
+
+let mk_css_atom s rs =
+   let cs = split s in
+   let map (b, (x1, x2)) = cs, b, x1, x2 in 
+   L.map map rs
+
+let mk_string_atom s rs =
+   let map (b, (x1, x2)) = s, b, x1, x2 in 
+   L.map map rs
+
+%}
+
+%token <int> NUM
+%token <string> TEXT 
+%token SPACE NAME TABLE CSS URI EXT SR OC CC OB CB PS CF OP CP AT EOF
+
+%start script
+%type <(string * string) list * (string * Table.table * Table.css Attr.atoms * Table.uri Attr.atoms * Table.ext Attr.atoms) list> script
+
+%%
+
+space:
+   | SPACE TEXT TEXT { $2, $3 }
+;
+
+spaces:
+   |              { []       }
+   | space spaces { $1 :: $2 }
+;
+
+text:
+   | TEXT                  { T.Plain $1             }
+   | AT OP TEXT TEXT CP    { T.Link (true, $3, $4)  }
+   | AT AT OP TEXT TEXT CP { T.Link (false, $4, $5) }
+   | AT TEXT               { T.Link (true, $2, $2)  }
+   | AT AT TEXT            { T.Link (false, $3, $3) }   
+;
+
+texts:
+  | text          { [$1]                    }
+  | text PS texts { $1 :: T.Plain " " :: $3 }
+  | text CF texts { $1 :: $3                }
+;
+
+key:
+   | texts { T.Text $1        }
+   | SR    { T.Glue None      }
+   | NUM   { T.Glue (Some $1) }
+;
+
+css:
+   |          { []       }
+   | CSS TEXT { split $2 }
+;
+
+uri:
+   |          { "" }
+   | URI TEXT { $2 }
+;
+
+ext:
+   |          { "" }
+   | EXT TEXT { $2 }
+;
+
+table:
+   | css uri ext name key     { T.mk_key        $5 $1 $2 $3 $4 }
+   | css uri ext OC tables CC { T.mk_line false $5 $1 $2 $3 "" }
+   | css uri ext OB tables CB { T.mk_line true  $5 $1 $2 $3 "" }
+;
+
+tables:
+   |              { []       }
+   | table tables { $1 :: $2 }
+;
+
+name:
+   |           { "" }
+   | NAME TEXT { $2 }
+;
+
+interval:
+   | NUM     { Some $1, Some $1 }
+   | SR      { None, None       } 
+   | NUM NUM { Some $1, Some $2 }
+   | NUM SR  { Some $1, None    }
+   | SR NUM  { None, Some $2    }
+   | SR SR   { None, None       }
+;
+
+range:
+   | OB interval CB { true, $2  }
+   | OC interval CC { false, $2 }
+;
+
+ranges:
+   |              { []       }
+   | range ranges { $1 :: $2 }
+;
+
+catom:
+   | CSS TEXT ranges { mk_css_atom $2 $3 }
+;
+
+catoms:
+   |              { []      }
+   | catom catoms { $1 @ $2 }
+;
+
+uatom:
+   | URI TEXT ranges { mk_string_atom $2 $3 }
+;
+
+uatoms:
+   |              { []      }
+   | uatom uatoms { $1 @ $2 }
+;
+
+xatom:
+   | EXT TEXT ranges { mk_string_atom $2 $3 }
+;
+
+xatoms:
+   |              { []      }
+   | xatom xatoms { $1 @ $2 }
+;
+
+directive:
+   | name TABLE table catoms uatoms xatoms { $1, $3, $4, $5, $6 }
+;
+
+directives:
+   |                      { []       }
+   | directive directives { $1 :: $2 }
+;
+
+script:
+   | spaces directives EOF { $1, $2 }
+;

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/textUnparser.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/textUnparser.ml
new file mode 100644 (file)
index 0000000..cf7724c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/textUnparser.ml
@@ -0,0 +1,101 @@
+module L = List
+module P = Printf
+module S = String
+
+module T = Table
+module F = Fold
+
+type status = {
+   i: int;              (* indentation *)
+   out: string -> unit; (* output function *)
+}
+
+let home = {
+   i = 0; out = print_string
+}
+
+let indent st =
+   S.make st.i ' '
+
+let add st = {st with i = st.i + 3}
+
+let sub st = {st with i = st.i - 3}
+
+let parent = function
+   | None       -> "key"
+   | Some false -> "col"
+   | Some true  -> "row"
+
+let size ts =
+   P.sprintf "(%u, %u); (%u+%u, %u+%u); %s"
+      ts.T.y ts.T.x ts.T.rf ts.T.ri ts.T.cf ts.T.ci (parent ts.T.p)
+
+let border tb =
+   let str = S.make 4 ' ' in
+   if tb.T.w then str.[0] <- 'W';
+   if tb.T.n then str.[1] <- 'N';
+   if tb.T.e then str.[2] <- 'E';
+   if tb.T.s then str.[3] <- 'S';
+   str
+
+let css tc =
+   P.sprintf "\"%s\"" (S.concat " " tc)
+
+let uri tu tx =
+   P.sprintf "@\"%s\" \"%s\"" tu tx
+
+let name tn =
+   P.sprintf "$\"%s\"" tn
+
+
+let text = function
+   | T.Plain s              -> P.sprintf "\"%s\"" s
+   | T.Link (true, uri, s)  -> P.sprintf "@(\"%s\" \"%s\")" uri s
+   | T.Link (false, uri, s) -> P.sprintf "@@(\"%s\" \"%s\")" uri s
+
+let key = function
+   | T.Text sl       -> S.concat " ^ " (L.map text sl)
+   | T.Glue None     -> "*"
+   | T.Glue (Some i) -> P.sprintf "%u" i
+
+let entry = function
+   | false -> "column"
+   | true  -> "row"
+
+(****************************************************************************)
+
+let open_table st t =
+   let str = 
+      P.sprintf "%s[{#%u: %s; %s; %s; %s; %s}\n"    
+         (indent st) t.T.ti (size t.T.ts) (border t.T.tb) (css t.T.tc) (uri t.T.tu t.T.tx) (name t.T.tn)
+   in
+   st.out str; add st
+
+let close_table st t =
+   let st = sub st in
+   let str = P.sprintf "%s]\n" (indent st) in
+   st.out str; st
+
+let map_key st k =
+   let str = P.sprintf "%s%s\n" (indent st) (key k) in
+   st.out str; st
+   
+let open_line b st =
+   let str = P.sprintf "%s%s\n" (indent st) (entry b) in
+   st.out str; st
+
+let close_line b st = st
+
+let open_entry b st = st
+
+let close_entry b st sst = st
+
+let cb = {
+   F.open_table = open_table; F.close_table = close_table;   
+   F.open_line = open_line; F.close_line = close_line;
+   F.open_entry = open_entry; F.close_entry = close_entry;
+   F.map_key = map_key;
+}
+
+let debug t =
+   let _ = F.fold_table cb home t in ()

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/xhtbl.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/xhtbl.ml
new file mode 100644 (file)
index 0000000..6c5f8b0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/xhtbl.ml
@@ -0,0 +1,77 @@
+module A = Arg
+module F = Filename
+module L = List
+
+module O  = Options 
+module TP = TextParser
+module TL = TextLexer
+module TU = TextUnparser
+module P1 = Pass1
+module P2 = Pass2
+module P3 = Pass3
+module M  = Matrix
+module XU = XmlUnparser
+
+let help    = "Usage: xhtbl [ -LX | -O <dir> | -d0 | -d1 | -d2 | -e1 | -e2 | -p0 | -p1 | -p2 | <file> ]*"
+let help_L  = " Output lexer tokens"
+let help_O  = "<dir>  Set this output directory"
+let help_X  = " Clear all options"
+let help_b  = "<uri>  Set this base uri for relative links"
+let help_d0 = " Output table contents after phase zero (parsing)"
+let help_d1 = " Output table contents after phase one (sizing)"
+let help_d2 = " Output table contents after phase two (filling)"
+let help_e1 = " Disabled"
+let help_e2 = " Output debug information during phase two (filling)"
+let help_p0 = " Process until phase zero (parsing)"
+let help_p1 = " Process until phase one (sizing)"
+let help_p2 = " Process until phase two (filling)"
+
+let hook = "xhtbl"
+
+let includes, tables = ref [], ref []
+
+let process_directive och bname (name, table, css, uri, ext) =
+   tables := name :: !tables;
+   if !O.d0 then TU.debug table;
+   if not !O.p0 then begin
+      let size = P1.process table in
+      if !O.d1 then TU.debug table;
+      if not !O.p1 then begin
+         let matrix = M.make size in
+         let _ = P2.process table matrix in
+         if !O.d2 then TU.debug table;
+         if not !O.p2 then P3.process css uri ext matrix;
+        let name = if name = "" then bname else name in
+         XU.output och name matrix
+      end
+   end
+
+let process_file fname =
+   let bname = F.chop_extension (F.basename fname) in
+   let ich = open_in fname in
+   let lexbuf = Lexing.from_channel ich in
+   let ns, ds = TP.script TL.token lexbuf in
+   close_in ich; includes := bname :: !includes;
+   let ns = ("", "http://www.w3.org/1999/xhtml") :: ns in
+   let och = XU.open_out bname ns in 
+   L.iter (process_directive och bname) ds;
+   XU.close_out och
+
+let main () =
+   A.parse [
+      "-L", A.Set O.debug_lexer, help_L;
+      "-O", A.String ((:=) O.output_dir), help_O; 
+      "-X", A.Unit O.clear, help_X;
+      "-b", A.String ((:=) O.baseuri), help_b;
+      "-d0", A.Set O.d0, help_d0;  
+      "-d1", A.Set O.d1, help_d1;  
+      "-d2", A.Set O.d2, help_d2;  
+      "-e1", A.Set O.e1, help_e1;  
+      "-e2", A.Set O.e2, help_e2;  
+      "-p0", A.Set O.p0, help_p0;  
+      "-p1", A.Set O.p1, help_p1;  
+      "-p2", A.Set O.p2, help_p2;  
+   ] process_file help;
+   XU.write_hook hook !includes !tables 
+
+let _ = main ()

diff --git a/matita/matita/contribs/lambdadelta/bin/xhtbl/xmlUnparser.ml b/matita/matita/contribs/lambdadelta/bin/xhtbl/xmlUnparser.ml
new file mode 100644 (file)
index 0000000..2f29e4b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/bin/xhtbl/xmlUnparser.ml
@@ -0,0 +1,104 @@
+module A = Array
+module F = Filename
+module L = List
+module P = Printf
+module S = String
+
+module O = Options
+module T = Table
+module M = Matrix
+
+let xhtbl = "xhtbl"
+
+let i = 0
+
+let myself = F.basename (Sys.argv.(0))
+
+let msg = P.sprintf "This file was generated by %s, do not edit" myself
+
+let compose uri ext =
+   if uri.[pred (S.length uri)] = '/' then uri else
+   try
+      let i = S.index uri '#' in
+      let uri, fragment = S.sub uri 0 i, S.sub uri i (S.length uri - i) in
+      uri ^ ext ^ fragment
+   with Not_found -> uri ^ ext
+
+let border cell =
+   let str = S.make 4 'n' in
+   if cell.M.cb.T.n then str.[0] <- 's';   
+   if cell.M.cb.T.e then str.[1] <- 's';
+   if cell.M.cb.T.s then str.[2] <- 's';
+   if cell.M.cb.T.w then str.[3] <- 's';
+   str :: cell.M.cc
+
+let text baseuri ext = function
+   | T.Plain s              -> s
+   | T.Link (true, uri, s)  -> P.sprintf "<a href=\"%s\">%s</a>" uri s
+   | T.Link (false, uri, s) -> 
+      let uri = !O.baseuri ^ baseuri ^ compose uri ext in
+      P.sprintf "<a href=\"%s\">%s</a>" uri s
+
+let name cell =
+   if cell.M.cn = "" then "" else P.sprintf " id=\"%s\"" cell.M.cn
+
+let key cell =
+   if cell.M.ck = [] then "<br/>" else S.concat "" (L.map (text cell.M.cu cell.M.cx) cell.M.ck)
+
+let ind i = S.make (2 * i) ' '
+
+let out_cell och cell =
+   let cc = xhtbl :: border cell in
+   P.fprintf och "%s<td class=\"%s\"%s>%s</td>\n"
+      (ind (i+3)) (S.concat " " cc) (name cell) (key cell)
+
+let out_row och row =
+   P.fprintf och "%s<tr class=\"%s\">\n" (ind (i+2)) xhtbl;
+   A.iter (out_cell och) row;
+   P.fprintf och "%s</tr>\n" (ind (i+2))
+
+let out_space och (name, uri) =
+   let name = if name = "" then name else ":" ^ name in
+   P.fprintf och "                xmlns%s=\"%s\"\n" name uri
+
+(****************************************************************************)
+
+let open_out name spaces =
+   let fname = F.concat !O.output_dir (P.sprintf "%s.xsl" name) in
+   let spaces = ("xsl", "http://www.w3.org/1999/XSL/Transform") :: spaces in
+   let och = open_out fname in
+   P.fprintf och "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n\n";
+   P.fprintf och "<!-- %s -->\n\n" msg;   
+   P.fprintf och "<xsl:stylesheet version=\"1.0\"\n";
+   L.iter (out_space och) spaces;
+   P.fprintf och ">\n\n";
+   och
+
+let output och name matrix =
+   P.fprintf och "<xsl:template name=\"%s\">\n" name;
+   P.fprintf och "%s<table class=\"%s\" cellpadding=\"4\" cellspacing=\"0\">\n" (ind (i+1)) xhtbl;
+   A.iter (out_row och) matrix.M.m;
+   P.fprintf och "%s</table>\n" (ind (i+1));
+   P.fprintf och "</xsl:template>\n\n"
+
+let close_out och =
+   P.fprintf och "</xsl:stylesheet>\n";
+   close_out och
+
+let map_incs och name =
+   P.fprintf och "<xsl:include href=\"%s.xsl\"/>\n" name
+
+let map_tbls och name =
+   P.fprintf och "%s<xsl:when test=\"@name='%s'\">\n" (ind (i+2)) name;
+   P.fprintf och "%s<xsl:call-template name=\"%s\"/>\n" (ind (i+3)) name;
+   P.fprintf och "%s</xsl:when>\n" (ind (i+2))
+
+let write_hook name incs tbls =
+   let och = open_out name [] in
+   L.iter (map_incs och) incs;
+   P.fprintf och "\n<xsl:template name=\"%s\">\n" name;
+   P.fprintf och "%s<xsl:choose>\n" (ind (i+1));
+   L.iter (map_tbls och) tbls;   
+   P.fprintf och "%s</xsl:choose>\n" (ind (i+1));
+   P.fprintf och "</xsl:template>\n\n";
+   close_out och

diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma
deleted file mode 100644 (file)
index 3b8f865..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/blt/defs.ma
+++ /dev/null
@@ -1,22 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/preamble.ma".
-
-rec definition blt (m: nat) (n: nat) on n: bool \def match n with [O 
-\Rightarrow false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S 
-m0) \Rightarrow (blt m0 n0)])].
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma b/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
deleted file mode 100644 (file)
index 7a6c3f2..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/blt/props.ma
+++ /dev/null
@@ -1,90 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/blt/defs.ma".
-
-lemma lt_blt:
- \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
-(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 
-\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 
-\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False | 
-(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) 
-H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def 
-(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt 
-y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: 
-nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n) 
-true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
-(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool 
-true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 
-with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda 
-(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
-
-lemma le_bge:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
-(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O 
-y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: 
-nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind 
-(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) 
-(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow 
-(\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: 
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
-in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow 
-(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
-in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in 
-(H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to 
-(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 
-(le_S_n n n0 H1))))) y)))) x).
-
-lemma blt_lt:
- \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
-y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) 
-true)).(let H0 \def (match H with [refl_equal \Rightarrow (\lambda (H0: (eq 
-bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: 
-bool).(match e with [true \Rightarrow False | false \Rightarrow True])) I 
-true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true))))) 
-(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true) 
-\to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt 
-n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n 
-(S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: 
-nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m) 
-\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool 
-(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
-
-lemma bge_le:
- \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
-y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) 
-false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq 
-bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda 
-(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: 
-(eq bool (blt O (S n)) false)).(let H1 \def (match H0 with [refl_equal 
-\Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind 
-bool (blt O (S n)) (\lambda (e: bool).(match e with [true \Rightarrow True | 
-false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in 
-(H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool 
-(blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) 
-(S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 
-H1))))))) y)))) x).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/definitions.ma b/matita/matita/contribs/lambdadelta/ground_1/definitions.ma
deleted file mode 100644 (file)
index 81a1e38..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/definitions.ma
+++ /dev/null
@@ -1,22 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/types/defs.ma".
-
-include "ground_1/blt/defs.ma".
-
-include "ground_1/plist/defs.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
deleted file mode 100644 (file)
index 724a347..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/ext/arith.ma
+++ /dev/null
@@ -1,592 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/preamble.ma".
-
-lemma nat_dec:
- \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to 
-(\forall (P: Prop).P))))
-\def
- \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat 
-n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: 
-nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall 
-(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: 
-Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) 
-((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq 
-nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S 
-n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match 
-ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H0) in 
-(False_ind P H1))))))) n2)) (\lambda (n: nat).(\lambda (H: ((\forall (n2: 
-nat).(or (eq nat n n2) ((eq nat n n2) \to (\forall (P: Prop).P)))))).(\lambda 
-(n2: nat).(nat_ind (\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) 
-n0) \to (\forall (P: Prop).P)))) (or_intror (eq nat (S n) O) ((eq nat (S n) 
-O) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat (S n) O)).(\lambda (P: 
-Prop).(let H1 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) 
-(\lambda (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to 
-(\forall (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall 
-(P: Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall 
-(P: Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 
-(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: 
-Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S 
-n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat 
-(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat 
-(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: 
-Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to 
-(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: 
-Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e with [O 
-\Rightarrow n | (S n3) \Rightarrow n3])) (S n) (S n0) H2) in (let H4 \def 
-(eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) \to (\forall (P0: 
-Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or 
-(eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3) 
-in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
-
-lemma simpl_plus_r:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) 
-(plus p n)) \to (eq nat m p))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat 
-(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda 
-(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: 
-nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m) 
-(plus_sym n m)))))).
-
-lemma minus_Sx_Sy:
- \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
-\def
- \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
-
-lemma minus_plus_r:
- \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
-\def
- \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: 
-nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
-
-lemma plus_permute_2_in_3:
- \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x 
-y) z) (plus (plus x z) y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x 
-(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat 
-(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind 
-nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) 
-(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z 
-y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
-
-lemma plus_permute_2_in_3_assoc:
- \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n 
-h) k) (plus n (plus k h)))))
-\def
- \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus 
-(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r 
-nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) 
-(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k 
-h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
-
-lemma plus_O:
- \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat 
-x O) (eq nat y O))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus 
-n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda 
-(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) 
-H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) 
-\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq 
-nat (plus (S n) y) O)).(let H1 \def (match H0 with [refl_equal \Rightarrow 
-(\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n) 
-y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in 
-(H1 (refl_equal nat O))))))) x).
-
-lemma minus_Sx_SO:
- \forall (x: nat).(eq nat (minus (S x) (S O)) x)
-\def
- \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal 
-nat x) (minus x O) (minus_n_O x)).
-
-lemma nat_dec_neg:
- \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq 
-nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or 
-(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) 
-(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq 
-nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) 
-(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq 
-nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat 
-(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S 
-n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or 
-(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq 
-nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda 
-(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S 
-n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not 
-(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H 
-n0)))) j)))) i).
-
-lemma neq_eq_e:
- \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) 
-\to P)) \to ((((eq nat i j) \to P)) \to P))))
-\def
- \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not 
-(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def 
-(nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
-
-lemma le_false:
- \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S 
-n) m) \to P))))
-\def
- \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: 
-Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: 
-Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match 
-H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def 
-(eq_ind nat (S n) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m0 H1) 
-\Rightarrow (\lambda (H2: (eq nat (S m0) O)).((let H3 \def (eq_ind nat (S m0) 
-(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 
-(refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: 
-nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda 
-(n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to 
-((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) 
-O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 with [le_n 
-\Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) 
-(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow (\lambda 
-(H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda (e: 
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) 
-in (False_ind ((le (S n) m0) \to P) H4)) H2))]) in (H2 (refl_equal nat 
-O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1) 
-\to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n) 
-(S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) 
-(le_S_n (S n1) n H2))))))) n0)))) m).
-
-lemma le_Sx_x:
- \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
-\def
- \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def 
-le_Sn_n in (False_ind P (H0 x H))))).
-
-lemma le_n_pred:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
-(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: 
-nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans 
-(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
-
-lemma minus_le:
- \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n 
-y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H: 
-((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda 
-(n0: nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: 
-nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow 
-(minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
-
-lemma le_plus_minus_sym:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) 
-n))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat 
-(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) 
-(plus (minus m n) n) (plus_sym (minus m n) n)))).
-
-lemma le_minus_minus:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) 
-\to (le (minus y x) (minus z x))))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: 
-nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) 
-(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat 
-z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z 
-(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
-
-lemma le_minus_plus:
- \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat 
-(minus (plus x y) z) (plus (minus x z) y)))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to 
-(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) 
-(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H with [le_n 
-\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: 
-nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) 
-(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) 
-(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq 
-nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: 
-nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O 
-m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) 
-in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: 
-nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus 
-(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S 
-z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n 
-(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def 
-(match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 
-\def (eq_ind nat (S z0) (\lambda (e: nat).(match e with [O \Rightarrow False 
-| (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y) 
-(S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda 
-(H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
-in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus 
-(minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: 
-nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus 
-n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S 
-n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
-
-lemma le_minus:
- \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to 
-(le x (minus z y)))))
-\def
- \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus 
-x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z 
-y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x 
-y))))).
-
-lemma le_trans_plus_r:
- \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to 
-(le y z))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus 
-x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
-
-lemma lt_x_O:
- \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
-\def
- \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def 
-(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match 
-ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in 
-(False_ind P H0))))).
-
-lemma le_gen_S:
- \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: 
-nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
-\def
- \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def 
-(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat 
-(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) 
-(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S 
-m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x 
-H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat 
-(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq 
-nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) 
-m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: 
-nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) 
-x H1 H0))]) in (H0 (refl_equal nat x))))).
-
-lemma lt_x_plus_x_Sy:
- \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: 
-nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) 
-(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
-
-lemma simpl_lt_plus_r:
- \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m 
-p)) \to (lt n m))))
-\def
- \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus 
-n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) 
-(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let 
-H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 
-(plus p m) (plus_sym m p)) in H1)))))).
-
-lemma minus_x_Sy:
- \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S 
-(minus x (S y))))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
-(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: 
-(lt y O)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat 
-(S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O 
-\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat 
-(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda 
-(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: 
-nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
-in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y))))) 
-H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: 
-((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S 
-y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
-(eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S 
-n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat 
-(S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0 
-(S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda 
-(H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) 
-y)))) x).
-
-lemma lt_plus_minus:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus 
-y (S x)))))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S 
-x) y H))).
-
-lemma lt_plus_minus_r:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y 
-(S x)) x)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat 
-(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x 
-y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
-
-lemma minus_x_SO:
- \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
-\def
- \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: 
-nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal 
-nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
-
-lemma le_x_pred_y:
- \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
-\def
- \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to 
-(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def 
-(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def 
-(eq_ind nat (S x) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
-_) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0) 
-\Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) 
-(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
-True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 
-(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt 
-x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S 
-n))).(le_S_n x n H0))))) y).
-
-lemma lt_le_minus:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S 
-O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) 
-(plus_sym x (S O)))))).
-
-lemma lt_le_e:
- \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) 
-\to ((((le d n) \to P)) \to P))))
-\def
- \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n 
-d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in 
-(or_ind (le d n) (lt n d) P H0 H H1)))))).
-
-lemma lt_eq_e:
- \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
-\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
-y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x 
-y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
-
-lemma lt_eq_gt_e:
- \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
-\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
-y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) 
-\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda 
-(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
-
-lemma lt_gen_xS:
- \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 
-nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S 
-n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) 
-(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S 
-n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) 
-(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda 
-(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda 
-(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: 
-nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat 
-(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) 
-(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt 
-m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
-
-lemma le_lt_false:
- \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: 
-Prop).P))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt 
-y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
-
-lemma lt_neq:
- \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
-\def
- \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq 
-nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in 
-(lt_n_n y H1))))).
-
-lemma arith0:
- \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) 
-\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
-\def
- \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le 
-(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) 
-h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 
-(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat 
-(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat 
-(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus 
-(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) 
-(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 
-d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) 
-(minus_plus h2 (plus d2 h1))))))).
-
-lemma O_minus:
- \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
-\def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
-(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O 
-y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: 
-nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind 
-(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
-x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) 
-O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 
-n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S 
-x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
-H1) in (False_ind (eq nat (S x0) O) H3))))) (le_gen_S x0 O H0))) (\lambda (n: 
-nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
-x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S 
-n))).(H n (le_S_n x0 n H1))))) y)))) x).
-
-lemma minus_minus:
- \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) 
-\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: 
-nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq 
-nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O 
-x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y 
-O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n 
-(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) 
-(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda 
-(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to 
-((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x 
-y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le 
-(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) 
-\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda 
-(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S 
-z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le 
-z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S 
-x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x0) 
-H2) in (False_ind (eq nat O y) H4))))) (le_gen_S z0 O H)))))) (\lambda (x0: 
-nat).(\lambda (_: ((\forall (y: nat).((le (S z0) x0) \to ((le (S z0) y) \to 
-((eq nat (minus x0 (S z0)) (minus y (S z0))) \to (eq nat x0 y))))))).(\lambda 
-(y: nat).(nat_ind (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S z0) n) 
-\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (eq nat (S x0) 
-n))))) (\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda 
-(_: (eq nat (minus (S x0) (S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0 
-x0 H) in (ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: 
-nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq nat O 
-(S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda (ee: 
-nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
-H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) (\lambda 
-(y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat 
-(minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda 
-(H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq 
-nat (minus (S x0) (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 
-(IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
-
-lemma plus_plus:
- \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: 
-nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z 
-x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
-\def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: 
-nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq 
-nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) 
-(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: 
-nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 
-O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat 
-(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O 
-(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def 
-(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus 
-O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) 
-(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: 
-nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to 
-((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus 
-x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: 
-nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) 
-\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S 
-z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: 
-nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O 
-(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus 
-(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda 
-(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O 
-(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y 
-\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
-nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq 
-nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) 
-in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) 
-H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: 
-nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus 
-(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) 
-\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: 
-nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda 
-(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O 
-x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
-nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
-y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 
-(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: 
-nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
-(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) 
-(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda 
-(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
-z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) 
-(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) 
-x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: 
-nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat 
-(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 
-y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: 
-nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S 
-z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) 
-\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda 
-(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S 
-z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let 
-H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) 
-(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) 
-y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O 
-z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 
-z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus 
-x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def 
-(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) 
-\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) 
-H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) 
-H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall 
-(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus 
-z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow 
-(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 
-y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S 
-z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 
-x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 
-y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) 
-x1)))) z).
-
-lemma le_S_minus:
- \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to 
-(le d (S (minus n h))))))
-\def
- \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus 
-d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 
-\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) 
-(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S 
-d (minus n h) (le_minus d n h H))))))).
-
-lemma lt_x_pred_y:
- \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
-\def
- \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred 
-n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) 
-(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda 
-(H0: (lt x n)).(lt_n_S x n H0)))) y)).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma b/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma
deleted file mode 100644 (file)
index c2dff18..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/ext/tactics.ma
+++ /dev/null
@@ -1,41 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/preamble.ma".
-
-lemma insert_eq:
- \forall (S: Type[0]).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall 
-(G: ((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) 
-\to ((P x) \to (G x))))))
-\def
- \lambda (S: Type[0]).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda 
-(G: ((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to 
-(G y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
-
-lemma unintro:
- \forall (A: Type[0]).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall 
-(x: A).(P x))) \to (P a))))
-\def
- \lambda (A: Type[0]).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda 
-(H: ((\forall (x: A).(P x)))).(H a)))).
-
-lemma xinduction:
- \forall (A: Type[0]).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall 
-(x: A).((eq A t x) \to (P x)))) \to (P t))))
-\def
- \lambda (A: Type[0]).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda 
-(H: ((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma
deleted file mode 100644 (file)
index 13a7bd0..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/plist/defs.ma
+++ /dev/null
@@ -1,34 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/preamble.ma".
-
-inductive PList: Type[0] \def
-| PNil: PList
-| PCons: nat \to (nat \to (PList \to PList)).
-
-rec definition PConsTail (hds: PList) on hds: nat \to (nat \to PList) \def 
-\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow (PCons 
-h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 h0 
-d0))])).
-
-rec definition Ss (hds: PList) on hds: PList \def match hds with [PNil 
-\Rightarrow PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))].
-
-rec definition papp (a: PList) on a: PList \to PList \def \lambda (b: 
-PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons 
-h d (papp a0 b))]).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma b/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma
deleted file mode 100644 (file)
index f80d0b0..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/plist/props.ma
+++ /dev/null
@@ -1,31 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/plist/defs.ma".
-
-lemma papp_ss:
- \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss 
-is2)) (Ss (papp is1 is2))))
-\def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
-PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: 
-PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp 
-(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList 
-(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n 
-(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p 
-is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/preamble.ma b/matita/matita/contribs/lambdadelta/ground_1/preamble.ma
deleted file mode 100644 (file)
index b19f254..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/preamble.ma
+++ /dev/null
@@ -1,15 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "legacy_1/theory.ma".

diff --git a/matita/matita/contribs/lambdadelta/ground_1/spare.ma b/matita/matita/contribs/lambdadelta/ground_1/spare.ma
deleted file mode 100644 (file)
index e3cba9b..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/spare.ma
+++ /dev/null
@@ -1,18 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/theory.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/theory.ma b/matita/matita/contribs/lambdadelta/ground_1/theory.ma
deleted file mode 100644 (file)
index 2f70654..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/theory.ma
+++ /dev/null
@@ -1,28 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/ext/tactics.ma".
-
-include "ground_1/types/fwd.ma".
-
-include "ground_1/types/props.ma".
-
-include "ground_1/ext/arith.ma".
-
-include "ground_1/blt/props.ma".
-
-include "ground_1/plist/props.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma b/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma
deleted file mode 100644 (file)
index 3a7dadb..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/types/defs.ma
+++ /dev/null
@@ -1,174 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/preamble.ma".
-
-inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
-| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
-
-inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
-| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
-
-inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop 
-\def
-| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 
-P4))))).
-
-inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
-| or3_intro0: P0 \to (or3 P0 P1 P2)
-| or3_intro1: P1 \to (or3 P0 P1 P2)
-| or3_intro2: P2 \to (or3 P0 P1 P2).
-
-inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
-| or4_intro0: P0 \to (or4 P0 P1 P2 P3)
-| or4_intro1: P1 \to (or4 P0 P1 P2 P3)
-| or4_intro2: P2 \to (or4 P0 P1 P2 P3)
-| or4_intro3: P3 \to (or4 P0 P1 P2 P3).
-
-inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop 
-\def
-| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4)
-| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4).
-
-inductive ex3 (A0: Type[0]) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to 
-Prop): Prop \def
-| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 
-P0 P1 P2)))).
-
-inductive ex4 (A0: Type[0]) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to 
-Prop) (P3: A0 \to Prop): Prop \def
-| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) 
-\to (ex4 A0 P0 P1 P2 P3))))).
-
-inductive ex_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)): Prop 
-\def
-| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 
-P0))).
-
-inductive ex2_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
-A0 \to (A1 \to Prop)): Prop \def
-| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
-\to (ex2_2 A0 A1 P0 P1)))).
-
-inductive ex3_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
-A0 \to (A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def
-| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
-\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))).
-
-inductive ex4_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
-A0 \to (A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): 
-Prop \def
-| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
-\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))).
-
-inductive ex_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
-(A2 \to Prop))): Prop \def
-| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 
-x2) \to (ex_3 A0 A1 A2 P0)))).
-
-inductive ex2_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
-(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def
-| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
-x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))).
-
-inductive ex3_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
-(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
-\to Prop))): Prop \def
-| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 
-P2)))))).
-
-inductive ex4_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
-(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
-\to Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def
-| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 
-A1 A2 P0 P1 P2 P3))))))).
-
-inductive ex5_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
-(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
-\to Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to 
-Prop))): Prop \def
-| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
-x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 
-x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))).
-
-inductive ex3_4 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (P0: 
-A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))) (P2: A0 \to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def
-| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
-(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))).
-
-inductive ex4_4 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (P0: 
-A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))) (P2: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 
-\to (A3 \to Prop)))): Prop \def
-| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
-((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))).
-
-inductive ex4_5 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
-Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to 
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop 
-\def
-| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
-((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 
-P2 P3))))))))).
-
-inductive ex5_5 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
-Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to 
-(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: 
-A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def
-| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
-((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to 
-(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))).
-
-inductive ex6_6 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
-Type[0]) (A5: Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
-Prop)))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) 
-(P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to 
-(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 
-\to (A3 \to (A4 \to (A5 \to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to (A5 \to Prop)))))): Prop \def
-| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 
-x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) 
-\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 
-A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))).
-
-inductive ex6_7 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
-Type[0]) (A5: Type[0]) (A6: Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to (A5 \to (A6 \to Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-(A5 \to (A6 \to Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
-\to (A6 \to Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
-(A6 \to Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
-\to Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
-Prop))))))): Prop \def
-| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
-(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 
-x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) 
-\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 
-x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 
-P5))))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma b/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma
deleted file mode 100644 (file)
index 7cb7c31..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/types/fwd.ma
+++ /dev/null
@@ -1,419 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/types/defs.ma".
-
-implied lemma and3_rect:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
-Type[0]).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
-Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to P))))).(\lambda (a: (and3 P0 
-P1 P2)).(match a with [(and3_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
-
-implied lemma and3_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
-Prop).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
-Prop).(and3_rect P0 P1 P2 P)))).
-
-implied lemma and4_rect:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to 
-((and4 P0 P1 P2 P3) \to P))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to (P3 \to 
-P)))))).(\lambda (a: (and4 P0 P1 P2 P3)).(match a with [(and4_intro x x0 x1 
-x2) \Rightarrow (f x x0 x1 x2)]))))))).
-
-implied lemma and4_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to ((and4 
-P0 P1 P2 P3) \to P))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P: Prop).(and4_rect P0 P1 P2 P3 P))))).
-
-implied lemma and5_rect:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P4: Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 
-\to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P4: Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to 
-(P2 \to (P3 \to (P4 \to P))))))).(\lambda (a: (and5 P0 P1 P2 P3 P4)).(match a 
-with [(and5_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
-
-implied lemma and5_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 
-\to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P4: Prop).(\lambda (P: Prop).(and5_rect P0 P1 P2 P3 P4 
-P)))))).
-
-implied lemma or3_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
-Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) \to ((or3 P0 P1 P2) 
-\to P)))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
-Prop).(\lambda (f: ((P0 \to P))).(\lambda (f0: ((P1 \to P))).(\lambda (f1: 
-((P2 \to P))).(\lambda (o: (or3 P0 P1 P2)).(match o with [(or3_intro0 x) 
-\Rightarrow (f x) | (or3_intro1 x) \Rightarrow (f0 x) | (or3_intro2 x) 
-\Rightarrow (f1 x)])))))))).
-
-implied lemma or4_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) 
-\to (((P3 \to P)) \to ((or4 P0 P1 P2 P3) \to P)))))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P: Prop).(\lambda (f: ((P0 \to P))).(\lambda (f0: ((P1 \to 
-P))).(\lambda (f1: ((P2 \to P))).(\lambda (f2: ((P3 \to P))).(\lambda (o: 
-(or4 P0 P1 P2 P3)).(match o with [(or4_intro0 x) \Rightarrow (f x) | 
-(or4_intro1 x) \Rightarrow (f0 x) | (or4_intro2 x) \Rightarrow (f1 x) | 
-(or4_intro3 x) \Rightarrow (f2 x)])))))))))).
-
-implied lemma or5_ind:
- \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
-Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) 
-\to (((P2 \to P)) \to (((P3 \to P)) \to (((P4 \to P)) \to ((or5 P0 P1 P2 P3 
-P4) \to P)))))))))))
-\def
- \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
-Prop).(\lambda (P4: Prop).(\lambda (P: Prop).(\lambda (f: ((P0 \to 
-P))).(\lambda (f0: ((P1 \to P))).(\lambda (f1: ((P2 \to P))).(\lambda (f2: 
-((P3 \to P))).(\lambda (f3: ((P4 \to P))).(\lambda (o: (or5 P0 P1 P2 P3 
-P4)).(match o with [(or5_intro0 x) \Rightarrow (f x) | (or5_intro1 x) 
-\Rightarrow (f0 x) | (or5_intro2 x) \Rightarrow (f1 x) | (or5_intro3 x) 
-\Rightarrow (f2 x) | (or5_intro4 x) \Rightarrow (f3 x)])))))))))))).
-
-implied lemma ex3_ind:
- \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
-Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P: Prop).(((\forall (x0: 
-A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to P))))) \to ((ex3 A0 P0 P1 P2) \to 
-P))))))
-\def
- \lambda (A0: Type[0]).(\lambda (P0: ((A0 \to Prop))).(\lambda (P1: ((A0 \to 
-Prop))).(\lambda (P2: ((A0 \to Prop))).(\lambda (P: Prop).(\lambda (f: 
-((\forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to P)))))).(\lambda 
-(e: (ex3 A0 P0 P1 P2)).(match e with [(ex3_intro x x0 x1 x2) \Rightarrow (f x 
-x0 x1 x2)]))))))).
-
-implied lemma ex4_ind:
- \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
-Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P3: ((A0 \to 
-Prop))).(\forall (P: Prop).(((\forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 
-x0) \to ((P3 x0) \to P)))))) \to ((ex4 A0 P0 P1 P2 P3) \to P)))))))
-\def
- \lambda (A0: Type[0]).(\lambda (P0: ((A0 \to Prop))).(\lambda (P1: ((A0 \to 
-Prop))).(\lambda (P2: ((A0 \to Prop))).(\lambda (P3: ((A0 \to 
-Prop))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).((P0 x0) \to ((P1 
-x0) \to ((P2 x0) \to ((P3 x0) \to P))))))).(\lambda (e: (ex4 A0 P0 P1 P2 
-P3)).(match e with [(ex4_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 
-x3)])))))))).
-
-implied lemma ex_2_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
-Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) 
-\to P)))) \to ((ex_2 A0 A1 P0) \to P)))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
-Prop)))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
-A1).((P0 x0 x1) \to P))))).(\lambda (e: (ex_2 A0 A1 P0)).(match e with 
-[(ex_2_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
-
-implied lemma ex2_2_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
-Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
-P))))) \to ((ex2_2 A0 A1 P0 P1) \to P))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
-Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P: Prop).(\lambda 
-(f: ((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
-P)))))).(\lambda (e: (ex2_2 A0 A1 P0 P1)).(match e with [(ex2_2_intro x x0 x1 
-x2) \Rightarrow (f x x0 x1 x2)]))))))).
-
-implied lemma ex3_2_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
-Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
-\to Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 
-x1) \to ((P1 x0 x1) \to ((P2 x0 x1) \to P)))))) \to ((ex3_2 A0 A1 P0 P1 P2) 
-\to P)))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
-Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P2: ((A0 \to (A1 
-\to Prop)))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
-A1).((P0 x0 x1) \to ((P1 x0 x1) \to ((P2 x0 x1) \to P))))))).(\lambda (e: 
-(ex3_2 A0 A1 P0 P1 P2)).(match e with [(ex3_2_intro x x0 x1 x2 x3) 
-\Rightarrow (f x x0 x1 x2 x3)])))))))).
-
-implied lemma ex4_2_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
-Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
-\to Prop)))).(\forall (P3: ((A0 \to (A1 \to Prop)))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
-((P2 x0 x1) \to ((P3 x0 x1) \to P))))))) \to ((ex4_2 A0 A1 P0 P1 P2 P3) \to 
-P))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
-Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P2: ((A0 \to (A1 
-\to Prop)))).(\lambda (P3: ((A0 \to (A1 \to Prop)))).(\lambda (P: 
-Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 
-x0 x1) \to ((P2 x0 x1) \to ((P3 x0 x1) \to P)))))))).(\lambda (e: (ex4_2 A0 
-A1 P0 P1 P2 P3)).(match e with [(ex4_2_intro x x0 x1 x2 x3 x4) \Rightarrow (f 
-x x0 x1 x2 x3 x4)]))))))))).
-
-implied lemma ex_3_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: Prop).(((\forall (x0: 
-A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to P))))) \to ((ex_3 
-A0 A1 A2 P0) \to P))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: Prop).(\lambda (f: 
-((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to 
-P)))))).(\lambda (e: (ex_3 A0 A1 A2 P0)).(match e with [(ex_3_intro x x0 x1 
-x2) \Rightarrow (f x x0 x1 x2)]))))))).
-
-implied lemma ex2_3_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: 
-A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to P)))))) \to 
-((ex2_3 A0 A1 A2 P0 P1) \to P)))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall 
-(x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to 
-P))))))).(\lambda (e: (ex2_3 A0 A1 A2 P0 P1)).(match e with [(ex2_3_intro x 
-x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
-
-implied lemma ex3_3_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) 
-\to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to P))))))) \to ((ex3_3 A0 A1 A2 P0 P1 
-P2) \to P))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: 
-Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
-A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to 
-P)))))))).(\lambda (e: (ex3_3 A0 A1 A2 P0 P1 P2)).(match e with [(ex3_3_intro 
-x x0 x1 x2 x3 x4) \Rightarrow (f x x0 x1 x2 x3 x4)]))))))))).
-
-implied lemma ex4_3_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
-((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: Prop).(((\forall (x0: 
-A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to 
-((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to P)))))))) \to ((ex4_3 A0 A1 A2 P0 P1 P2 
-P3) \to P)))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P3: 
-((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall 
-(x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 
-x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to P))))))))).(\lambda (e: (ex4_3 
-A0 A1 A2 P0 P1 P2 P3)).(match e with [(ex4_3_intro x x0 x1 x2 x3 x4 x5) 
-\Rightarrow (f x x0 x1 x2 x3 x4 x5)])))))))))).
-
-implied lemma ex5_3_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
-((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P4: ((A0 \to (A1 \to (A2 \to 
-Prop))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
-(x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 
-x2) \to ((P4 x0 x1 x2) \to P))))))))) \to ((ex5_3 A0 A1 A2 P0 P1 P2 P3 P4) 
-\to P))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
-\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P3: 
-((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P4: ((A0 \to (A1 \to (A2 \to 
-Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
-A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) 
-\to ((P3 x0 x1 x2) \to ((P4 x0 x1 x2) \to P)))))))))).(\lambda (e: (ex5_3 A0 
-A1 A2 P0 P1 P2 P3 P4)).(match e with [(ex5_3_intro x x0 x1 x2 x3 x4 x5 x6) 
-\Rightarrow (f x x0 x1 x2 x3 x4 x5 x6)]))))))))))).
-
-implied lemma ex3_4_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
-(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
-A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
-P)))))))) \to ((ex3_4 A0 A1 A2 A3 P0 P1 P2) \to P)))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda 
-(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda (P: Prop).(\lambda 
-(f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
-A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
-P))))))))).(\lambda (e: (ex3_4 A0 A1 A2 A3 P0 P1 P2)).(match e with 
-[(ex3_4_intro x x0 x1 x2 x3 x4 x5) \Rightarrow (f x x0 x1 x2 x3 x4 
-x5)])))))))))).
-
-implied lemma ex4_4_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
-(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall (P3: ((A0 \to (A1 
-\to (A2 \to (A3 \to Prop)))))).(\forall (P: Prop).(((\forall (x0: 
-A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: A3).((P0 x0 x1 x2 x3) 
-\to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to ((P3 x0 x1 x2 x3) \to 
-P))))))))) \to ((ex4_4 A0 A1 A2 A3 P0 P1 P2 P3) \to P))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
-Prop)))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda 
-(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda (P3: ((A0 \to (A1 
-\to (A2 \to (A3 \to Prop)))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: 
-A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: A3).((P0 x0 x1 x2 x3) 
-\to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to ((P3 x0 x1 x2 x3) \to 
-P)))))))))).(\lambda (e: (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)).(match e with 
-[(ex4_4_intro x x0 x1 x2 x3 x4 x5 x6) \Rightarrow (f x x0 x1 x2 x3 x4 x5 
-x6)]))))))))))).
-
-implied lemma ex4_5_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
-(A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
-(x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 
-x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to 
-P)))))))))) \to ((ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3) \to P)))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to 
-(A3 \to (A4 \to Prop))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
-A1).(\forall (x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 
-x4) \to ((P1 x0 x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 
-x4) \to P))))))))))).(\lambda (e: (ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3)).(match 
-e with [(ex4_5_intro x x0 x1 x2 x3 x4 x5 x6 x7) \Rightarrow (f x x0 x1 x2 x3 
-x4 x5 x6 x7)])))))))))))).
-
-implied lemma ex5_5_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
-(A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\forall (P4: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
-(x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 
-x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 
-x2 x3 x4) \to P))))))))))) \to ((ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4) \to 
-P))))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to 
-(A3 \to (A4 \to Prop))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
-(A4 \to Prop))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\lambda (P4: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
-Prop))))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
-A1).(\forall (x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 
-x4) \to ((P1 x0 x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 
-x4) \to ((P4 x0 x1 x2 x3 x4) \to P)))))))))))).(\lambda (e: (ex5_5 A0 A1 A2 
-A3 A4 P0 P1 P2 P3 P4)).(match e with [(ex5_5_intro x x0 x1 x2 x3 x4 x5 x6 x7 
-x8) \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))))))))).
-
-implied lemma ex6_6_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (P0: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P1: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P2: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P3: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P4: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P5: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
-A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 x0 
-x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) \to 
-((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to P))))))))))))) \to 
-((ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4 P5) \to P))))))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (A5: Type[0]).(\lambda (P0: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P1: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P2: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P3: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P4: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P5: 
-((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P: 
-Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
-A2).(\forall (x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 
-x5) \to ((P1 x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 
-x3 x4 x5) \to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to 
-P)))))))))))))).(\lambda (e: (ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4 
-P5)).(match e with [(ex6_6_intro x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) 
-\Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)]))))))))))))))).
-
-implied lemma ex6_7_ind:
- \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
-(A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (A6: 
-Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
-\to Prop))))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
-\to (A6 \to Prop))))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to (A5 \to (A6 \to Prop))))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 
-\to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P4: ((A0 \to (A1 \to (A2 
-\to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P5: ((A0 \to (A1 
-\to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P: 
-Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
-A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 x3 x4 
-x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) \to ((P3 
-x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 x2 x3 x4 
-x5 x6) \to P)))))))))))))) \to ((ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 
-P5) \to P)))))))))))))))
-\def
- \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
-(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (A5: Type[0]).(\lambda (A6: 
-Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
-\to Prop))))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
-\to (A6 \to Prop))))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 
-\to (A5 \to (A6 \to Prop))))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 
-\to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P4: ((A0 \to (A1 \to (A2 
-\to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P5: ((A0 \to (A1 
-\to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P: 
-Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
-A2).(\forall (x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: 
-A6).((P0 x0 x1 x2 x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 
-x2 x3 x4 x5 x6) \to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) 
-\to ((P5 x0 x1 x2 x3 x4 x5 x6) \to P))))))))))))))).(\lambda (e: (ex6_7 A0 A1 
-A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 P5)).(match e with [(ex6_7_intro x x0 x1 x2 x3 
-x4 x5 x6 x7 x8 x9 x10 x11) \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 
-x11)])))))))))))))))).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1/types/props.ma b/matita/matita/contribs/lambdadelta/ground_1/types/props.ma
deleted file mode 100644 (file)
index 79919dc..0000000
--- a/matita/matita/contribs/lambdadelta/ground_1/types/props.ma
+++ /dev/null
@@ -1,30 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "ground_1/types/defs.ma".
-
-lemma ex2_sym:
- \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
-Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A 
-(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))))))
-\def
- \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to 
-Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q 
-x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A 
-(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: 
-(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda 
-(x0: A).(P x0)) x H1 H0)))) H)))).
-

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/blt/defs.ma b/matita/matita/contribs/lambdadelta/ground_1A/blt/defs.ma
new file mode 100644 (file)
index 0000000..07fc374
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/blt/defs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/preamble.ma".
+
+rec definition blt (m: nat) (n: nat) on n: bool \def match n with [O 
+\Rightarrow false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S 
+m0) \Rightarrow (blt m0 n0)])].
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/blt/props.ma b/matita/matita/contribs/lambdadelta/ground_1A/blt/props.ma
new file mode 100644 (file)
index 0000000..f6094dc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/blt/props.ma
@@ -0,0 +1,90 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/blt/defs.ma".
+
+lemma lt_blt:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 
+\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 
+\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False | 
+(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) 
+H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def 
+(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt 
+y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: 
+nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n) 
+true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
+(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool 
+true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 
+with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda 
+(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
+
+lemma le_bge:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
+(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O 
+y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: 
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind 
+(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) 
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow 
+(\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
+in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow 
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
+in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in 
+(H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to 
+(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 
+(le_S_n n n0 H1))))) y)))) x).
+
+lemma blt_lt:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
+y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) 
+true)).(let H0 \def (match H with [refl_equal \Rightarrow (\lambda (H0: (eq 
+bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: 
+bool).(match e with [true \Rightarrow False | false \Rightarrow True])) I 
+true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true))))) 
+(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true) 
+\to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt 
+n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n 
+(S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: 
+nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m) 
+\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool 
+(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
+
+lemma bge_le:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt 
+y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) 
+false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq 
+bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda 
+(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: 
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 with [refl_equal 
+\Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind 
+bool (blt O (S n)) (\lambda (e: bool).(match e with [true \Rightarrow True | 
+false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in 
+(H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool 
+(blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) 
+(S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 
+H1))))))) y)))) x).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/definitions.ma b/matita/matita/contribs/lambdadelta/ground_1A/definitions.ma
new file mode 100644 (file)
index 0000000..639fcd2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/definitions.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/types/defs.ma".
+
+include "ground_1A/blt/defs.ma".
+
+include "ground_1A/plist/defs.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/ext/arith.ma b/matita/matita/contribs/lambdadelta/ground_1A/ext/arith.ma
new file mode 100644 (file)
index 0000000..7bbf6cc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/ext/arith.ma
@@ -0,0 +1,592 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/preamble.ma".
+
+lemma nat_dec:
+ \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to 
+(\forall (P: Prop).P))))
+\def
+ \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat 
+n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: 
+nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall 
+(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: 
+Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) 
+((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq 
+nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S 
+n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match 
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H0) in 
+(False_ind P H1))))))) n2)) (\lambda (n: nat).(\lambda (H: ((\forall (n2: 
+nat).(or (eq nat n n2) ((eq nat n n2) \to (\forall (P: Prop).P)))))).(\lambda 
+(n2: nat).(nat_ind (\lambda (n0: nat).(or (eq nat (S n) n0) ((eq nat (S n) 
+n0) \to (\forall (P: Prop).P)))) (or_intror (eq nat (S n) O) ((eq nat (S n) 
+O) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat (S n) O)).(\lambda (P: 
+Prop).(let H1 \def (eq_ind nat (S n) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) 
+(\lambda (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to 
+(\forall (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall 
+(P: Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall 
+(P: Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 
+(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: 
+Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S 
+n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat 
+(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat 
+(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: 
+Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to 
+(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: 
+Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e with [O 
+\Rightarrow n | (S n3) \Rightarrow n3])) (S n) (S n0) H2) in (let H4 \def 
+(eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) \to (\forall (P0: 
+Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 (\lambda (n3: nat).(or 
+(eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: Prop).P0)))) H0 n H3) 
+in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) n1).
+
+lemma simpl_plus_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) 
+(plus p n)) \to (eq nat m p))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat 
+(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda 
+(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: 
+nat).(eq nat n0 (plus n p))) (plus_sym p n) (plus m n) H) (plus n m) 
+(plus_sym n m)))))).
+
+lemma minus_Sx_Sy:
+ \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
+
+lemma minus_plus_r:
+ \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
+\def
+ \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: 
+nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
+
+lemma plus_permute_2_in_3:
+ \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x 
+y) z) (plus (plus x z) y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x 
+(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat 
+(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind 
+nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) 
+(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z 
+y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
+
+lemma plus_permute_2_in_3_assoc:
+ \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n 
+h) k) (plus n (plus k h)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus 
+(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r 
+nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) 
+(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k 
+h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
+
+lemma plus_O:
+ \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat 
+x O) (eq nat y O))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus 
+n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda 
+(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) 
+H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) 
+\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq 
+nat (plus (S n) y) O)).(let H1 \def (match H0 with [refl_equal \Rightarrow 
+(\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n) 
+y) (\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in 
+(H1 (refl_equal nat O))))))) x).
+
+lemma minus_Sx_SO:
+ \forall (x: nat).(eq nat (minus (S x) (S O)) x)
+\def
+ \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal 
+nat x) (minus x O) (minus_n_O x)).
+
+lemma nat_dec_neg:
+ \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq 
+nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or 
+(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) 
+(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq 
+nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) 
+(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq 
+nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat 
+(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S 
+n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or 
+(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq 
+nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda 
+(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S 
+n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not 
+(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H 
+n0)))) j)))) i).
+
+lemma neq_eq_e:
+ \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) 
+\to P)) \to ((((eq nat i j) \to P)) \to P))))
+\def
+ \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not 
+(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def 
+(nat_dec_neg i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
+
+lemma le_false:
+ \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S 
+n) m) \to P))))
+\def
+ \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: 
+Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: 
+Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match 
+H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def 
+(eq_ind nat (S n) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m0 H1) 
+\Rightarrow (\lambda (H2: (eq nat (S m0) O)).((let H3 \def (eq_ind nat (S m0) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 
+(refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: 
+nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda 
+(n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to 
+((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) 
+O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 with [le_n 
+\Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow (\lambda 
+(H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) 
+in (False_ind ((le (S n) m0) \to P) H4)) H2))]) in (H2 (refl_equal nat 
+O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1) 
+\to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n) 
+(S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) 
+(le_S_n (S n1) n H2))))))) n0)))) m).
+
+lemma le_Sx_x:
+ \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
+\def
+ \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def 
+le_Sn_n in (False_ind P (H0 x H))))).
+
+lemma le_n_pred:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0: 
+nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans 
+(pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
+
+lemma minus_le:
+ \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n 
+y) n))) (\lambda (_: nat).(le_O_n O)) (\lambda (n: nat).(\lambda (H: 
+((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda 
+(n0: nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: 
+nat).(\lambda (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow 
+(minus n l)]) (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
+
+lemma le_plus_minus_sym:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) 
+n))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat 
+(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) 
+(plus (minus m n) n) (plus_sym (minus m n) n)))).
+
+lemma le_minus_minus:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) 
+\to (le (minus y x) (minus z x))))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: 
+nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x) 
+(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat 
+z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z 
+(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
+
+lemma le_minus_plus:
+ \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat 
+(minus (plus x y) z) (plus (minus x z) y)))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to 
+(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) 
+(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H with [le_n 
+\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: 
+nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) 
+(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) 
+(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq 
+nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: 
+nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O 
+m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) 
+in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: 
+nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus 
+(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S 
+z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n 
+(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def 
+(match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 
+\def (eq_ind nat (S z0) (\lambda (e: nat).(match e with [O \Rightarrow False 
+| (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y) 
+(S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda 
+(H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) 
+in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus 
+(minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: 
+nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus 
+n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S 
+n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
+
+lemma le_minus:
+ \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to 
+(le x (minus z y)))))
+\def
+ \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus 
+x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z 
+y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x 
+y))))).
+
+lemma le_trans_plus_r:
+ \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to 
+(le y z))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus 
+x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
+
+lemma lt_x_O:
+ \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
+\def
+ \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def 
+(le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match 
+ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x) H_y) in 
+(False_ind P H0))))).
+
+lemma le_gen_S:
+ \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: 
+nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
+\def
+ \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def 
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat 
+(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) 
+(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S 
+m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x 
+H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat 
+(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq 
+nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) 
+m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: 
+nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) 
+x H1 H0))]) in (H0 (refl_equal nat x))))).
+
+lemma lt_x_plus_x_Sy:
+ \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: 
+nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) 
+(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
+
+lemma simpl_lt_plus_r:
+ \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m 
+p)) \to (lt n m))))
+\def
+ \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus 
+n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p) 
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let 
+H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 
+(plus p m) (plus_sym m p)) in H1)))))).
+
+lemma minus_x_Sy:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S 
+(minus x (S y))))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to 
+(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: 
+(lt y O)).(let H0 \def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat 
+(S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat 
+(minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda 
+(H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: 
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) 
+in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y))))) 
+H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: 
+((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S 
+y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to 
+(eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S 
+n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat 
+(S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0 
+(S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda 
+(H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) 
+y)))) x).
+
+lemma lt_plus_minus:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus 
+y (S x)))))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S 
+x) y H))).
+
+lemma lt_plus_minus_r:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y 
+(S x)) x)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat 
+(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x 
+y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
+
+lemma minus_x_SO:
+ \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
+\def
+ \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: 
+nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal 
+nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
+
+lemma le_x_pred_y:
+ \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
+\def
+ \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to 
+(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def 
+(match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def 
+(eq_ind nat (S x) (\lambda (e: nat).(match e with [O \Rightarrow False | (S 
+_) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0) 
+\Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) 
+(\lambda (e: nat).(match e with [O \Rightarrow False | (S _) \Rightarrow 
+True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 
+(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt 
+x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S 
+n))).(le_S_n x n H0))))) y).
+
+lemma lt_le_minus:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S 
+O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) 
+(plus_sym x (S O)))))).
+
+lemma lt_le_e:
+ \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) 
+\to ((((le d n) \to P)) \to P))))
+\def
+ \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n 
+d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in 
+(or_ind (le d n) (lt n d) P H0 H H1)))))).
+
+lemma lt_eq_e:
+ \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
+\to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
+y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x 
+y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
+
+lemma lt_eq_gt_e:
+ \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) 
+\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x 
+y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) 
+\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda 
+(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
+
+lemma lt_gen_xS:
+ \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 
+nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S 
+n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) 
+(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S 
+n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) 
+(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda 
+(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda 
+(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: 
+nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat 
+(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) 
+(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt 
+m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
+
+lemma le_lt_false:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: 
+Prop).P))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt 
+y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
+
+lemma lt_neq:
+ \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq 
+nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in 
+(lt_n_n y H1))))).
+
+lemma arith0:
+ \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) 
+\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
+\def
+ \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le 
+(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) 
+h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 
+(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat 
+(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat 
+(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus 
+(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1) 
+(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2 
+d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1) 
+(minus_plus h2 (plus d2 h1))))))).
+
+lemma O_minus:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to 
+(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O 
+y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: 
+nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind 
+(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
+x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) 
+O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 
+n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S 
+x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
+H1) in (False_ind (eq nat (S x0) O) H3))))) (le_gen_S x0 O H0))) (\lambda (n: 
+nat).(\lambda (_: (((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S 
+x0) | (S l) \Rightarrow (minus x0 l)]) O)))).(\lambda (H1: (le (S x0) (S 
+n))).(H n (le_S_n x0 n H1))))) y)))) x).
+
+lemma minus_minus:
+ \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) 
+\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: 
+nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq 
+nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O 
+x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y 
+O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n 
+(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) 
+(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda 
+(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to 
+((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x 
+y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le 
+(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) 
+\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda 
+(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S 
+z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le 
+z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S 
+x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x0) 
+H2) in (False_ind (eq nat O y) H4))))) (le_gen_S z0 O H)))))) (\lambda (x0: 
+nat).(\lambda (_: ((\forall (y: nat).((le (S z0) x0) \to ((le (S z0) y) \to 
+((eq nat (minus x0 (S z0)) (minus y (S z0))) \to (eq nat x0 y))))))).(\lambda 
+(y: nat).(nat_ind (\lambda (n: nat).((le (S z0) (S x0)) \to ((le (S z0) n) 
+\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0))) \to (eq nat (S x0) 
+n))))) (\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) O)).(\lambda 
+(_: (eq nat (minus (S x0) (S z0)) (minus O (S z0)))).(let H_y \def (le_S_n z0 
+x0 H) in (ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: 
+nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq nat O 
+(S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda (ee: 
+nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) 
+H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0)))))) (\lambda 
+(y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat 
+(minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda 
+(H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq 
+nat (minus (S x0) (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 
+(IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
+
+lemma plus_plus:
+ \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: 
+nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z 
+x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
+\def
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq 
+nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) 
+(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: 
+nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 
+O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat 
+(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O 
+(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def 
+(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus 
+O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) 
+(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to 
+((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus 
+x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: 
+nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) 
+\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S 
+z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: 
+nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O 
+(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus 
+(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda 
+(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O 
+(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y 
+\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
+nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq 
+nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) 
+in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2) 
+H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: 
+nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus 
+(match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2)) 
+\to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: 
+nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda 
+(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O 
+x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S 
+y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 
+(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: 
+nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus 
+(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) 
+(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda 
+(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus 
+z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) 
+(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) 
+x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: 
+nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat 
+(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 
+y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: 
+nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S 
+z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) 
+\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda 
+(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S 
+z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let 
+H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) 
+(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) 
+y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O 
+z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 
+z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus 
+x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def 
+(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) 
+\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) 
+H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) 
+H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall 
+(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus 
+z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow 
+(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 
+y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S 
+z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 
+x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 
+y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) 
+x1)))) z).
+
+lemma le_S_minus:
+ \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to 
+(le d (S (minus n h))))))
+\def
+ \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus 
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 
+\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) 
+(le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S 
+d (minus n h) (le_minus d n h H))))))).
+
+lemma lt_x_pred_y:
+ \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
+\def
+ \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred 
+n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O))) 
+(\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda 
+(H0: (lt x n)).(lt_n_S x n H0)))) y)).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/ext/tactics.ma b/matita/matita/contribs/lambdadelta/ground_1A/ext/tactics.ma
new file mode 100644 (file)
index 0000000..06c0c39
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/ext/tactics.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/preamble.ma".
+
+lemma insert_eq:
+ \forall (S: Type[0]).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall 
+(G: ((S \to Prop))).(((\forall (y: S).((P y) \to ((eq S y x) \to (G y))))) 
+\to ((P x) \to (G x))))))
+\def
+ \lambda (S: Type[0]).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda 
+(G: ((S \to Prop))).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to 
+(G y)))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))).
+
+lemma unintro:
+ \forall (A: Type[0]).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall 
+(x: A).(P x))) \to (P a))))
+\def
+ \lambda (A: Type[0]).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda 
+(H: ((\forall (x: A).(P x)))).(H a)))).
+
+lemma xinduction:
+ \forall (A: Type[0]).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall 
+(x: A).((eq A t x) \to (P x)))) \to (P t))))
+\def
+ \lambda (A: Type[0]).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda 
+(H: ((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/plist/defs.ma b/matita/matita/contribs/lambdadelta/ground_1A/plist/defs.ma
new file mode 100644 (file)
index 0000000..a13d54f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/plist/defs.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/preamble.ma".
+
+inductive PList: Type[0] \def
+| PNil: PList
+| PCons: nat \to (nat \to (PList \to PList)).
+
+rec definition PConsTail (hds: PList) on hds: nat \to (nat \to PList) \def 
+\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow (PCons 
+h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 h0 
+d0))])).
+
+rec definition Ss (hds: PList) on hds: PList \def match hds with [PNil 
+\Rightarrow PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))].
+
+rec definition papp (a: PList) on a: PList \to PList \def \lambda (b: 
+PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons 
+h d (papp a0 b))]).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/plist/props.ma b/matita/matita/contribs/lambdadelta/ground_1A/plist/props.ma
new file mode 100644 (file)
index 0000000..9e9c7fe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/plist/props.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/plist/defs.ma".
+
+lemma papp_ss:
+ \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss 
+is2)) (Ss (papp is1 is2))))
+\def
+ \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
+PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: 
+PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: 
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp 
+(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList 
+(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n 
+(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p 
+is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/preamble.ma b/matita/matita/contribs/lambdadelta/ground_1A/preamble.ma
new file mode 100644 (file)
index 0000000..17a034f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/preamble.ma
@@ -0,0 +1,15 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "legacy_1A/theory.ma".

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/spare.ma b/matita/matita/contribs/lambdadelta/ground_1A/spare.ma
new file mode 100644 (file)
index 0000000..b45fecc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/spare.ma
@@ -0,0 +1,18 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/theory.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/theory.ma b/matita/matita/contribs/lambdadelta/ground_1A/theory.ma
new file mode 100644 (file)
index 0000000..c0c5a77
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/theory.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/ext/tactics.ma".
+
+include "ground_1A/types/fwd.ma".
+
+include "ground_1A/types/props.ma".
+
+include "ground_1A/ext/arith.ma".
+
+include "ground_1A/blt/props.ma".
+
+include "ground_1A/plist/props.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/types/defs.ma b/matita/matita/contribs/lambdadelta/ground_1A/types/defs.ma
new file mode 100644 (file)
index 0000000..b9984a6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/types/defs.ma
@@ -0,0 +1,174 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/preamble.ma".
+
+inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
+| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))).
+
+inductive and4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
+| and4_intro: P0 \to (P1 \to (P2 \to (P3 \to (and4 P0 P1 P2 P3)))).
+
+inductive and5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop 
+\def
+| and5_intro: P0 \to (P1 \to (P2 \to (P3 \to (P4 \to (and5 P0 P1 P2 P3 
+P4))))).
+
+inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def
+| or3_intro0: P0 \to (or3 P0 P1 P2)
+| or3_intro1: P1 \to (or3 P0 P1 P2)
+| or3_intro2: P2 \to (or3 P0 P1 P2).
+
+inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def
+| or4_intro0: P0 \to (or4 P0 P1 P2 P3)
+| or4_intro1: P1 \to (or4 P0 P1 P2 P3)
+| or4_intro2: P2 \to (or4 P0 P1 P2 P3)
+| or4_intro3: P3 \to (or4 P0 P1 P2 P3).
+
+inductive or5 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop) (P4: Prop): Prop 
+\def
+| or5_intro0: P0 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro1: P1 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro2: P2 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro3: P3 \to (or5 P0 P1 P2 P3 P4)
+| or5_intro4: P4 \to (or5 P0 P1 P2 P3 P4).
+
+inductive ex3 (A0: Type[0]) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to 
+Prop): Prop \def
+| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 
+P0 P1 P2)))).
+
+inductive ex4 (A0: Type[0]) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to 
+Prop) (P3: A0 \to Prop): Prop \def
+| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) 
+\to (ex4 A0 P0 P1 P2 P3))))).
+
+inductive ex_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)): Prop 
+\def
+| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 
+P0))).
+
+inductive ex2_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
+A0 \to (A1 \to Prop)): Prop \def
+| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
+\to (ex2_2 A0 A1 P0 P1)))).
+
+inductive ex3_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
+A0 \to (A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def
+| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
+\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))).
+
+inductive ex4_2 (A0: Type[0]) (A1: Type[0]) (P0: A0 \to (A1 \to Prop)) (P1: 
+A0 \to (A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): 
+Prop \def
+| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) 
+\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))).
+
+inductive ex_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
+(A2 \to Prop))): Prop \def
+| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 
+x2) \to (ex_3 A0 A1 A2 P0)))).
+
+inductive ex2_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
+(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def
+| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
+x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))).
+
+inductive ex3_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
+(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
+\to Prop))): Prop \def
+| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 
+P2)))))).
+
+inductive ex4_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
+(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
+\to Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def
+| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 
+A1 A2 P0 P1 P2 P3))))))).
+
+inductive ex5_3 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (P0: A0 \to (A1 \to 
+(A2 \to Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 
+\to Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))) (P4: A0 \to (A1 \to (A2 \to 
+Prop))): Prop \def
+| ex5_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 
+x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to ((P4 x0 
+x1 x2) \to (ex5_3 A0 A1 A2 P0 P1 P2 P3 P4)))))))).
+
+inductive ex3_4 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (P0: 
+A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))) (P2: A0 \to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def
+| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
+(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))).
+
+inductive ex4_4 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (P0: 
+A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))) (P2: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 
+\to (A3 \to Prop)))): Prop \def
+| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
+((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))).
+
+inductive ex4_5 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
+Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to 
+(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop 
+\def
+| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
+((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 
+P2 P3))))))))).
+
+inductive ex5_5 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
+Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to 
+(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: 
+A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def
+| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to 
+((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to 
+(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))).
+
+inductive ex6_6 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
+Type[0]) (A5: Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
+Prop)))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) 
+(P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to 
+(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 
+\to (A3 \to (A4 \to (A5 \to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to (A5 \to Prop)))))): Prop \def
+| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 
+x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) 
+\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 
+A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))).
+
+inductive ex6_7 (A0: Type[0]) (A1: Type[0]) (A2: Type[0]) (A3: Type[0]) (A4: 
+Type[0]) (A5: Type[0]) (A6: Type[0]) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 
+\to (A5 \to (A6 \to Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+(A5 \to (A6 \to Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
+\to (A6 \to Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to 
+(A6 \to Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
+\to Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to 
+Prop))))))): Prop \def
+| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall 
+(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 
+x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) 
+\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 
+x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 
+P5))))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/types/fwd.ma b/matita/matita/contribs/lambdadelta/ground_1A/types/fwd.ma
new file mode 100644 (file)
index 0000000..862e581
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/types/fwd.ma
@@ -0,0 +1,419 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/types/defs.ma".
+
+implied lemma and3_rect:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
+Type[0]).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
+Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to P))))).(\lambda (a: (and3 P0 
+P1 P2)).(match a with [(and3_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
+
+implied lemma and3_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
+Prop).(((P0 \to (P1 \to (P2 \to P)))) \to ((and3 P0 P1 P2) \to P)))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
+Prop).(and3_rect P0 P1 P2 P)))).
+
+implied lemma and4_rect:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to 
+((and4 P0 P1 P2 P3) \to P))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to (P2 \to (P3 \to 
+P)))))).(\lambda (a: (and4 P0 P1 P2 P3)).(match a with [(and4_intro x x0 x1 
+x2) \Rightarrow (f x x0 x1 x2)]))))))).
+
+implied lemma and4_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 \to P))))) \to ((and4 
+P0 P1 P2 P3) \to P))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P: Prop).(and4_rect P0 P1 P2 P3 P))))).
+
+implied lemma and5_rect:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P4: Prop).(\forall (P: Type[0]).(((P0 \to (P1 \to (P2 \to (P3 
+\to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P4: Prop).(\lambda (P: Type[0]).(\lambda (f: ((P0 \to (P1 \to 
+(P2 \to (P3 \to (P4 \to P))))))).(\lambda (a: (and5 P0 P1 P2 P3 P4)).(match a 
+with [(and5_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
+
+implied lemma and5_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to (P1 \to (P2 \to (P3 
+\to (P4 \to P)))))) \to ((and5 P0 P1 P2 P3 P4) \to P)))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P4: Prop).(\lambda (P: Prop).(and5_rect P0 P1 P2 P3 P4 
+P)))))).
+
+implied lemma or3_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P: 
+Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) \to ((or3 P0 P1 P2) 
+\to P)))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P: 
+Prop).(\lambda (f: ((P0 \to P))).(\lambda (f0: ((P1 \to P))).(\lambda (f1: 
+((P2 \to P))).(\lambda (o: (or3 P0 P1 P2)).(match o with [(or3_intro0 x) 
+\Rightarrow (f x) | (or3_intro1 x) \Rightarrow (f0 x) | (or3_intro2 x) 
+\Rightarrow (f1 x)])))))))).
+
+implied lemma or4_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) \to (((P2 \to P)) 
+\to (((P3 \to P)) \to ((or4 P0 P1 P2 P3) \to P)))))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P: Prop).(\lambda (f: ((P0 \to P))).(\lambda (f0: ((P1 \to 
+P))).(\lambda (f1: ((P2 \to P))).(\lambda (f2: ((P3 \to P))).(\lambda (o: 
+(or4 P0 P1 P2 P3)).(match o with [(or4_intro0 x) \Rightarrow (f x) | 
+(or4_intro1 x) \Rightarrow (f0 x) | (or4_intro2 x) \Rightarrow (f1 x) | 
+(or4_intro3 x) \Rightarrow (f2 x)])))))))))).
+
+implied lemma or5_ind:
+ \forall (P0: Prop).(\forall (P1: Prop).(\forall (P2: Prop).(\forall (P3: 
+Prop).(\forall (P4: Prop).(\forall (P: Prop).(((P0 \to P)) \to (((P1 \to P)) 
+\to (((P2 \to P)) \to (((P3 \to P)) \to (((P4 \to P)) \to ((or5 P0 P1 P2 P3 
+P4) \to P)))))))))))
+\def
+ \lambda (P0: Prop).(\lambda (P1: Prop).(\lambda (P2: Prop).(\lambda (P3: 
+Prop).(\lambda (P4: Prop).(\lambda (P: Prop).(\lambda (f: ((P0 \to 
+P))).(\lambda (f0: ((P1 \to P))).(\lambda (f1: ((P2 \to P))).(\lambda (f2: 
+((P3 \to P))).(\lambda (f3: ((P4 \to P))).(\lambda (o: (or5 P0 P1 P2 P3 
+P4)).(match o with [(or5_intro0 x) \Rightarrow (f x) | (or5_intro1 x) 
+\Rightarrow (f0 x) | (or5_intro2 x) \Rightarrow (f1 x) | (or5_intro3 x) 
+\Rightarrow (f2 x) | (or5_intro4 x) \Rightarrow (f3 x)])))))))))))).
+
+implied lemma ex3_ind:
+ \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
+Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P: Prop).(((\forall (x0: 
+A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to P))))) \to ((ex3 A0 P0 P1 P2) \to 
+P))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (P0: ((A0 \to Prop))).(\lambda (P1: ((A0 \to 
+Prop))).(\lambda (P2: ((A0 \to Prop))).(\lambda (P: Prop).(\lambda (f: 
+((\forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to P)))))).(\lambda 
+(e: (ex3 A0 P0 P1 P2)).(match e with [(ex3_intro x x0 x1 x2) \Rightarrow (f x 
+x0 x1 x2)]))))))).
+
+implied lemma ex4_ind:
+ \forall (A0: Type[0]).(\forall (P0: ((A0 \to Prop))).(\forall (P1: ((A0 \to 
+Prop))).(\forall (P2: ((A0 \to Prop))).(\forall (P3: ((A0 \to 
+Prop))).(\forall (P: Prop).(((\forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 
+x0) \to ((P3 x0) \to P)))))) \to ((ex4 A0 P0 P1 P2 P3) \to P)))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (P0: ((A0 \to Prop))).(\lambda (P1: ((A0 \to 
+Prop))).(\lambda (P2: ((A0 \to Prop))).(\lambda (P3: ((A0 \to 
+Prop))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).((P0 x0) \to ((P1 
+x0) \to ((P2 x0) \to ((P3 x0) \to P))))))).(\lambda (e: (ex4 A0 P0 P1 P2 
+P3)).(match e with [(ex4_intro x x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 
+x3)])))))))).
+
+implied lemma ex_2_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
+Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) 
+\to P)))) \to ((ex_2 A0 A1 P0) \to P)))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
+Prop)))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
+A1).((P0 x0 x1) \to P))))).(\lambda (e: (ex_2 A0 A1 P0)).(match e with 
+[(ex_2_intro x x0 x1) \Rightarrow (f x x0 x1)])))))).
+
+implied lemma ex2_2_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
+Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
+P))))) \to ((ex2_2 A0 A1 P0 P1) \to P))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
+Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P: Prop).(\lambda 
+(f: ((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
+P)))))).(\lambda (e: (ex2_2 A0 A1 P0 P1)).(match e with [(ex2_2_intro x x0 x1 
+x2) \Rightarrow (f x x0 x1 x2)]))))))).
+
+implied lemma ex3_2_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
+Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
+\to Prop)))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 
+x1) \to ((P1 x0 x1) \to ((P2 x0 x1) \to P)))))) \to ((ex3_2 A0 A1 P0 P1 P2) 
+\to P)))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
+Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P2: ((A0 \to (A1 
+\to Prop)))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
+A1).((P0 x0 x1) \to ((P1 x0 x1) \to ((P2 x0 x1) \to P))))))).(\lambda (e: 
+(ex3_2 A0 A1 P0 P1 P2)).(match e with [(ex3_2_intro x x0 x1 x2 x3) 
+\Rightarrow (f x x0 x1 x2 x3)])))))))).
+
+implied lemma ex4_2_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (P0: ((A0 \to (A1 \to 
+Prop)))).(\forall (P1: ((A0 \to (A1 \to Prop)))).(\forall (P2: ((A0 \to (A1 
+\to Prop)))).(\forall (P3: ((A0 \to (A1 \to Prop)))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) \to 
+((P2 x0 x1) \to ((P3 x0 x1) \to P))))))) \to ((ex4_2 A0 A1 P0 P1 P2 P3) \to 
+P))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (P0: ((A0 \to (A1 \to 
+Prop)))).(\lambda (P1: ((A0 \to (A1 \to Prop)))).(\lambda (P2: ((A0 \to (A1 
+\to Prop)))).(\lambda (P3: ((A0 \to (A1 \to Prop)))).(\lambda (P: 
+Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 
+x0 x1) \to ((P2 x0 x1) \to ((P3 x0 x1) \to P)))))))).(\lambda (e: (ex4_2 A0 
+A1 P0 P1 P2 P3)).(match e with [(ex4_2_intro x x0 x1 x2 x3 x4) \Rightarrow (f 
+x x0 x1 x2 x3 x4)]))))))))).
+
+implied lemma ex_3_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: Prop).(((\forall (x0: 
+A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to P))))) \to ((ex_3 
+A0 A1 A2 P0) \to P))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: Prop).(\lambda (f: 
+((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to 
+P)))))).(\lambda (e: (ex_3 A0 A1 A2 P0)).(match e with [(ex_3_intro x x0 x1 
+x2) \Rightarrow (f x x0 x1 x2)]))))))).
+
+implied lemma ex2_3_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: 
+A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to P)))))) \to 
+((ex2_3 A0 A1 A2 P0 P1) \to P)))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall 
+(x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to 
+P))))))).(\lambda (e: (ex2_3 A0 A1 A2 P0 P1)).(match e with [(ex2_3_intro x 
+x0 x1 x2 x3) \Rightarrow (f x x0 x1 x2 x3)])))))))).
+
+implied lemma ex3_3_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) 
+\to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to P))))))) \to ((ex3_3 A0 A1 A2 P0 P1 
+P2) \to P))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: 
+Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
+A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to 
+P)))))))).(\lambda (e: (ex3_3 A0 A1 A2 P0 P1 P2)).(match e with [(ex3_3_intro 
+x x0 x1 x2 x3 x4) \Rightarrow (f x x0 x1 x2 x3 x4)]))))))))).
+
+implied lemma ex4_3_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
+((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P: Prop).(((\forall (x0: 
+A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to 
+((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to P)))))))) \to ((ex4_3 A0 A1 A2 P0 P1 P2 
+P3) \to P)))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P3: 
+((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall 
+(x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 
+x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to P))))))))).(\lambda (e: (ex4_3 
+A0 A1 A2 P0 P1 P2 P3)).(match e with [(ex4_3_intro x x0 x1 x2 x3 x4 x5) 
+\Rightarrow (f x x0 x1 x2 x3 x4 x5)])))))))))).
+
+implied lemma ex5_3_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P3: 
+((A0 \to (A1 \to (A2 \to Prop))))).(\forall (P4: ((A0 \to (A1 \to (A2 \to 
+Prop))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
+(x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 
+x2) \to ((P4 x0 x1 x2) \to P))))))))) \to ((ex5_3 A0 A1 A2 P0 P1 P2 P3 P4) 
+\to P))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(P0: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P1: ((A0 \to (A1 \to (A2 
+\to Prop))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P3: 
+((A0 \to (A1 \to (A2 \to Prop))))).(\lambda (P4: ((A0 \to (A1 \to (A2 \to 
+Prop))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
+A1).(\forall (x2: A2).((P0 x0 x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) 
+\to ((P3 x0 x1 x2) \to ((P4 x0 x1 x2) \to P)))))))))).(\lambda (e: (ex5_3 A0 
+A1 A2 P0 P1 P2 P3 P4)).(match e with [(ex5_3_intro x x0 x1 x2 x3 x4 x5 x6) 
+\Rightarrow (f x x0 x1 x2 x3 x4 x5 x6)]))))))))))).
+
+implied lemma ex3_4_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
+(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
+A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
+P)))))))) \to ((ex3_4 A0 A1 A2 A3 P0 P1 P2) \to P)))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda 
+(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda (P: Prop).(\lambda 
+(f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
+A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to 
+P))))))))).(\lambda (e: (ex3_4 A0 A1 A2 A3 P0 P1 P2)).(match e with 
+[(ex3_4_intro x x0 x1 x2 x3 x4 x5) \Rightarrow (f x x0 x1 x2 x3 x4 
+x5)])))))))))).
+
+implied lemma ex4_4_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall 
+(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\forall (P3: ((A0 \to (A1 
+\to (A2 \to (A3 \to Prop)))))).(\forall (P: Prop).(((\forall (x0: 
+A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: A3).((P0 x0 x1 x2 x3) 
+\to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to ((P3 x0 x1 x2 x3) \to 
+P))))))))) \to ((ex4_4 A0 A1 A2 A3 P0 P1 P2 P3) \to P))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to 
+Prop)))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda 
+(P2: ((A0 \to (A1 \to (A2 \to (A3 \to Prop)))))).(\lambda (P3: ((A0 \to (A1 
+\to (A2 \to (A3 \to Prop)))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: 
+A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: A3).((P0 x0 x1 x2 x3) 
+\to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to ((P3 x0 x1 x2 x3) \to 
+P)))))))))).(\lambda (e: (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)).(match e with 
+[(ex4_4_intro x x0 x1 x2 x3 x4 x5 x6) \Rightarrow (f x x0 x1 x2 x3 x4 x5 
+x6)]))))))))))).
+
+implied lemma ex4_5_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
+(x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 
+x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to 
+P)))))))))) \to ((ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3) \to P)))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
+A1).(\forall (x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 
+x4) \to ((P1 x0 x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 
+x4) \to P))))))))))).(\lambda (e: (ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3)).(match 
+e with [(ex4_5_intro x x0 x1 x2 x3 x4 x5 x6 x7) \Rightarrow (f x x0 x1 x2 x3 
+x4 x5 x6 x7)])))))))))))).
+
+implied lemma ex5_5_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (A4: Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\forall (P4: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\forall (P: Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall 
+(x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 
+x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 
+x2 x3 x4) \to P))))))))))) \to ((ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4) \to 
+P))))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to 
+(A3 \to (A4 \to Prop))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to 
+(A4 \to Prop))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\lambda (P4: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to 
+Prop))))))).(\lambda (P: Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: 
+A1).(\forall (x2: A2).(\forall (x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 
+x4) \to ((P1 x0 x1 x2 x3 x4) \to ((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 
+x4) \to ((P4 x0 x1 x2 x3 x4) \to P)))))))))))).(\lambda (e: (ex5_5 A0 A1 A2 
+A3 A4 P0 P1 P2 P3 P4)).(match e with [(ex5_5_intro x x0 x1 x2 x3 x4 x5 x6 x7 
+x8) \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))))))))).
+
+implied lemma ex6_6_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (P0: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P1: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P2: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P3: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P4: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P5: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
+A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 x0 
+x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) \to 
+((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to P))))))))))))) \to 
+((ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4 P5) \to P))))))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (A5: Type[0]).(\lambda (P0: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P1: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P2: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P3: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P4: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P5: 
+((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))))).(\lambda (P: 
+Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
+A2).(\forall (x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 
+x5) \to ((P1 x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 
+x3 x4 x5) \to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to 
+P)))))))))))))).(\lambda (e: (ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4 
+P5)).(match e with [(ex6_6_intro x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10) 
+\Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10)]))))))))))))))).
+
+implied lemma ex6_7_ind:
+ \forall (A0: Type[0]).(\forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall 
+(A3: Type[0]).(\forall (A4: Type[0]).(\forall (A5: Type[0]).(\forall (A6: 
+Type[0]).(\forall (P0: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
+\to Prop))))))))).(\forall (P1: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
+\to (A6 \to Prop))))))))).(\forall (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 
+\to (A5 \to (A6 \to Prop))))))))).(\forall (P3: ((A0 \to (A1 \to (A2 \to (A3 
+\to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P4: ((A0 \to (A1 \to (A2 
+\to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P5: ((A0 \to (A1 
+\to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\forall (P: 
+Prop).(((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall (x3: 
+A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 x3 x4 
+x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) \to ((P3 
+x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 x2 x3 x4 
+x5 x6) \to P)))))))))))))) \to ((ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 
+P5) \to P)))))))))))))))
+\def
+ \lambda (A0: Type[0]).(\lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda 
+(A3: Type[0]).(\lambda (A4: Type[0]).(\lambda (A5: Type[0]).(\lambda (A6: 
+Type[0]).(\lambda (P0: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 
+\to Prop))))))))).(\lambda (P1: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 
+\to (A6 \to Prop))))))))).(\lambda (P2: ((A0 \to (A1 \to (A2 \to (A3 \to (A4 
+\to (A5 \to (A6 \to Prop))))))))).(\lambda (P3: ((A0 \to (A1 \to (A2 \to (A3 
+\to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P4: ((A0 \to (A1 \to (A2 
+\to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P5: ((A0 \to (A1 
+\to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to Prop))))))))).(\lambda (P: 
+Prop).(\lambda (f: ((\forall (x0: A0).(\forall (x1: A1).(\forall (x2: 
+A2).(\forall (x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: 
+A6).((P0 x0 x1 x2 x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 
+x2 x3 x4 x5 x6) \to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) 
+\to ((P5 x0 x1 x2 x3 x4 x5 x6) \to P))))))))))))))).(\lambda (e: (ex6_7 A0 A1 
+A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 P5)).(match e with [(ex6_7_intro x x0 x1 x2 x3 
+x4 x5 x6 x7 x8 x9 x10 x11) \Rightarrow (f x x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 
+x11)])))))))))))))))).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_1A/types/props.ma b/matita/matita/contribs/lambdadelta/ground_1A/types/props.ma
new file mode 100644 (file)
index 0000000..b8cc67a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_1A/types/props.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/types/defs.ma".
+
+lemma ex2_sym:
+ \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
+Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A 
+(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))))))
+\def
+ \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to 
+Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q 
+x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A 
+(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: 
+(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda 
+(x0: A).(P x0)) x H1 H0)))) H)))).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma
new file mode 100644 (file)
index 0000000..0c1a561
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/lib/arith.ma
@@ -0,0 +1,208 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "arithmetics/nat.ma".
+include "ground_2A/lib/star.ma".
+
+(* ARITHMETICAL PROPERTIES **************************************************)
+
+(* Equations ****************************************************************)
+
+lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
+// qed-.
+
+(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
+lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
+#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
+qed-.
+
+fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
+/2 width=1 by plus_minus_minus_be/ qed-.
+
+lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
+// qed.
+
+lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
+/2 by plus_minus/ qed.
+
+lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
+/2 by plus_minus/ qed.
+
+lemma minus_minus_comm3: ∀n,x,y,z. n-x-y-z = n-y-z-x.
+// qed.
+
+lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
+#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
+qed.
+
+lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
+#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2 width=1 by arith_b1/
+qed.
+
+lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
+/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
+
+lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
+                a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
+#a1 #a2 #b #c1 #H1 #H2 >plus_minus /2 width=1 by arith_b2/
+qed.
+
+lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
+/2 width=1 by plus_minus/ qed-.
+
+(* Properties ***************************************************************)
+
+lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
+#n1 elim n1 -n1 [| #n1 #IHn1 ] * [2,4: #n2 ]
+[1,4: @or_intror #H destruct
+| elim (IHn1 n2) -IHn1 /3 width=1 by or_intror, or_introl/
+| /2 width=1 by or_introl/
+]
+qed-.
+
+lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
+#m #n elim (lt_or_ge m n) /2 width=1 by or3_intro0/
+#H elim H -m /2 width=1 by or3_intro1/
+#m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
+qed-.
+
+fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
+// qed-.
+
+fact le_repl_sn_trans_aux: ∀x,y,z:nat. x ≤ z → y = x → y ≤ z.
+// qed-.
+
+lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
+/3 width=1 by monotonic_le_minus_l/ qed.
+
+(* Note: this might interfere with nat.ma *)
+lemma monotonic_lt_pred: ∀m,n. m < n → O < m → pred m < pred n.
+#m #n #Hmn #Hm whd >(S_pred … Hm)
+@le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
+qed.
+
+lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
+/3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
+
+lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-1+n ≤ y-z-1.
+#z #x #y #n #Hzx #Hxny
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-1 ≤ x-z-1+n.
+#z #x #y #n #Hzx #Hyxn
+>plus_minus [2: /2 width=1 by monotonic_le_minus_r/ ]
+>plus_minus [2: /2 width=1 by lt_to_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+(* Inversion & forward lemmas ***********************************************)
+
+lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
+// qed-.
+
+lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
+/2 width=1 by monotonic_pred/ qed-.
+
+lemma lt_refl_false: ∀n. n < n → ⊥.
+#n #H elim (lt_to_not_eq … H) -H /2 width=1 by/
+qed-.
+
+lemma lt_zero_false: ∀n. n < 0 → ⊥.
+#n #H elim (lt_to_not_le … H) -H /2 width=1 by/
+qed-.
+
+lemma pred_inv_refl: ∀m. pred m = m → m = 0.
+* // normalize #m #H elim (lt_refl_false m) //
+qed-.
+
+lemma le_plus_xSy_O_false: ∀x,y. x + S y ≤ 0 → ⊥.
+#x #y #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
+qed-.
+
+lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥.
+#y #z #x elim x -x /3 width=1 by le_S_S_to_le/
+#H elim (le_plus_xSy_O_false … H)
+qed-.
+
+lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
+/2 width=4 by le_plus_xySz_x_false/ qed-.
+
+lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
+/2 width=4 by plus_xySz_x_false/ qed-.
+
+(* Note this should go in nat.ma *)
+lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
+#x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
+#x #_ #y @(nat_ind_plus … y) -y /2 width=1 by or_intror/
+#y #_ >minus_plus_plus_l
+#H lapply (discr_plus_xy_minus_xz … H) -H
+#H destruct
+qed-.
+
+lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
+* /2 width=1 by conj/ #x #y normalize #H destruct
+qed-.
+
+(* Iterators ****************************************************************)
+
+(* Note: see also: lib/arithemetics/bigops.ma *)
+let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
+  match n with
+   [ O   ⇒ nil
+   | S k ⇒ op (iter k B op nil)
+   ].
+
+interpretation "iterated function" 'exp op n = (iter n ? op).
+
+lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
+#B #f #b #l >commutative_plus //
+qed.
+
+lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
+#B #f #b #l elim l -l normalize //
+qed.
+
+lemma iter_plus: ∀B:Type[0]. ∀f:B→B. ∀b,l1,l2. f^(l1+l2) b = f^l1 (f^l2 b).
+#B #f #b #l1 elim l1 -l1 normalize //
+qed.
+
+(* Trichotomy operator ******************************************************)
+
+(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
+let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
+  match n1 with
+  [ O    ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
+  | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
+  ].
+
+lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
+#A #a1 #a2 #a3 #n2 elim n2 -n2
+[ #n1 #H elim (lt_zero_false … H)
+| #n2 #IH #n1 elim n1 -n1 /3 width=1 by monotonic_lt_pred/
+]
+qed.
+
+lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
+#A #a1 #a2 #a3 #n elim n -n normalize //
+qed.
+
+lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
+#A #a1 #a2 #a3 #n1 elim n1 -n1
+[ #n2 #H elim (lt_zero_false … H)
+| #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma
new file mode 100644 (file)
index 0000000..293940b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/lib/bool.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/bool.ma".
+include "ground_2A/lib/star.ma".
+include "ground_2A/notation/constructors/no_0.ma".
+include "ground_2A/notation/constructors/yes_0.ma".
+
+(* BOOLEAN PROPERTIES *******************************************************)
+
+interpretation "boolean false" 'no = false.
+
+interpretation "boolean true" 'yes = true.
+
+(* Basic properties *********************************************************)
+
+lemma orb_false_r: ∀b1,b2:bool. (b1 ∨ b2) = false → b1 = false ∧ b2 = false.
+* normalize /2 width=1 by conj/ #b2 #H destruct
+qed-.
+
+lemma commutative_orb: commutative … orb.
+* * // qed.
+
+lemma eq_bool_dec: ∀b1,b2:bool. Decidable (b1 = b2).
+* * /2 width=1 by or_introl/
+@or_intror #H destruct
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma
new file mode 100644 (file)
index 0000000..f572a3f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/lib/list.ma
@@ -0,0 +1,59 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/notation/constructors/nil_0.ma".
+include "ground_2A/notation/constructors/cons_2.ma".
+include "ground_2A/notation/constructors/cons_3.ma".
+include "ground_2A/notation/functions/append_2.ma".
+include "ground_2A/lib/arith.ma".
+
+(* LISTS ********************************************************************)
+
+inductive list (A:Type[0]) : Type[0] :=
+  | nil : list A
+  | cons: A → list A → list A.
+
+interpretation "nil (list)" 'Nil = (nil ?).
+
+interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl).
+
+let rec all A (R:predicate A) (l:list A) on l ≝
+  match l with
+  [ nil        ⇒ ⊤
+  | cons hd tl ⇒ R hd ∧ all A R tl
+  ].
+
+inductive list2 (A1,A2:Type[0]) : Type[0] :=
+  | nil2 : list2 A1 A2
+  | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
+
+interpretation "nil (list of pairs)" 'Nil = (nil2 ? ?).
+
+interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
+
+let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with
+[ nil2           ⇒ l2
+| cons2 a1 a2 tl ⇒ {a1, a2} @ append2 A1 A2 tl l2
+].
+
+interpretation "append (list of pairs)"
+   'Append l1 l2 = (append2 ? ? l1 l2).
+
+let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with
+[ nil2        ⇒ 0
+| cons2 _ _ l ⇒ length2 A1 A2 l + 1
+].
+
+interpretation "length (list of pairs)"
+   'card l = (length2 ? ? l).

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma
new file mode 100644 (file)
index 0000000..ae707f2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/lib/lstar.ma
@@ -0,0 +1,20 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "arithmetics/lstar.ma".
+
+(* PROPERTIES OF NAT-LABELED REFLEXIVE AND TRANSITIVE CLOSURE ***************)
+
+definition llstar: ∀A:Type[0]. ∀B. (A→relation B) → nat → (A→relation B) ≝
+                   λA,B,R,l,a. lstar … (R a) l.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma
new file mode 100644 (file)
index 0000000..0f193f0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma
@@ -0,0 +1,329 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/star1.ma".
+include "ground_2A/xoa/xoa_props.ma".
+
+(* PROPERTIES OF RELATIONS **************************************************)
+
+definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
+
+definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
+                       ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+
+definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
+                       ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
+                       ∃∃a. R2 a1 a & R1 a2 a.
+
+definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
+                        ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
+                        ∃∃a. R2 a1 a & R1 a a2.
+
+definition bi_confluent:  ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
+                          ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
+                          ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
+
+definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
+                λA,B,R,a. TC … (R a).
+
+definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
+                       ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
+
+definition s_r_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
+                           ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
+
+definition s_rs_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
+                            ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
+
+definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
+                           ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
+
+lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
+                 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
+                 ∃∃a. R2 a1 a & TC … R1 a2 a.
+#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
+[ #a1 #Ha01 #a2 #Ha02
+  elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+  elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+  elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5 by step, ex2_intro/
+]
+qed.
+
+lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
+                 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
+                 ∃∃a. TC … R2 a1 a & R1 a2 a.
+#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
+[ #a2 #Ha02 #a1 #Ha01
+  elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
+  elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
+  elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3 by step, ex2_intro/
+]
+qed.
+
+lemma TC_confluent2: ∀A,R1,R2.
+                     confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
+#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
+[ #a1 #Ha01 #a2 #Ha02
+  elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+  elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+  elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5 by step, ex2_intro/
+]
+qed.
+
+lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
+                 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
+                 ∃∃a. R2 a1 a & TC … R1 a a2.
+#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
+[ #a0 #Ha10 #a2 #Ha02
+  elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+  elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
+  elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
+]
+qed.
+
+lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
+                 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
+                 ∃∃a. TC … R2 a1 a & R1 a a2.
+#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
+[ #a2 #Ha02 #a1 #Ha10
+  elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
+  elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
+  elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3 by step, ex2_intro/
+]
+qed.
+
+lemma TC_transitive2: ∀A,R1,R2.
+                      transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
+#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
+[ #a0 #Ha10 #a2 #Ha02
+  elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+  elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
+  elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
+]
+qed.
+
+definition NF: ∀A. relation A → relation A → predicate A ≝
+   λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
+
+definition NF_dec: ∀A. relation A → relation A → Prop ≝
+                   λA,R,S. ∀a1. NF A R S a1 ∨
+                   ∃∃a2. R … a1 a2 & (S a2 a1 → ⊥).
+
+inductive SN (A) (R,S:relation A): predicate A ≝
+| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
+.
+
+lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
+#A #R #S #a1 #Ha1
+@SN_intro #a2 #HRa12 #HSa12
+elim HSa12 -HSa12 /2 width=1 by/
+qed.
+
+lemma SN_to_NF: ∀A,R,S. NF_dec A R S →
+                ∀a1. SN A R S a1 →
+                ∃∃a2. star … R a1 a2 & NF A R S a2.
+#A #R #S #HRS #a1 #H elim H -a1
+#a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3 by srefl, ex2_intro/
+* #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/
+qed-.
+
+definition NF_sn: ∀A. relation A → relation A → predicate A ≝
+   λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
+
+inductive SN_sn (A) (R,S:relation A): predicate A ≝
+| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
+.
+
+lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
+#A #R #S #a2 #Ha2
+@SN_sn_intro #a1 #HRa12 #HSa12
+elim HSa12 -HSa12 /2 width=1 by/
+qed.
+
+lemma LTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
+#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 /3 width=3 by inj/
+#T #T2 #_ #HT2 #IHT1 #L1 #HL12
+lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3 by step/
+qed-.
+
+lemma s_r_conf1_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (LTC … S) R.
+#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/
+qed-.
+
+lemma s_r_trans_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R →
+                      s_r_transitive A B S R → s_rs_transitive A B S R.
+#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/
+#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1
+/4 width=5 by s_r_conf1_LTC1, trans_TC/
+qed-.
+
+lemma s_r_trans_LTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (LTC … R).
+#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/
+qed-.
+
+lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (LTC … S) R →
+                         s_rs_transitive A B S R.
+#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
+elim (TC_idem … (S L1) …  T1 T2)
+#_ #H @H @HSR //
+qed-.
+
+lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R →
+                         s_r_transitive A B (LTC … S) R.
+#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
+elim (TC_idem … (S L1) …  T1 T2)
+#H #_ @H @HSR //
+qed-.
+
+lemma s_rs_trans_TC1: ∀A,B,S,R. s_rs_transitive A B S R →
+                      s_rs_transitive A B (LTC … S) R.
+#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
+elim (TC_idem … (S L1) …  T1 T2)
+elim (TC_idem … (S L2) …  T1 T2)
+#_ #H1 #H2 #_ @H2 @HSR /3 width=3 by/
+qed-.
+
+(* relations on unboxed pairs ***********************************************)
+
+lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
+                   ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
+                   ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
+#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
+[ #a2 #b2 #H02
+  elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
+| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
+  elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4 by ex2_2_intro, bi_step/
+]
+qed.
+
+lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
+                       bi_confluent A B (bi_TC … R).
+#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
+[ #a1 #b1 #H01 #a2 #b2 #H02
+  elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
+| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
+  elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
+  elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7 by ex2_2_intro, bi_step/
+]
+qed.
+
+lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
+                      ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+                      R a1 b1 a2 b2 ∨
+                      ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4 by ex2_2_intro, or_intror/
+qed-.
+
+lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
+                      ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+                      R a1 b1 a2 b2 ∨
+                      ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
+[ /2 width=1 by or_introl/
+| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4 by ex2_2_intro, or_intror/ (**) (* auto fails without #_ *)
+]
+qed-.
+
+(* relations on unboxed triples *********************************************)
+
+definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
+                   λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
+                   ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
+
+lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
+/3 width=1 by and3_intro, or_intror/ qed.
+
+definition tri_star: ∀A,B,C,R. tri_relation A B C ≝
+                     λA,B,C,R. tri_RC A B C (tri_TC … R).
+
+lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R).
+/2 width=1 by/ qed.
+
+lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
+                          tri_TC A B C R a1 b1 c1 a2 b2 c2 →
+                          tri_star A B C R a1 b1 c1 a2 b2 c2.
+/2 width=1 by or_introl/ qed.
+
+lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
+                         R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+/3 width=1 by tri_TC_to_tri_star, tri_inj/ qed.
+
+lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                       tri_star A B C R a1 b1 c1 a b c →
+                       R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
+[ /3 width=5 by tri_TC_to_tri_star, tri_step/
+| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
+]
+qed.
+
+lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c →
+                       tri_star A B C R a b c a2 b2 c2 →
+                       tri_star A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
+[ /3 width=5 by tri_TC_to_tri_star, tri_TC_strap/
+| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
+]
+qed.
+
+lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                                    tri_star A B C R a1 b1 c1 a b c →
+                                    tri_TC A B C R a b c a2 b2 c2 →
+                                    tri_TC A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
+[ /2 width=5 by tri_TC_transitive/
+| * #H1 #H2 #H3 destruct /2 width=1 by/
+]
+qed.
+
+lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                                    tri_TC A B C R a1 b1 c1 a b c →
+                                    tri_star A B C R a b c a2 b2 c2 →
+                                    tri_TC A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
+[ /2 width=5 by tri_TC_transitive/
+| * #H1 #H2 #H3 destruct /2 width=1 by/
+]
+qed.
+
+lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R).
+#A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 *
+[ /3 width=5 by tri_star_to_tri_TC_to_tri_TC, tri_TC_to_tri_star/
+| * #H1 #H2 #H3 destruct /2 width=1 by/
+]
+qed.
+
+lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 →
+                    (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
+                    ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
+#A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 *
+[ #H12 elim H12 -a2 -b2 -c2 /3 width=6 by tri_TC_to_tri_star/
+| * #H1 #H2 #H3 destruct //
+]
+qed-.
+
+lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
+                       (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
+                       ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
+#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 *
+[ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/
+| * #H1 #H2 #H3 destruct //
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma
new file mode 100644 (file)
index 0000000..c95c57d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( hd @ break tl )"
+  right associative with precedence 47
+  for @{ 'Cons $hd $tl }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma
new file mode 100644 (file)
index 0000000..cfa556e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/cons_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( { term 46 hd1 , break term 46 hd2 } @ break term 46 tl )"
+  non associative with precedence 47
+  for @{ 'Cons $hd1 $hd2 $tl }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma
new file mode 100644 (file)
index 0000000..f5c8491
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/infinity_0.ma
@@ -0,0 +1,20 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "∞"
+   non associative with precedence 55
+   for @{ 'Infinity }.
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma
new file mode 100644 (file)
index 0000000..6ea5151
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/nil_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "◊"
+  non associative with precedence 46
+  for @{ 'Nil }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma
new file mode 100644 (file)
index 0000000..af69221
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/no_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "Ⓕ"
+  non associative with precedence 55
+  for @{'no}.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma
new file mode 100644 (file)
index 0000000..c321749
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/constructors/yes_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "Ⓣ"
+  non associative with precedence 55
+  for @{'yes}.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma
new file mode 100644 (file)
index 0000000..f6d9518
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/append_2.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( l1 @@ break l2 )"
+  right associative with precedence 47
+  for @{ 'Append $l1 $l2 }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma
new file mode 100644 (file)
index 0000000..cf94d04
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/predecessor_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( ⫰ term 70 T )" 
+   non associative with precedence 70
+   for @{ 'Predecessor $T }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma
new file mode 100644 (file)
index 0000000..05e2c31
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/functions/successor_1.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( ⫯ term 70 T )" 
+   non associative with precedence 70
+   for @{ 'Successor $T }.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma
new file mode 100644 (file)
index 0000000..b964325
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/false_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "⊥"
+  non associative with precedence 19
+  for @{'false}.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma
new file mode 100644 (file)
index 0000000..7a9ad43
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa/true_0.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "⊤"
+  non associative with precedence 19
+  for @{'true}.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma
new file mode 100644 (file)
index 0000000..a7651b7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa2_notation.ma
@@ -0,0 +1,16 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma
new file mode 100644 (file)
index 0000000..6054aa3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/notation/xoa_notation.ma
@@ -0,0 +1,326 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (1, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }.
+
+(* multiple existental quantifier (1, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) }.
+
+(* multiple existental quantifier (2, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }.
+
+(* multiple existental quantifier (2, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) }.
+
+(* multiple existental quantifier (3, 1) *)
+
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
+
+(* multiple existental quantifier (3, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }.
+
+(* multiple existental quantifier (3, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }.
+
+(* multiple existental quantifier (3, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) }.
+
+(* multiple existental quantifier (4, 1) *)
+
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) (λ${ident x0}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) (λ${ident x0}:$T0.$P3) }.
+
+(* multiple existental quantifier (4, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }.
+
+(* multiple existental quantifier (4, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }.
+
+(* multiple existental quantifier (4, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }.
+
+(* multiple existental quantifier (4, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) }.
+
+(* multiple existental quantifier (5, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }.
+
+(* multiple existental quantifier (5, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }.
+
+(* multiple existental quantifier (5, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
+
+(* multiple existental quantifier (5, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }.
+
+(* multiple existental quantifier (5, 6) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) }.
+
+(* multiple existental quantifier (6, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) }.
+
+(* multiple existental quantifier (6, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }.
+
+(* multiple existental quantifier (6, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }.
+
+(* multiple existental quantifier (6, 6) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }.
+
+(* multiple existental quantifier (6, 7) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }.
+
+(* multiple existental quantifier (7, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P6) }.
+
+(* multiple existental quantifier (7, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) }.
+
+(* multiple existental quantifier (7, 7) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.λ${ident x6}.$P6) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P6) }.
+
+(* multiple existental quantifier (8, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P7) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P7) }.
+
+(* multiple existental quantifier (8, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P6) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P7) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P7) }.
+
+(* multiple disjunction connective (3) *)
+
+notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)"
+ non associative with precedence 30
+ for @{ 'Or $P0 $P1 $P2 }.
+
+(* multiple disjunction connective (4) *)
+
+notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)"
+ non associative with precedence 30
+ for @{ 'Or $P0 $P1 $P2 $P3 }.
+
+(* multiple disjunction connective (5) *)
+
+notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3 break | term 29 P4)"
+ non associative with precedence 30
+ for @{ 'Or $P0 $P1 $P2 $P3 $P4 }.
+
+(* multiple conjunction connective (3) *)
+
+notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)"
+ non associative with precedence 35
+ for @{ 'And $P0 $P1 $P2 }.
+
+(* multiple conjunction connective (4) *)
+
+notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2 break & term 34 P3)"
+ non associative with precedence 35
+ for @{ 'And $P0 $P1 $P2 $P3 }.
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma
new file mode 100644 (file)
index 0000000..27da042
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa.ma
@@ -0,0 +1,293 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2A/notation/xoa_notation.ma".
+
+(* multiple existental quantifier (1, 2) *)
+
+inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝
+   | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ?
+.
+
+interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
+
+(* multiple existental quantifier (1, 3) *)
+
+inductive ex1_3 (A0,A1,A2:Type[0]) (P0:A0→A1→A2→Prop) : Prop ≝
+   | ex1_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → ex1_3 ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0).
+
+(* multiple existental quantifier (2, 2) *)
+
+inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝
+   | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
+
+(* multiple existental quantifier (2, 3) *)
+
+inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝
+   | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1).
+
+(* multiple existental quantifier (3, 1) *)
+
+inductive ex3 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
+   | ex3_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3 ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3 ? P0 P1 P2).
+
+(* multiple existental quantifier (3, 2) *)
+
+inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝
+   | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
+
+(* multiple existental quantifier (3, 3) *)
+
+inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝
+   | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
+
+(* multiple existental quantifier (3, 4) *)
+
+inductive ex3_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex3_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → ex3_4 ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2).
+
+(* multiple existental quantifier (4, 1) *)
+
+inductive ex4 (A0:Type[0]) (P0,P1,P2,P3:A0→Prop) : Prop ≝
+   | ex4_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → P3 x0 → ex4 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4 ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (4, 2) *)
+
+inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝
+   | ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (4, 3) *)
+
+inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝
+   | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (4, 4) *)
+
+inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (4, 5) *)
+
+inductive ex4_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→A4→Prop) : Prop ≝
+   | ex4_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → ex4_5 ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3).
+
+(* multiple existental quantifier (5, 2) *)
+
+inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝
+   | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (5, 3) *)
+
+inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝
+   | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (5, 4) *)
+
+inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (5, 5) *)
+
+inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝
+   | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (5, 6) *)
+
+inductive ex5_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
+   | ex5_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → ex5_6 ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 6)" 'Ex P0 P1 P2 P3 P4 = (ex5_6 ? ? ? ? ? ? P0 P1 P2 P3 P4).
+
+(* multiple existental quantifier (6, 3) *)
+
+inductive ex6_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→Prop) : Prop ≝
+   | ex6_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → ex6_3 ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 3)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (6, 4) *)
+
+inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (6, 5) *)
+
+inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝
+   | ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (6, 6) *)
+
+inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
+   | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (6, 7) *)
+
+inductive ex6_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
+   | ex6_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → ex6_7 ? ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+
+(* multiple existental quantifier (7, 3) *)
+
+inductive ex7_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→Prop) : Prop ≝
+   | ex7_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → P6 x0 x1 x2 → ex7_3 ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 3)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_3 ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
+(* multiple existental quantifier (7, 4) *)
+
+inductive ex7_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex7_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → ex7_4 ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
+(* multiple existental quantifier (7, 7) *)
+
+inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
+   | ex7_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → P6 x0 x1 x2 x3 x4 x5 x6 → ex7_7 ? ? ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+
+(* multiple existental quantifier (8, 4) *)
+
+inductive ex8_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→Prop) : Prop ≝
+   | ex8_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → P7 x0 x1 x2 x3 → ex8_4 ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (8, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
+
+(* multiple existental quantifier (8, 5) *)
+
+inductive ex8_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→A4→Prop) : Prop ≝
+   | ex8_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → P6 x0 x1 x2 x3 x4 → P7 x0 x1 x2 x3 x4 → ex8_5 ? ? ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (8, 5)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
+
+(* multiple disjunction connective (3) *)
+
+inductive or3 (P0,P1,P2:Prop) : Prop ≝
+   | or3_intro0: P0 → or3 ? ? ?
+   | or3_intro1: P1 → or3 ? ? ?
+   | or3_intro2: P2 → or3 ? ? ?
+.
+
+interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2).
+
+(* multiple disjunction connective (4) *)
+
+inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝
+   | or4_intro0: P0 → or4 ? ? ? ?
+   | or4_intro1: P1 → or4 ? ? ? ?
+   | or4_intro2: P2 → or4 ? ? ? ?
+   | or4_intro3: P3 → or4 ? ? ? ?
+.
+
+interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3).
+
+(* multiple disjunction connective (5) *)
+
+inductive or5 (P0,P1,P2,P3,P4:Prop) : Prop ≝
+   | or5_intro0: P0 → or5 ? ? ? ? ?
+   | or5_intro1: P1 → or5 ? ? ? ? ?
+   | or5_intro2: P2 → or5 ? ? ? ? ?
+   | or5_intro3: P3 → or5 ? ? ? ? ?
+   | or5_intro4: P4 → or5 ? ? ? ? ?
+.
+
+interpretation "multiple disjunction connective (5)" 'Or P0 P1 P2 P3 P4 = (or5 P0 P1 P2 P3 P4).
+
+(* multiple conjunction connective (3) *)
+
+inductive and3 (P0,P1,P2:Prop) : Prop ≝
+   | and3_intro: P0 → P1 → P2 → and3 ? ? ?
+.
+
+interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2).
+
+(* multiple conjunction connective (4) *)
+
+inductive and4 (P0,P1,P2,P3:Prop) : Prop ≝
+   | and4_intro: P0 → P1 → P2 → P3 → and4 ? ? ? ?
+.
+
+interpretation "multiple conjunction connective (4)" 'And P0 P1 P2 P3 = (and4 P0 P1 P2 P3).
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma
new file mode 100644 (file)
index 0000000..3975bd4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa2.ma
@@ -0,0 +1,20 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2A/notation/xoa2_notation.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma
new file mode 100644 (file)
index 0000000..8ab935a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/xoa/xoa_props.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/logic.ma".
+include "ground_2A/notation/xoa/false_0.ma".
+include "ground_2A/notation/xoa/true_0.ma".
+include "ground_2A/xoa/xoa.ma".
+
+interpretation "logical false" 'false = False.
+
+interpretation "logical true" 'true = True.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma
new file mode 100644 (file)
index 0000000..1d802e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat.ma
@@ -0,0 +1,34 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "arithmetics/nat.ma".
+include "ground_2A/notation/constructors/infinity_0.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the type of natural numbers with infinity *)
+inductive ynat: Type[0] ≝
+| yinj: nat → ynat
+| Y   : ynat
+.
+
+coercion yinj.
+
+interpretation "ynat infinity" 'Infinity = Y.
+
+(* Inversion lemmas *********************************************************)
+
+lemma yinj_inj: ∀m,n. yinj m = yinj n → m = n.
+#m #n #H destruct //
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma
new file mode 100644 (file)
index 0000000..4a8e89e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma
@@ -0,0 +1,136 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_succ.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* order relation *)
+inductive yle: relation ynat ≝
+| yle_inj: ∀m,n. m ≤ n → yle m n
+| yle_Y  : ∀m. yle m (∞)
+.
+
+interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact yle_inv_inj2_aux: ∀x,y. x ≤ y → ∀n. y = yinj n →
+                       ∃∃m. m ≤ n & x = yinj m.
+#x #y * -x -y
+[ #x #y #Hxy #n #Hy destruct /2 width=3 by ex2_intro/
+| #x #n #Hy destruct
+]
+qed-.
+
+lemma yle_inv_inj2: ∀x,n. x ≤ yinj n → ∃∃m. m ≤ n & x = yinj m.
+/2 width=3 by yle_inv_inj2_aux/ qed-.
+
+lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
+#m #n #H elim (yle_inv_inj2 … H) -H
+#x #Hxn #H destruct //
+qed-.
+
+fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
+#m #x * -m -x
+[ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
+| #m #H destruct
+] 
+qed-.
+
+lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
+/2 width =3 by yle_inv_O2_aux/ qed-.
+
+fact yle_inv_Y1_aux: ∀x,n. x ≤ n → x = ∞ → n = ∞.
+#x #n * -x -n //
+#x #n #_ #H destruct
+qed-.
+
+lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
+/2 width=3 by yle_inv_Y1_aux/ qed-.
+
+(* Inversion lemmas on successor ********************************************)
+
+fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
+#x #y * -x -y
+[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
+  #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
+  #m #Hnm #H destruct /3 width=1 by yle_inj, conj/
+| #x #y #H destruct /2 width=1 by yle_Y, conj/
+]
+qed-.
+
+lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
+/2 width=3 by yle_inv_succ1_aux/ qed-.
+
+lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
+#m #n #H elim (yle_inv_succ1 … H) -H //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma le_O1: ∀n:ynat. 0 ≤ n.
+* /2 width=1 by yle_inj/
+qed.
+
+lemma yle_refl: reflexive … yle.
+* /2 width=1 by le_n, yle_inj/
+qed.
+
+lemma yle_split: ∀x,y:ynat. x ≤ y ∨ y ≤ x.
+* /2 width=1 by or_intror/
+#x * /2 width=1 by or_introl/
+#y elim (le_or_ge x y) /3 width=1 by yle_inj, or_introl, or_intror/
+qed-.
+
+(* Properties on predecessor ************************************************)
+
+lemma yle_pred_sn: ∀m,n. m ≤ n → ⫰m ≤ n.
+#m #n * -m -n /3 width=3 by transitive_le, yle_inj/
+qed.
+
+lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
+/2 width=1 by yle_refl, yle_pred_sn/ qed.
+
+lemma yle_pred: ∀m,n. m ≤ n → ⫰m ≤ ⫰n.
+#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
+qed.
+
+(* Properties on successor **************************************************)
+
+lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
+#m #n * -m -n /3 width=1 by yle_inj, le_S_S/
+qed.
+
+lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
+#m #n * -m -n /3 width=1 by le_S, yle_inj/
+qed.
+
+lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
+/2 width=1 by yle_succ_dx/ qed.
+
+lemma yle_refl_SP_dx: ∀x. x ≤ ⫯⫰x.
+* // * //
+qed. 
+
+(* Main properties **********************************************************)
+
+theorem yle_trans: Transitive … yle.
+#x #y * -x -y
+[ #x #y #Hxy * //
+  #z #H lapply (yle_inv_inj … H) -H
+  /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
+| #x #z #H lapply (yle_inv_Y1 … H) //
+]
+qed-. 

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma
new file mode 100644 (file)
index 0000000..850fae1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma
@@ -0,0 +1,182 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_le.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* strict order relation *)
+inductive ylt: relation ynat ≝
+| ylt_inj: ∀m,n. m < n → ylt m n
+| ylt_Y  : ∀m:nat. ylt m (∞)
+.
+
+interpretation "ynat 'less than'" 'lt x y = (ylt x y).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
+#x #y * -x -y /2 width=2 by ex_intro/
+qed-.
+
+lemma ylt_fwd_le_succ: ∀x,y. x < y → ⫯x ≤ y.
+#x #y * -x -y /2 width=1 by yle_inj/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
+                       ∃∃m. m < n & x = yinj m.
+#x #y * -x -y
+[ #x #y #Hxy #n #Hy elim (le_inv_S1 … Hxy) -Hxy
+  #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/
+| #x #n #Hy destruct
+]
+qed-.
+
+lemma ylt_inv_inj2: ∀x,n. x < yinj n →
+                    ∃∃m. m < n & x = yinj m.
+/2 width=3 by ylt_inv_inj2_aux/ qed-.
+
+lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n.
+#m #n #H elim (ylt_inv_inj2 … H) -H
+#x #Hx #H destruct //
+qed-.
+
+lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
+#n #H elim (ylt_fwd_gen … H) -H
+#y #H destruct
+qed-.
+
+lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
+* // #n #H lapply (ylt_inv_inj … H) -H normalize
+/3 width=1 by S_pred, eq_f/
+qed-.
+
+(* Inversion lemmas on successor ********************************************)
+
+fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
+#x #y * -x -y
+[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
+  #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
+  #m #Hnm #H destruct /3 width=1 by ylt_inj, conj/
+| #x #y #H elim (ysucc_inv_inj_sn … H) -H
+  #m #H #_ destruct /2 width=1 by ylt_Y, conj/
+]
+qed-.
+
+lemma ylt_inv_succ1: ∀m,y. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
+/2 width=3 by ylt_inv_succ1_aux/ qed-.
+
+lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
+#m #n #H elim (ylt_inv_succ1 … H) -H //
+qed-.
+
+(* Forward lemmas on successor **********************************************)
+
+fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
+#x #y * -x -y
+[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
+  #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/
+| #x #n #H lapply (ysucc_inv_Y_sn … H) -H //
+]
+qed-.
+
+lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
+/2 width=3 by ylt_fwd_succ2_aux/ qed-.
+
+(* inversion and forward lemmas on yle **************************************)
+
+lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
+#m #n * -m -n /2 width=1 by yle_inj/
+qed-.
+
+lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
+#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/
+qed-.
+
+lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
+#m #n * -m -n
+[ #m #n #Hmn #H lapply (yle_inv_inj … H) -H
+  #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/
+| #m #H lapply (yle_inv_Y1 … H) -H
+  #H destruct
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ylt_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x.
+* /2 width=1 by/ normalize
+#H destruct
+qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma ylt_pred: ∀m,n. m < n → 0 < m → ⫰m < ⫰n.
+#m #n * -m -n
+/4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
+qed.
+
+(* Properties on successor **************************************************)
+
+lemma ylt_O_succ: ∀n. 0 < ⫯n.
+* /2 width=1 by ylt_inj/
+qed.
+
+lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
+#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
+qed.
+
+(* Properties on order ******************************************************)
+
+lemma yle_split_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
+#m #n * -m -n
+[ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
+  /3 width=1 by or_introl, ylt_inj/
+| * /2 width=1 by or_introl, ylt_Y/
+]
+qed-.
+
+lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m..
+#m #n elim (yle_split m n) /2 width=1 by or_intror/
+#H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/
+qed-. 
+
+lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
+#x #y #z * -y -z
+[ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H
+  #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/
+| #y * //
+]
+qed-.
+
+lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x < z.
+#x #y #z * -y -z
+[ #y #z #Hyz #H elim (yle_inv_inj2 … H) -H
+  #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/
+| #y #H elim (yle_inv_inj2 … H) -H //
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem ylt_trans: Transitive … ylt.
+#x #y * -x -y
+[ #x #y #Hxy * //
+  #z #H lapply (ylt_inv_inj … H) -H
+  /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *)
+| #x #z #H elim (ylt_yle_false … H) //
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma
new file mode 100644 (file)
index 0000000..05ce327
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_max.ma
@@ -0,0 +1,58 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+lemma ymax_pre_dx: ∀x,y. x ≤ y → x - y + y = y.
+#x #y * -x -y //
+#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
+qed-.
+
+lemma ymax_pre_sn: ∀x,y. y ≤ x → x - y + y = x.
+#x #y * -x -y
+[ #x #y #Hxy >yminus_inj /3 width=3 by plus_minus, eq_f/
+| * //
+]
+qed-.
+
+lemma ymax_pre_i_dx: ∀y,x. y ≤ x - y + y.
+// qed.
+
+lemma ymax_pre_i_sn: ∀y,x. x ≤ x - y + y.
+* // #y * /2 width=1 by yle_inj/
+qed.
+
+lemma ymax_pre_e: ∀x,z. x ≤ z → ∀y. y ≤ z → x - y + y ≤ z.
+#x #z #Hxz #y #Hyz elim (yle_split x y)
+[ #Hxy >(ymax_pre_dx … Hxy) -x //
+| #Hyx >(ymax_pre_sn … Hyx) -y //
+]
+qed.
+
+lemma ymax_pre_dx_comm: ∀x,y. x ≤ y → y + (x - y) = y.
+/2 width=1 by ymax_pre_dx/ qed-.
+
+lemma ymax_pre_sn_comm: ∀x,y. y ≤ x → y + (x - y) = x.
+/2 width=1 by ymax_pre_sn/ qed-.
+
+lemma ymax_pre_i_dx_comm: ∀y,x. y ≤ y + (x - y).
+// qed.
+
+lemma ymax_pre_i_sn_comm: ∀y,x. x ≤ y + (x - y).
+/2 width=1 by ymax_pre_i_sn/ qed.
+
+lemma ymax_pre_e_comm: ∀x,z. x ≤ z → ∀y. y ≤ z → y + (x - y) ≤ z.
+/2 width=1 by ymax_pre_e/ qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma
new file mode 100644 (file)
index 0000000..4ec861b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_min.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_plus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+fact ymin_pre_dx_aux: ∀x,y. y ≤ x → x - (x - y) ≤ y.
+#x #y * -x -y
+[ #x #y #Hxy >yminus_inj
+ /3 width=4 by yle_inj, monotonic_le_minus_l/
+| * //
+]
+qed-.
+
+lemma ymin_pre_sn: ∀x,y. x ≤ y → x - (x - y) = x.
+#x #y * -x -y //
+#x #y #Hxy >yminus_inj >(eq_minus_O … Hxy) -Hxy //
+qed-.
+
+lemma ymin_pre_i_dx: ∀x,y. x - (x - y) ≤ y.
+#x #y elim (yle_split x y) /2 width=1 by ymin_pre_dx_aux/
+#Hxy >(ymin_pre_sn … Hxy) //
+qed.
+
+lemma ymin_pre_i_sn: ∀x,y. x - (x - y) ≤ x.
+// qed.
+
+lemma ymin_pre_dx: ∀x,y. y ≤ yinj x → yinj x - (yinj x - y) = y.
+#x #y #H elim (yle_inv_inj2 … H) -H
+#z #Hzx #H destruct >yminus_inj
+/3 width=4 by minus_le_minus_minus_comm, eq_f/
+qed-.
+
+lemma ymin_pre_e: ∀z,x. z ≤ yinj x → ∀y. z ≤ y →
+                  z ≤ yinj x - (yinj x - y).
+#z #x #Hzx #y #Hzy elim (yle_split x y)
+[ #H >(ymin_pre_sn … H) -y //
+| #H >(ymin_pre_dx … H) -x //
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma
new file mode 100644 (file)
index 0000000..e4763f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_minus.ma
@@ -0,0 +1,109 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_lt.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* subtraction *)
+definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
+[ yinj n ⇒ ypred^n x
+| Y      ⇒ yinj 0
+].
+
+interpretation "ynat minus" 'minus x y = (yminus x y).
+
+(* Basic properties *********************************************************)
+
+lemma yminus_inj: ∀n,m. yinj m - yinj n = yinj (m - n).
+#n elim n -n /2 width=3 by trans_eq/
+qed.
+
+lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
+#n elim n -n // normalize
+#n #IHn >IHn //
+qed.
+
+lemma yminus_O1: ∀x:ynat. 0 - x = 0.
+* // qed.
+
+lemma yminus_refl: ∀x:ynat. x - x = 0.
+* // qed.
+
+lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
+* #y [ * #z [ * // ] ] >yminus_O1 //
+qed.
+
+(* Properties on predecessor ************************************************)
+
+lemma yminus_SO2: ∀m. m - 1 = ⫰m.
+* //
+qed.
+
+lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
+* // #n *
+[ #m #Hm #Hn >yminus_inj >yminus_inj
+  /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+(* Properties on successor **************************************************)
+
+lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
+* // #n * [2: >yminus_Y_inj // ]
+#m >yminus_inj //
+qed.
+
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
+#n *
+[ #m #Hmn >yminus_inj >yminus_inj
+  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
+* //
+qed.
+
+(* Properties on order ******************************************************)
+
+lemma yle_minus_sn: ∀n,m. m - n ≤ m.
+* // #n * /2 width=1 by yle_inj/
+qed.
+
+lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
+#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
+qed-.
+
+lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
+* // #n *
+[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+  /2 width=1 by yle_inj/
+| >yminus_Y_inj #H destruct
+]
+qed.
+
+lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
+#x #y #Hxy * //
+#z elim z -z /3 width=1 by yle_pred/
+qed.
+
+(* Properties on strict order ***********************************************)
+
+lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
+#x #y * -x -y
+/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
+qed.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma
new file mode 100644 (file)
index 0000000..761fc99
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_plus.ma
@@ -0,0 +1,203 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/ynat/ynat_minus.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* addition *)
+definition yplus: ynat → ynat → ynat ≝ λx,y. match y with
+[ yinj n ⇒ ysucc^n x
+| Y      ⇒ Y
+].
+
+interpretation "ynat plus" 'plus x y = (yplus x y).
+
+(* Properties on successor **************************************************)
+
+lemma yplus_succ2: ∀m,n. m + ⫯n = ⫯(m + n).
+#m * //
+qed.
+
+lemma yplus_succ1: ∀m,n. ⫯m + n = ⫯(m + n).
+#m * normalize //
+qed.
+
+lemma yplus_succ_swap: ∀m,n. m + ⫯n = ⫯m + n.
+// qed.
+
+lemma yplus_SO2: ∀m. m + 1 = ⫯m.
+* //
+qed.
+
+(* Basic properties *********************************************************)
+
+lemma yplus_inj: ∀n,m. yinj m + yinj n = yinj (m + n).
+#n elim n -n [ normalize // ]
+#n #IHn #m >(yplus_succ2 ? n) >IHn -IHn
+<plus_n_Sm //
+qed.
+
+lemma yplus_Y1: ∀m. ∞ + m = ∞.
+* normalize //
+qed.
+
+lemma yplus_comm: commutative … yplus.
+* [ #m ] * [1,3: #n ] //
+normalize >ysucc_iter_Y //
+qed.
+
+lemma yplus_assoc: associative … yplus.
+#x #y * // #z cases y -y
+[ #y >yplus_inj whd in ⊢ (??%%); <iter_plus //
+| >yplus_Y1 //
+]
+qed.
+
+lemma yplus_O1: ∀n:ynat. 0 + n = n.
+#n >yplus_comm // qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
+                     ∃∃m,n. m + n = z & x = yinj m & y = yinj n.
+#z * [2: normalize #x #H destruct ]
+#y * [2: >yplus_Y1 #H destruct ]
+/3 width=5 by yinj_inj, ex3_2_intro/
+qed-.
+
+(* Properties on order ******************************************************)
+
+lemma yle_plus_dx2: ∀n,m. n ≤ m + n.
+* //
+#n elim n -n //
+#n #IHn #m >(yplus_succ2 ? n) @(yle_succ n) // (**) (* full auto fails *)
+qed.
+
+lemma yle_plus_dx1: ∀n,m. m ≤ m + n.
+// qed.
+
+lemma yle_plus_dx1_trans: ∀x,z. z ≤ x → ∀y. z ≤ x + y.
+/2 width=3 by yle_trans/ qed.
+
+lemma yle_plus_dx2_trans: ∀y,z. z ≤ y → ∀x. z ≤ x + y.
+/2 width=3 by yle_trans/ qed.
+
+lemma monotonic_yle_plus_dx: ∀x,y. x ≤ y → ∀z. x + z ≤ y + z.
+#x #y #Hxy * //
+#z elim z -z /3 width=1 by yle_succ/
+qed.
+
+lemma monotonic_yle_plus_sn: ∀x,y. x ≤ y → ∀z. z + x ≤ z + y.
+/2 width=1 by monotonic_yle_plus_dx/ qed.
+
+lemma monotonic_yle_plus: ∀x1,y1. x1 ≤ y1 → ∀x2,y2. x2 ≤ y2 →
+                          x1 + x2 ≤ y1 + y2.
+/3 width=3 by monotonic_yle_plus_dx, yle_trans/ qed.
+
+(* Forward lemmas on order **************************************************)
+
+lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
+/2 width=3 by yle_trans/ qed-.
+
+lemma yle_fwd_plus_sn1: ∀x,y,z. x + y ≤ z → x ≤ z.
+/2 width=3 by yle_trans/ qed-.
+
+lemma yle_inv_monotonic_plus_dx: ∀x,y:ynat.∀z:nat. x + z ≤ y + z → x ≤ y.
+#x #y #z elim z -z /3 width=1 by yle_inv_succ/
+qed-.
+
+lemma yle_inv_monotonic_plus_sn: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
+/2 width=2 by yle_inv_monotonic_plus_dx/ qed-.
+
+lemma yle_fwd_plus_ge: ∀m1,m2:nat. m2 ≤ m1 → ∀n1,n2:ynat. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m1 #m2 #Hm12 #n1 #n2 #H
+lapply (monotonic_yle_plus … Hm12 … H) -Hm12 -H
+/2 width=2 by yle_inv_monotonic_plus_sn/
+qed-.
+
+lemma yle_fwd_plus_ge_inj: ∀m1:nat. ∀m2,n1,n2:ynat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m2 #m1 #n1 #n2 #H elim (yle_inv_inj2 … H) -H
+#x #H0 #H destruct /3 width=4 by yle_fwd_plus_ge, yle_inj/
+qed-.
+
+(* Forward lemmas on strict order *******************************************)
+
+lemma ylt_inv_monotonic_plus_dx: ∀x,y,z. x + z < y + z → x < y.
+* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 … H) ]
+#x * // #y * [2: #H elim (ylt_inv_Y1 … H) ]
+/4 width=3 by ylt_inv_inj, ylt_inj, lt_plus_to_lt_l/
+qed-.
+
+(* Properties on strict order ***********************************************)
+
+lemma ylt_plus_dx1_trans: ∀x,z. z < x → ∀y. z < x + yinj y.
+/2 width=3 by ylt_yle_trans/ qed.
+
+lemma ylt_plus_dx2_trans: ∀y,z. z < y → ∀x. z < yinj x + y.
+/2 width=3 by ylt_yle_trans/ qed.
+
+lemma monotonic_ylt_plus_dx: ∀x,y. x < y → ∀z:nat. x + yinj z < y + yinj z.
+#x #y #Hxy #z elim z -z /3 width=1 by ylt_succ/
+qed.
+
+lemma monotonic_ylt_plus_sn: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
+/2 width=1 by monotonic_ylt_plus_dx/ qed.
+
+(* Properties on minus ******************************************************)
+
+lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
+#m #n elim n -n //
+#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
+qed.
+
+lemma yplus_minus: ∀m,n. m + n - n ≤ m.
+#m * //
+qed.
+
+(* Forward lemmas on minus **************************************************)
+
+lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
+
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+
+lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
+#x *
+[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
+  /4 width=1 by yle_inv_inj, plus_minus, eq_f/
+| >yminus_Y_inj //
+]
+qed-.
+
+lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
+#y * // #x * //
+#z #Hxy >yplus_inj >yminus_inj <plus_minus
+/2 width=1 by yle_inv_inj/
+qed-.
+
+(* Inversion lemmas on minus ************************************************)
+
+lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
+/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
+
+lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
+/2 width=1 by yle_inv_plus_inj2/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma
new file mode 100644 (file)
index 0000000..0ce5dae
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/notation/functions/predecessor_1.ma".
+include "ground_2A/lib/arith.ma".
+include "ground_2A/ynat/ynat.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the predecessor function *)
+definition ypred: ynat → ynat ≝ λm. match m with
+[ yinj m ⇒ pred m
+| Y      ⇒ Y
+].
+
+interpretation "ynat predecessor" 'Predecessor m = (ypred m).
+
+(* Properties ***************************************************************)
+
+lemma ypred_inj_rew: ∀m:nat. ⫰m = pred m.
+// qed.
+
+(* Inversion lemmas *********************************************************)
+
+lemma ypred_inv_refl: ∀m. ⫰m = m → m = 0 ∨ m = ∞.
+* // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+/4 width=1 by pred_inv_refl, or_introl, eq_f/
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_succ.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_succ.ma
new file mode 100644 (file)
index 0000000..23d5658
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_succ.ma
@@ -0,0 +1,84 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2A/notation/functions/successor_1.ma".
+include "ground_2A/ynat/ynat_pred.ma".
+
+(* NATURAL NUMBERS WITH INFINITY ********************************************)
+
+(* the successor function *)
+definition ysucc: ynat → ynat ≝ λm. match m with
+[ yinj m ⇒ S m
+| Y      ⇒ Y
+].
+
+interpretation "ynat successor" 'Successor m = (ysucc m).
+
+(* Properties ***************************************************************)
+
+lemma ypred_succ: ∀m. ⫰⫯m = m.
+* // qed.
+
+lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m. n = ⫯m.
+*
+[ * /2 width=1 by or_introl/
+  #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
+| @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
+]
+qed-.
+
+lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
+#m elim m -m //
+#m #IHm whd in ⊢ (??%?); >IHm //
+qed.
+
+(* Inversion lemmas *********************************************************)
+
+lemma ysucc_inj: ∀m,n. ⫯m = ⫯n → m = n.
+#m #n #H <(ypred_succ m) <(ypred_succ n) //
+qed-.
+
+lemma ysucc_inv_refl: ∀m. ⫯m = m → m = ∞.
+* // normalize
+#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
+#H elim (lt_refl_false m) //
+qed-.
+
+lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ⫯n1 →
+                        ∃∃m1. n1 = yinj m1 & m2 = S m1.
+#m2 * normalize
+[ #n1 #H destruct /2 width=3 by ex2_intro/
+| #H destruct
+]
+qed-.
+
+lemma ysucc_inv_inj_dx: ∀m2,n1. ⫯n1 = yinj m2  →
+                        ∃∃m1. n1 = yinj m1 & m2 = S m1.
+/2 width=1 by ysucc_inv_inj_sn/ qed-.
+
+lemma ysucc_inv_Y_sn: ∀m. ∞ = ⫯m → m = ∞.
+* // normalize
+#m #H destruct
+qed-.
+
+lemma ysucc_inv_Y_dx: ∀m. ⫯m = ∞ → m = ∞.
+/2 width=1 by ysucc_inv_Y_sn/ qed-.
+
+lemma ysucc_inv_O_sn: ∀m. yinj 0 = ⫯m → ⊥. (**) (* explicit coercion *)
+#m #H elim (ysucc_inv_inj_sn … H) -H
+#n #_ #H destruct
+qed-.
+
+lemma ysucc_inv_O_dx: ∀m. ⫯m = 0 → ⊥.
+/2 width=2 by ysucc_inv_O_sn/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/hls.ml b/matita/matita/contribs/lambdadelta/hls.ml
deleted file mode 100644 (file)
index d796a4a..0000000
--- a/matita/matita/contribs/lambdadelta/hls.ml
+++ /dev/null
@@ -1,57 +0,0 @@
-let cols =
-   try int_of_string (Sys.getenv "COLUMNS")
-   with Not_found -> failwith "environment variable COLUMNS not visible"
-
-let hl = ref []
-
-let normal = "\x1B[0m"
-
-let color = "\x1B[32m"
-
-let add s =
-   if s = "" then false else
-   begin hl := s :: !hl; true end
-
-let rec read ich =
-   if Scanf.fscanf ich "%s " add then read ich
-
-let length l s = max l (String.length s)
-
-let split s =
-try 
-   let i = String.rindex s '.' in
-   if i = 0 then s, "" else
-   String.sub s 0 i, String.sub s i (String.length s - i)    
-with Not_found -> s, ""
-
-let compare s1 s2 =
-   let n1, e1 = split s1 in
-   let n2, e2 = split s2 in
-   let e = String.compare e1 e2 in
-   if e = 0 then String.compare n1 n2 else e
-
-let write l c s =
-   let pre, post = if List.mem s !hl then color, normal else "", "" in
-   let spc = String.make (l - String.length s) ' ' in
-   let bol, ret =
-       if c = 0 || c = cols then "", l else
-       if c + l < cols then " ", c + succ l else "\n", l
-   in
-   Printf.printf "%s%s%s%s%s" bol pre s post spc;
-   ret
-
-let process fname =
-   let ich = open_in fname in
-   read ich; close_in ich
-
-let help = ""
-
-let main =
-   Arg.parse [] process help;
-   let files = Sys.readdir "." in
-   let l = Array.fold_left length 0 files in
-   if cols < l then failwith "window too small";
-   Array.fast_sort compare files;
-   let c = Array.fold_left (write l) 0 files in
-   if 0 < c && c < cols then print_newline ();
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma
deleted file mode 100644 (file)
index 497403d..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/defs.ma
+++ /dev/null
@@ -1,93 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/preamble.ma".
-
-inductive eq (A: Type[0]) (x: A): A \to Prop \def
-| refl_equal: eq A x x.
-
-inductive True: Prop \def
-| I: True.
-
-inductive land (A: Prop) (B: Prop): Prop \def
-| conj: A \to (B \to (land A B)).
-
-inductive or (A: Prop) (B: Prop): Prop \def
-| or_introl: A \to (or A B)
-| or_intror: B \to (or A B).
-
-inductive ex (A: Type[0]) (P: A \to Prop): Prop \def
-| ex_intro: \forall (x: A).((P x) \to (ex A P)).
-
-inductive ex2 (A: Type[0]) (P: A \to Prop) (Q: A \to Prop): Prop \def
-| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))).
-
-definition not:
- Prop \to Prop
-\def
- \lambda (A: Prop).(A \to False).
-
-inductive bool: Type[0] \def
-| true: bool
-| false: bool.
-
-inductive nat: Type[0] \def
-| O: nat
-| S: nat \to nat.
-
-inductive le (n: nat): nat \to Prop \def
-| le_n: le n n
-| le_S: \forall (m: nat).((le n m) \to (le n (S m))).
-
-definition lt:
- nat \to (nat \to Prop)
-\def
- \lambda (n: nat).(\lambda (m: nat).(le (S n) m)).
-
-definition IsSucc:
- nat \to Prop
-\def
- \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow 
-True]).
-
-definition pred:
- nat \to nat
-\def
- \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]).
-
-rec definition plus (n: nat) on n: nat \to nat \def \lambda (m: nat).(match n 
-with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))]).
-
-rec definition minus (n: nat) on n: nat \to nat \def \lambda (m: nat).(match 
-n with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S 
-k) | (S l) \Rightarrow (minus k l)])]).
-
-inductive Acc (A: Type[0]) (R: A \to (A \to Prop)): A \to Prop \def
-| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to 
-(Acc A R x)).
-
-definition well_founded:
- \forall (A: Type[0]).(((A \to (A \to Prop))) \to Prop)
-\def
- \lambda (A: Type[0]).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: 
-A).(Acc A R a))).
-
-definition ltof:
- \forall (A: Type[0]).(((A \to nat)) \to (A \to (A \to Prop)))
-\def
- \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda 
-(b: A).(lt (f a) (f b))))).
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma
deleted file mode 100644 (file)
index c11c7d7..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/fwd.ma
+++ /dev/null
@@ -1,94 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/coq/defs.ma".
-
-implied lemma False_rect:
- \forall (P: Type[0]).(False \to P)
-\def
- \lambda (P: Type[0]).(\lambda (f: False).(match f in False with [])).
-
-implied lemma False_ind:
- \forall (P: Prop).(False \to P)
-\def
- \lambda (P: Prop).(False_rect P).
-
-implied lemma land_rect:
- \forall (A: Prop).(\forall (B: Prop).(\forall (P: Type[0]).(((A \to (B \to 
-P))) \to ((land A B) \to P))))
-\def
- \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Type[0]).(\lambda (f: ((A 
-\to (B \to P)))).(\lambda (a: (land A B)).(match a with [(conj x x0) 
-\Rightarrow (f x x0)]))))).
-
-implied lemma land_ind:
- \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to (B \to P))) 
-\to ((land A B) \to P))))
-\def
- \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Prop).(land_rect A B P))).
-
-implied lemma or_ind:
- \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to P)) \to 
-(((B \to P)) \to ((or A B) \to P)))))
-\def
- \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Prop).(\lambda (f: ((A \to 
-P))).(\lambda (f0: ((B \to P))).(\lambda (o: (or A B)).(match o with 
-[(or_introl x) \Rightarrow (f x) | (or_intror x) \Rightarrow (f0 x)])))))).
-
-implied lemma ex_ind:
- \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (P0: 
-Prop).(((\forall (x: A).((P x) \to P0))) \to ((ex A P) \to P0))))
-\def
- \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (P0: 
-Prop).(\lambda (f: ((\forall (x: A).((P x) \to P0)))).(\lambda (e: (ex A 
-P)).(match e with [(ex_intro x x0) \Rightarrow (f x x0)]))))).
-
-implied lemma ex2_ind:
- \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
-Prop))).(\forall (P0: Prop).(((\forall (x: A).((P x) \to ((Q x) \to P0)))) 
-\to ((ex2 A P Q) \to P0)))))
-\def
- \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to 
-Prop))).(\lambda (P0: Prop).(\lambda (f: ((\forall (x: A).((P x) \to ((Q x) 
-\to P0))))).(\lambda (e: (ex2 A P Q)).(match e with [(ex_intro2 x x0 x1) 
-\Rightarrow (f x x0 x1)])))))).
-
-implied lemma eq_rect:
- \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Type[0]))).((P x) 
-\to (\forall (y: A).((eq A x y) \to (P y))))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to 
-Type[0]))).(\lambda (f: (P x)).(\lambda (y: A).(\lambda (e: (eq A x 
-y)).(match e with [refl_equal \Rightarrow f])))))).
-
-implied lemma eq_ind:
- \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
-(\forall (y: A).((eq A x y) \to (P y))))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(eq_rect A 
-x P))).
-
-implied rec lemma le_ind (n: nat) (P: (nat \to Prop)) (f: P n) (f0: (\forall 
-(m: nat).((le n m) \to ((P m) \to (P (S m)))))) (n0: nat) (l: le n n0) on l: 
-P n0 \def match l with [le_n \Rightarrow f | (le_S m l0) \Rightarrow (f0 m l0 
-((le_ind n P f f0) m l0))].
-
-implied rec lemma Acc_ind (A: Type[0]) (R: (A \to (A \to Prop))) (P: (A \to 
-Prop)) (f: (\forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to 
-(((\forall (y: A).((R y x) \to (P y)))) \to (P x))))) (a: A) (a0: Acc A R a) 
-on a0: P a \def match a0 with [(Acc_intro x a1) \Rightarrow (f x a1 (\lambda 
-(y: A).(\lambda (r0: (R y x)).((Acc_ind A R P f) y (a1 y r0)))))].
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma b/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
deleted file mode 100644 (file)
index b5069fd..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/coq/props.ma
+++ /dev/null
@@ -1,597 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/coq/fwd.ma".
-
-lemma f_equal:
- \forall (A: Type[0]).(\forall (B: Type[0]).(\forall (f: ((A \to 
-B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
-\def
- \lambda (A: Type[0]).(\lambda (B: Type[0]).(\lambda (f: ((A \to 
-B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x 
-(\lambda (a: A).(eq B (f x) (f a))) (refl_equal B (f x)) y H)))))).
-
-lemma f_equal2:
- \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (B: Type[0]).(\forall 
-(f: ((A1 \to (A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: 
-A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) 
-(f y1 y2)))))))))))
-\def
- \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (B: Type[0]).(\lambda 
-(f: ((A1 \to (A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: 
-A2).(\lambda (y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
-A1).((eq A2 x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 
-y2)).(eq_ind A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B 
-(f x1 x2)) y2 H0)) y1 H))))))))).
-
-lemma f_equal3:
- \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (A3: Type[0]).(\forall 
-(B: Type[0]).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: 
-A1).(\forall (y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: 
-A3).(\forall (y3: A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) 
-\to (eq B (f x1 x2 x3) (f y1 y2 y3)))))))))))))))
-\def
- \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (A3: Type[0]).(\lambda 
-(B: Type[0]).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: 
-A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: 
-A3).(\lambda (y3: A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
-A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) 
-(\lambda (H0: (eq A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) 
-\to (eq B (f x1 x2 x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 
-x3 (\lambda (a: A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 
-x3)) y3 H1)) y2 H0)) y1 H)))))))))))).
-
-lemma sym_eq:
- \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y 
-x))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x 
-y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
-
-lemma eq_ind_r:
- \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
-(\forall (y: A).((eq A y x) \to (P y))))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda 
-(H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) 
-with [refl_equal \Rightarrow H])))))).
-
-lemma trans_eq:
- \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A 
-x y) \to ((eq A y z) \to (eq A x z))))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (z: 
-A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda 
-(a: A).(eq A x a)) H z H0)))))).
-
-lemma sym_not_eq:
- \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to 
-(not (eq A y x)))))
-\def
- \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq 
-A x y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a 
-y)) (refl_equal A y) x h2)))))).
-
-lemma nat_double_ind:
- \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to 
-(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: 
-nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: 
-nat).(R n m))))))
-\def
- \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O 
-n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall 
-(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: 
-nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: 
-nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind 
-(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R 
-(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
-
-lemma eq_add_S:
- \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S 
-m))).(f_equal nat nat pred (S n) (S m) H))).
-
-lemma O_S:
- \forall (n: nat).(not (eq nat O (S n)))
-\def
- \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda 
-(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
-
-lemma not_eq_S:
- \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S 
-n) (S m)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda 
-(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
-
-lemma pred_Sn:
- \forall (m: nat).(eq nat m (pred (S m)))
-\def
- \lambda (m: nat).(refl_equal nat (pred (S m))).
-
-lemma S_pred:
- \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) 
-(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S 
-m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 
-(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
-
-lemma le_trans:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) 
-\to (le n p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H 
-(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n 
-m0 IHle)))) p H0))))).
-
-lemma le_trans_S:
- \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S 
-n) m (le_S n n (le_n n)) H))).
-
-lemma le_n_S:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
-(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le 
-n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
-
-lemma le_O_n:
- \forall (n: nat).(le O n)
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda 
-(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
-
-lemma le_S_n:
- \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S 
-n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: 
-nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S 
-n m0 H0)))) (S m) H))).
-
-lemma le_Sn_O:
- \forall (n: nat).(not (le (S n) O))
-\def
- \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: 
-nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: 
-(IsSucc m)).I))) O H)).
-
-lemma le_Sn_n:
- \forall (n: nat).(not (le (S n) n))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O 
-O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le 
-(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
-
-lemma le_antisym:
- \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n 
-m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda 
-(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal 
-nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) 
-\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S 
-m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) 
-in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
-
-lemma le_n_O_eq:
- \forall (n: nat).((le n O) \to (eq nat O n))
-\def
- \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
-
-lemma le_elim_rel:
- \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to 
-(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S 
-q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
-\def
- \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O 
-p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p 
-q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: 
-nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda 
-(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: 
-nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) 
-m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn 
-n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: 
-(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 
-H1)))))) m Le))))) n)))).
-
-lemma lt_n_n:
- \forall (n: nat).(not (lt n n))
-\def
- le_Sn_n.
-
-lemma lt_n_S:
- \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m 
-H))).
-
-lemma lt_n_Sn:
- \forall (n: nat).(lt n (S n))
-\def
- \lambda (n: nat).(le_n (S n)).
-
-lemma lt_S_n:
- \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S 
-n) m H))).
-
-lemma lt_n_O:
- \forall (n: nat).(not (lt n O))
-\def
- le_Sn_O.
-
-lemma lt_trans:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) 
-\to (lt n p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S 
-(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt 
-n m0)).(le_S (S n) m0 IHle)))) p H0))))).
-
-lemma lt_O_Sn:
- \forall (n: nat).(lt O (S n))
-\def
- \lambda (n: nat).(le_n_S O n (le_O_n n)).
-
-lemma lt_le_S:
- \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
-\def
- \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
-
-lemma le_not_lt:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
-(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n 
-m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle 
-(le_trans_S (S m0) n H1)))))) m H))).
-
-lemma le_lt_n_Sm:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
-
-lemma le_lt_trans:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) 
-\to (lt n p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) 
-(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
-(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
-
-lemma lt_le_trans:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) 
-\to (lt n p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H 
-(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S 
-(S n) m0 IHle)))) p H0))))).
-
-lemma lt_le_weak:
- \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m 
-H))).
-
-lemma lt_n_Sm_le:
- \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m 
-H))).
-
-lemma le_lt_or_eq:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
-(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) 
-(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: 
-(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) 
-(le_n_S n m0 H0))))) m H))).
-
-lemma le_or_lt:
- \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: 
-nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: 
-nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: 
-nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) 
-(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 
-n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S 
-n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S 
-n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S 
-m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
-
-lemma plus_n_O:
- \forall (n: nat).(eq nat n (plus n O))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) 
-(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
-O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
-
-lemma plus_n_Sm:
- \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
-\def
- \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S 
-(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: 
-nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat 
-S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
-
-lemma plus_sym:
- \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus 
-n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus 
-y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S 
-(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) 
-(plus_n_Sm m y)))) n)).
-
-lemma plus_Snm_nSm:
- \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: 
-nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda 
-(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n 
-(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
-
-lemma plus_assoc_l:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m 
-p)) (plus (plus n m) p))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
-nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat 
-(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) 
-(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 
-m) p) H))) n))).
-
-lemma plus_assoc_r:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n 
-m) p) (plus n (plus m p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n 
-(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
-
-lemma simpl_plus_l:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) 
-(plus n p)) \to (eq nat m p))))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: 
-nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: 
-nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: 
-nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) 
-(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: 
-nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn 
-(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) 
-(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
-
-lemma minus_n_O:
- \forall (n: nat).(eq nat n (minus n O))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) 
-(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
-O))).(refl_equal nat (S n0)))) n).
-
-lemma minus_n_n:
- \forall (n: nat).(eq nat O (minus n n))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) 
-(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
-n0))).IHn)) n).
-
-lemma minus_Sn_m:
- \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) 
-(minus (S n) m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel 
-(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) 
-n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p 
-(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: 
-(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow 
-(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
-
-lemma plus_minus:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) 
-\to (eq nat p (minus n m)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind 
-(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p 
-(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat 
-n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) 
-(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 
-p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 
-p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda 
-(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq 
-nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H 
-(eq_add_S m0 (plus n0 p) H0)))))) m n))).
-
-lemma minus_plus:
- \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
-\def
- \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) 
-(plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
-
-lemma le_pred_n:
- \forall (n: nat).(le (pred n) n)
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) 
-(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 
-(le_n n0)))) n).
-
-lemma le_plus_l:
- \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
-\def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus 
-n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: 
-((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus 
-n0 m) (IHn m))))) n).
-
-lemma le_plus_r:
- \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus 
-n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m 
-(plus n0 m) H))) n)).
-
-lemma simpl_le_plus_l:
- \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p 
-m)) \to (le n m))))
-\def
- \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: 
-nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: 
-nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: 
-nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) 
-(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda 
-(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus 
-p0 m) H))))))) p).
-
-lemma le_plus_trans:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n 
-(plus m p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
-m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
-
-lemma le_reg_l:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus 
-p n) (plus p m)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
-nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) 
-(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 
-m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
-p))).
-
-lemma le_plus_plus:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
-n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: 
-nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: 
-nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S 
-(plus n p) (plus m0 q) H2)))) m H)))))).
-
-lemma le_plus_minus:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m 
-n)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel 
-(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) 
-(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda 
-(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat 
-S q (plus p (minus q p)) H0))))) n m Le))).
-
-lemma le_plus_minus_r:
- \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m 
-n)) m)))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m 
-(plus n (minus m n)) (le_plus_minus n m H)))).
-
-lemma simpl_lt_plus_l:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p 
-m)) \to (lt n m))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
-nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) 
-(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n 
-m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S 
-(plus p0 n)) (plus p0 m) H))))) p))).
-
-lemma lt_reg_l:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
-p n) (plus p m)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
-nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) 
-(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 
-m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
-p))).
-
-lemma lt_reg_r:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
-n p) (plus m p)))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
-m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r 
-nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: 
-nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt 
-(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m 
-p)) (plus n p) (plus_sym n p))))).
-
-lemma le_lt_plus_plus:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
-n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n 
-(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) 
-(plus (S n) p) (plus_Snm_nSm n p))))))).
-
-lemma lt_le_plus_plus:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
-n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m 
-p q H H0)))))).
-
-lemma lt_plus_plus:
- \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
-n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
-\def
- \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
-nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q 
-H (lt_le_weak p q H0))))))).
-
-lemma well_founded_ltof:
- \forall (A: Type[0]).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
-\def
- \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: 
-nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A 
-(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A 
-(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: 
-(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall 
-(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda 
-(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda 
-(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb 
-(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a 
-(le_n (S (f a))))))).
-
-lemma lt_wf:
- well_founded nat lt
-\def
- well_founded_ltof nat (\lambda (m: nat).m).
-
-lemma lt_wf_ind:
- \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: 
-nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
-\def
- \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: 
-nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt 
-(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: 
-nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y 
-x) \to (P y))))).(H x H1)))) p (lt_wf p)))).
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma b/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma
deleted file mode 100644 (file)
index bcdc2a0..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/definitions.ma
+++ /dev/null
@@ -1,18 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/coq/defs.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma b/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma
deleted file mode 100644 (file)
index 5466cb1..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/preamble.ma
+++ /dev/null
@@ -1,17 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basics/pts.ma".
-
-inductive False: Prop \def .

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/spare.ma b/matita/matita/contribs/lambdadelta/legacy_1/spare.ma
deleted file mode 100644 (file)
index b276321..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/spare.ma
+++ /dev/null
@@ -1,18 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/theory.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1/theory.ma b/matita/matita/contribs/lambdadelta/legacy_1/theory.ma
deleted file mode 100644 (file)
index 17a1ec9..0000000
--- a/matita/matita/contribs/lambdadelta/legacy_1/theory.ma
+++ /dev/null
@@ -1,18 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "legacy_1/coq/props.ma".
-

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/coq/defs.ma b/matita/matita/contribs/lambdadelta/legacy_1A/coq/defs.ma
new file mode 100644 (file)
index 0000000..0eb631b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/coq/defs.ma
@@ -0,0 +1,93 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/preamble.ma".
+
+inductive eq (A: Type[0]) (x: A): A \to Prop \def
+| refl_equal: eq A x x.
+
+inductive True: Prop \def
+| I: True.
+
+inductive land (A: Prop) (B: Prop): Prop \def
+| conj: A \to (B \to (land A B)).
+
+inductive or (A: Prop) (B: Prop): Prop \def
+| or_introl: A \to (or A B)
+| or_intror: B \to (or A B).
+
+inductive ex (A: Type[0]) (P: A \to Prop): Prop \def
+| ex_intro: \forall (x: A).((P x) \to (ex A P)).
+
+inductive ex2 (A: Type[0]) (P: A \to Prop) (Q: A \to Prop): Prop \def
+| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))).
+
+definition not:
+ Prop \to Prop
+\def
+ \lambda (A: Prop).(A \to False).
+
+inductive bool: Type[0] \def
+| true: bool
+| false: bool.
+
+inductive nat: Type[0] \def
+| O: nat
+| S: nat \to nat.
+
+inductive le (n: nat): nat \to Prop \def
+| le_n: le n n
+| le_S: \forall (m: nat).((le n m) \to (le n (S m))).
+
+definition lt:
+ nat \to (nat \to Prop)
+\def
+ \lambda (n: nat).(\lambda (m: nat).(le (S n) m)).
+
+definition IsSucc:
+ nat \to Prop
+\def
+ \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow 
+True]).
+
+definition pred:
+ nat \to nat
+\def
+ \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]).
+
+rec definition plus (n: nat) on n: nat \to nat \def \lambda (m: nat).(match n 
+with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))]).
+
+rec definition minus (n: nat) on n: nat \to nat \def \lambda (m: nat).(match 
+n with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S 
+k) | (S l) \Rightarrow (minus k l)])]).
+
+inductive Acc (A: Type[0]) (R: A \to (A \to Prop)): A \to Prop \def
+| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to 
+(Acc A R x)).
+
+definition well_founded:
+ \forall (A: Type[0]).(((A \to (A \to Prop))) \to Prop)
+\def
+ \lambda (A: Type[0]).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: 
+A).(Acc A R a))).
+
+definition ltof:
+ \forall (A: Type[0]).(((A \to nat)) \to (A \to (A \to Prop)))
+\def
+ \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda 
+(b: A).(lt (f a) (f b))))).
+

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/coq/fwd.ma b/matita/matita/contribs/lambdadelta/legacy_1A/coq/fwd.ma
new file mode 100644 (file)
index 0000000..fbd8a53
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/coq/fwd.ma
@@ -0,0 +1,94 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/coq/defs.ma".
+
+implied lemma False_rect:
+ \forall (P: Type[0]).(False \to P)
+\def
+ \lambda (P: Type[0]).(\lambda (f: False).(match f in False with [])).
+
+implied lemma False_ind:
+ \forall (P: Prop).(False \to P)
+\def
+ \lambda (P: Prop).(False_rect P).
+
+implied lemma land_rect:
+ \forall (A: Prop).(\forall (B: Prop).(\forall (P: Type[0]).(((A \to (B \to 
+P))) \to ((land A B) \to P))))
+\def
+ \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Type[0]).(\lambda (f: ((A 
+\to (B \to P)))).(\lambda (a: (land A B)).(match a with [(conj x x0) 
+\Rightarrow (f x x0)]))))).
+
+implied lemma land_ind:
+ \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to (B \to P))) 
+\to ((land A B) \to P))))
+\def
+ \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Prop).(land_rect A B P))).
+
+implied lemma or_ind:
+ \forall (A: Prop).(\forall (B: Prop).(\forall (P: Prop).(((A \to P)) \to 
+(((B \to P)) \to ((or A B) \to P)))))
+\def
+ \lambda (A: Prop).(\lambda (B: Prop).(\lambda (P: Prop).(\lambda (f: ((A \to 
+P))).(\lambda (f0: ((B \to P))).(\lambda (o: (or A B)).(match o with 
+[(or_introl x) \Rightarrow (f x) | (or_intror x) \Rightarrow (f0 x)])))))).
+
+implied lemma ex_ind:
+ \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (P0: 
+Prop).(((\forall (x: A).((P x) \to P0))) \to ((ex A P) \to P0))))
+\def
+ \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (P0: 
+Prop).(\lambda (f: ((\forall (x: A).((P x) \to P0)))).(\lambda (e: (ex A 
+P)).(match e with [(ex_intro x x0) \Rightarrow (f x x0)]))))).
+
+implied lemma ex2_ind:
+ \forall (A: Type[0]).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to 
+Prop))).(\forall (P0: Prop).(((\forall (x: A).((P x) \to ((Q x) \to P0)))) 
+\to ((ex2 A P Q) \to P0)))))
+\def
+ \lambda (A: Type[0]).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to 
+Prop))).(\lambda (P0: Prop).(\lambda (f: ((\forall (x: A).((P x) \to ((Q x) 
+\to P0))))).(\lambda (e: (ex2 A P Q)).(match e with [(ex_intro2 x x0 x1) 
+\Rightarrow (f x x0 x1)])))))).
+
+implied lemma eq_rect:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Type[0]))).((P x) 
+\to (\forall (y: A).((eq A x y) \to (P y))))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to 
+Type[0]))).(\lambda (f: (P x)).(\lambda (y: A).(\lambda (e: (eq A x 
+y)).(match e with [refl_equal \Rightarrow f])))))).
+
+implied lemma eq_ind:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
+(\forall (y: A).((eq A x y) \to (P y))))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(eq_rect A 
+x P))).
+
+implied rec lemma le_ind (n: nat) (P: (nat \to Prop)) (f: P n) (f0: (\forall 
+(m: nat).((le n m) \to ((P m) \to (P (S m)))))) (n0: nat) (l: le n n0) on l: 
+P n0 \def match l with [le_n \Rightarrow f | (le_S m l0) \Rightarrow (f0 m l0 
+((le_ind n P f f0) m l0))].
+
+implied rec lemma Acc_ind (A: Type[0]) (R: (A \to (A \to Prop))) (P: (A \to 
+Prop)) (f: (\forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to 
+(((\forall (y: A).((R y x) \to (P y)))) \to (P x))))) (a: A) (a0: Acc A R a) 
+on a0: P a \def match a0 with [(Acc_intro x a1) \Rightarrow (f x a1 (\lambda 
+(y: A).(\lambda (r0: (R y x)).((Acc_ind A R P f) y (a1 y r0)))))].
+

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/coq/props.ma b/matita/matita/contribs/lambdadelta/legacy_1A/coq/props.ma
new file mode 100644 (file)
index 0000000..8e169b3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/coq/props.ma
@@ -0,0 +1,597 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/coq/fwd.ma".
+
+lemma f_equal:
+ \forall (A: Type[0]).(\forall (B: Type[0]).(\forall (f: ((A \to 
+B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
+\def
+ \lambda (A: Type[0]).(\lambda (B: Type[0]).(\lambda (f: ((A \to 
+B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x 
+(\lambda (a: A).(eq B (f x) (f a))) (refl_equal B (f x)) y H)))))).
+
+lemma f_equal2:
+ \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (B: Type[0]).(\forall 
+(f: ((A1 \to (A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: 
+A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) 
+(f y1 y2)))))))))))
+\def
+ \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (B: Type[0]).(\lambda 
+(f: ((A1 \to (A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: 
+A2).(\lambda (y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
+A1).((eq A2 x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 
+y2)).(eq_ind A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B 
+(f x1 x2)) y2 H0)) y1 H))))))))).
+
+lemma f_equal3:
+ \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (A3: Type[0]).(\forall 
+(B: Type[0]).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: 
+A1).(\forall (y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: 
+A3).(\forall (y3: A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) 
+\to (eq B (f x1 x2 x3) (f y1 y2 y3)))))))))))))))
+\def
+ \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (A3: Type[0]).(\lambda 
+(B: Type[0]).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: 
+A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: 
+A3).(\lambda (y3: A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: 
+A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) 
+(\lambda (H0: (eq A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) 
+\to (eq B (f x1 x2 x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 
+x3 (\lambda (a: A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 
+x3)) y3 H1)) y2 H0)) y1 H)))))))))))).
+
+lemma sym_eq:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y 
+x))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x 
+y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
+
+lemma eq_ind_r:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to 
+(\forall (y: A).((eq A y x) \to (P y))))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda 
+(H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) 
+with [refl_equal \Rightarrow H])))))).
+
+lemma trans_eq:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A 
+x y) \to ((eq A y z) \to (eq A x z))))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (z: 
+A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda 
+(a: A).(eq A x a)) H z H0)))))).
+
+lemma sym_not_eq:
+ \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to 
+(not (eq A y x)))))
+\def
+ \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq 
+A x y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a 
+y)) (refl_equal A y) x h2)))))).
+
+lemma nat_double_ind:
+ \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to 
+(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m: 
+nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m: 
+nat).(R n m))))))
+\def
+ \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O 
+n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall 
+(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n: 
+nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0: 
+nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind 
+(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R 
+(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
+
+lemma eq_add_S:
+ \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S 
+m))).(f_equal nat nat pred (S n) (S m) H))).
+
+lemma O_S:
+ \forall (n: nat).(not (eq nat O (S n)))
+\def
+ \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda 
+(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
+
+lemma not_eq_S:
+ \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S 
+n) (S m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda 
+(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
+
+lemma pred_Sn:
+ \forall (m: nat).(eq nat m (pred (S m)))
+\def
+ \lambda (m: nat).(refl_equal nat (pred (S m))).
+
+lemma S_pred:
+ \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m) 
+(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S 
+m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0 
+(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
+
+lemma le_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p) 
+\to (le n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H 
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n 
+m0 IHle)))) p H0))))).
+
+lemma le_trans_S:
+ \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S 
+n) m (le_S n n (le_n n)) H))).
+
+lemma le_n_S:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le 
+n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
+
+lemma le_O_n:
+ \forall (n: nat).(le O n)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda 
+(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
+
+lemma le_S_n:
+ \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S 
+n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0: 
+nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S 
+n m0 H0)))) (S m) H))).
+
+lemma le_Sn_O:
+ \forall (n: nat).(not (le (S n) O))
+\def
+ \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0: 
+nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_: 
+(IsSucc m)).I))) O H)).
+
+lemma le_Sn_n:
+ \forall (n: nat).(not (le (S n) n))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O 
+O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le 
+(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
+
+lemma le_antisym:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n 
+m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda 
+(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal 
+nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) 
+\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S 
+m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0) 
+in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
+
+lemma le_n_O_eq:
+ \forall (n: nat).((le n O) \to (eq nat O n))
+\def
+ \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
+
+lemma le_elim_rel:
+ \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to 
+(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S 
+q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
+\def
+ \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O 
+p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p 
+q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0: 
+nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda 
+(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m: 
+nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0) 
+m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn 
+n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: 
+(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0 
+H1)))))) m Le))))) n)))).
+
+lemma lt_n_n:
+ \forall (n: nat).(not (lt n n))
+\def
+ le_Sn_n.
+
+lemma lt_n_S:
+ \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m 
+H))).
+
+lemma lt_n_Sn:
+ \forall (n: nat).(lt n (S n))
+\def
+ \lambda (n: nat).(le_n (S n)).
+
+lemma lt_S_n:
+ \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S 
+n) m H))).
+
+lemma lt_n_O:
+ \forall (n: nat).(not (lt n O))
+\def
+ le_Sn_O.
+
+lemma lt_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p) 
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S 
+(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt 
+n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+
+lemma lt_O_Sn:
+ \forall (n: nat).(lt O (S n))
+\def
+ \lambda (n: nat).(le_n_S O n (le_O_n n)).
+
+lemma lt_le_S:
+ \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
+\def
+ \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
+
+lemma le_not_lt:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n 
+m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle 
+(le_trans_S (S m0) n H1)))))) m H))).
+
+lemma le_lt_n_Sm:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
+
+lemma le_lt_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p) 
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) 
+(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: 
+(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+
+lemma lt_le_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p) 
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H 
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S 
+(S n) m0 IHle)))) p H0))))).
+
+lemma lt_le_weak:
+ \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m 
+H))).
+
+lemma lt_n_Sm_le:
+ \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m 
+H))).
+
+lemma le_lt_or_eq:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda 
+(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n) 
+(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_: 
+(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0)) 
+(le_n_S n m0 H0))))) m H))).
+
+lemma le_or_lt:
+ \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0: 
+nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0: 
+nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0: 
+nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0)))) 
+(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 
+n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S 
+n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S 
+n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S 
+m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
+
+lemma plus_n_O:
+ \forall (n: nat).(eq nat n (plus n O))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 
+O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
+
+lemma plus_n_Sm:
+ \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
+\def
+ \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S 
+(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0: 
+nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat 
+S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
+
+lemma plus_sym:
+ \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus 
+n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus 
+y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S 
+(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y)) 
+(plus_n_Sm m y)))) n)).
+
+lemma plus_Snm_nSm:
+ \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0: 
+nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda 
+(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n 
+(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
+
+lemma plus_assoc_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m 
+p)) (plus (plus n m) p))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat 
+(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p)) 
+(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0 
+m) p) H))) n))).
+
+lemma plus_assoc_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n 
+m) p) (plus n (plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n 
+(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
+
+lemma simpl_plus_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m) 
+(plus n p)) \to (eq nat m p))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p: 
+nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m: 
+nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0: 
+nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m) 
+(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p: 
+nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn 
+(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p) 
+(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
+
+lemma minus_n_O:
+ \forall (n: nat).(eq nat n (minus n O))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 
+O))).(refl_equal nat (S n0)))) n).
+
+lemma minus_n_n:
+ \forall (n: nat).(eq nat O (minus n n))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0))) 
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 
+n0))).IHn)) n).
+
+lemma minus_Sn_m:
+ \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m)) 
+(minus (S n) m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel 
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1) 
+n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p 
+(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_: 
+(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow 
+(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
+
+lemma plus_minus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p)) 
+\to (eq nat p (minus n m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind 
+(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p 
+(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat 
+n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O) 
+(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0 
+p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0 
+p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda 
+(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq 
+nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H 
+(eq_add_S m0 (plus n0 p) H0)))))) m n))).
+
+lemma minus_plus:
+ \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n) 
+(plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
+
+lemma le_pred_n:
+ \forall (n: nat).(le (pred n) n)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O) 
+(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0 
+(le_n n0)))) n).
+
+lemma le_plus_l:
+ \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus 
+n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn: 
+((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus 
+n0 m) (IHn m))))) n).
+
+lemma le_plus_r:
+ \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus 
+n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m 
+(plus n0 m) H))) n)).
+
+lemma simpl_le_plus_l:
+ \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p 
+m)) \to (le n m))))
+\def
+ \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m: 
+nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n: 
+nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0: 
+nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n) 
+(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda 
+(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus 
+p0 m) H))))))) p).
+
+lemma le_plus_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n 
+(plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n 
+m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
+
+lemma le_reg_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus 
+p n) (plus p m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0 
+m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
+p))).
+
+lemma le_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
+n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0: 
+nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0: 
+nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S 
+(plus n p) (plus m0 q) H2)))) m H)))))).
+
+lemma le_plus_minus:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m 
+n)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel 
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0))))) 
+(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda 
+(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat 
+S q (plus p (minus q p)) H0))))) n m Le))).
+
+lemma le_plus_minus_r:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m 
+n)) m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m 
+(plus n (minus m n)) (le_plus_minus n m H)))).
+
+lemma simpl_lt_plus_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p 
+m)) \to (lt n m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n 
+m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S 
+(plus p0 n)) (plus p0 m) H))))) p))).
+
+lemma lt_reg_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
+p n) (plus p m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0: 
+nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H) 
+(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0 
+m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H))))) 
+p))).
+
+lemma lt_reg_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus 
+n p) (plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n 
+m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r 
+nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0: 
+nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt 
+(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m 
+p)) (plus n p) (plus_sym n p))))).
+
+lemma le_lt_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le 
+n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n 
+(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0) 
+(plus (S n) p) (plus_Snm_nSm n p))))))).
+
+lemma lt_le_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
+n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
+nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m 
+p q H H0)))))).
+
+lemma lt_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt 
+n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q: 
+nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q 
+H (lt_le_weak p q H0))))))).
+
+lemma well_founded_ltof:
+ \forall (A: Type[0]).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
+\def
+ \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n: 
+nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A 
+(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A 
+(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2: 
+(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall 
+(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda 
+(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda 
+(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb 
+(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a 
+(le_n (S (f a))))))).
+
+lemma lt_wf:
+ well_founded nat lt
+\def
+ well_founded_ltof nat (\lambda (m: nat).m).
+
+lemma lt_wf_ind:
+ \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n: 
+nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
+\def
+ \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n: 
+nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt 
+(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y: 
+nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y 
+x) \to (P y))))).(H x H1)))) p (lt_wf p)))).
+

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/definitions.ma b/matita/matita/contribs/lambdadelta/legacy_1A/definitions.ma
new file mode 100644 (file)
index 0000000..ad3e144
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/definitions.ma
@@ -0,0 +1,18 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/coq/defs.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/preamble.ma b/matita/matita/contribs/lambdadelta/legacy_1A/preamble.ma
new file mode 100644 (file)
index 0000000..5466cb1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/preamble.ma
@@ -0,0 +1,17 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/pts.ma".
+
+inductive False: Prop \def .

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/spare.ma b/matita/matita/contribs/lambdadelta/legacy_1A/spare.ma
new file mode 100644 (file)
index 0000000..2bcbf4c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/spare.ma
@@ -0,0 +1,18 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/theory.ma".
+

diff --git a/matita/matita/contribs/lambdadelta/legacy_1A/theory.ma b/matita/matita/contribs/lambdadelta/legacy_1A/theory.ma
new file mode 100644 (file)
index 0000000..6d89251
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/legacy_1A/theory.ma
@@ -0,0 +1,18 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "legacy_1A/coq/props.ma".
+